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The Asymptotic Statistics of Random Covering Surfaces

Published online by Cambridge University Press:  15 May 2023

Michael Magee
Affiliation:
Department of Mathematical Sciences, Durham University, United Kingdom; E-mail: michael.r.magee@durham.ac.uk
Doron Puder*
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 6997801, Israel

Abstract

Let $\Gamma _{g}$ be the fundamental group of a closed connected orientable surface of genus $g\geq 2$ . We develop a new method for integrating over the representation space $\mathbb {X}_{g,n}=\mathrm {Hom}(\Gamma _{g},S_{n})$ , where $S_{n}$ is the symmetric group of permutations of $\{1,\ldots ,n\}$ . Equivalently, this is the space of all vertex-labeled, n-sheeted covering spaces of the closed surface of genus g.

Given $\phi \in \mathbb {X}_{g,n}$ and $\gamma \in \Gamma _{g}$ , we let $\mathsf {fix}_{\gamma }(\phi )$ be the number of fixed points of the permutation $\phi (\gamma )$ . The function $\mathsf {fix}_{\gamma }$ is a special case of a natural family of functions on $\mathbb {X}_{g,n}$ called Wilson loops. Our new methodology leads to an asymptotic formula, as $n\to \infty $ , for the expectation of $\mathsf {fix}_{\gamma }$ with respect to the uniform probability measure on $\mathbb {X}_{g,n}$ , which is denoted by $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$ . We prove that if $\gamma \in \Gamma _{g}$ is not the identity and q is maximal such that $\gamma $ is a q th power in $\Gamma _{g}$ , then

$$\begin{align*}\mathbb{E}_{g,n}\left[\mathsf{fix}_{\gamma}\right]=d(q)+O(n^{-1}) \end{align*}$$

as $n\to \infty $ , where $d\left (q\right )$ is the number of divisors of q. Even the weaker corollary that $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]=o(n)$ as $n\to \infty $ is a new result of this paper. We also prove that $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$ can be approximated to any order $O(n^{-M})$ by a polynomial in $n^{-1}$ .

Type
Algebra
Creative Commons
Creative Common License - CCCreative Common License - BY
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

1 Introduction

Let $g\geq 2$ , and let $\Sigma _{g}$ be a closed orientable topological surface of genus g. We fix a base point $o\in \Sigma _{g}$ and let

(1.1) $$ \begin{align} \Gamma_{g}\stackrel{\mathrm{def}}{=}\pi_{1}\left(\Sigma_{g},o\right)\cong\left\langle a_{1},b_{1},\ldots,a_{g},b_{g}\,\middle|\,\left[a_{1},b_{1}\right]\cdots\left[a_{g},b_{g}\right]\right\rangle \end{align} $$

be the fundamental group of $\Sigma _{g}$ . Denote by

$$\begin{align*}\mathbb{X}_{g,n}\stackrel{\mathrm{def}}{=}\mathrm{Hom}\left(\Gamma_{g},S_{n}\right) \end{align*}$$

the representation space of all homomorphisms from $\Gamma _{g}$ to $S_{n}$ , where $S_{n}$ is the symmetric group of permutations of $\{1,\ldots ,n\}$ . From another point of view, the space $\mathbb {X}_{g,n}$ can be viewed as the space of degree-n covering maps of $\Sigma _{g}$ . Indeed, for every not-necessarily-connected degree-n covering map

$$\begin{align*}p\colon X\twoheadrightarrow\Sigma_{g}, \end{align*}$$

we may identify the fiber $p^{-1}\left (o\right )$ with $\left \{ 1,\ldots ,n\right \} $ , and the monodromy action of $\pi _{1}\left (\Sigma _{g},o\right )$ on the fiber then gives rise to a homomorphism $\phi \in \mathrm {Hom}(\Gamma _{g},S_{n})$ . This gives a one-to-one correspondence between $\mathbb {X}_{g,n}$ and degree-n covering maps with $p^{-1}\left (o\right )=\left \{ 1,\ldots ,n\right \} $ . This correspondence is discussed in more detail in $\S \S $ 2.2.

The space $\mathbb {X}_{g,n}$ was studied by Liebeck and Shalev [Reference Liebeck and ShalevLS04], who showed that a uniformly random homomorphism $\phi \colon \Gamma _{g}\to S_{n}$ satisfies $\phi \left (\Gamma _{g}\right )\supseteq A_{n}$ a.a.s. (asymptotically almost surely, namely, with probability tending to $1$ as $n\to \infty $ ) [Reference Liebeck and ShalevLS04, Thm. 1.12].Footnote 1 In particular the image is a.a.s. transitive, or, equivalently, the corresponding random degree-n covering space is a.a.s. connected. When $\Gamma _{g}$ is replaced by a nonabelian free group, the analogous result holds by the famous theorem of Dixon [Reference DixonDix69] that two random permutations in $S_{n}$ a.a.s. generate $S_{n}$ or $A_{n}$ .

In the current work we address the problem of integration over the space $\mathbb {X}_{g,n}$ . Namely, our goal is to analyze the expected value $\mathbb {E}_{g,n}\left [f\right ]$ of functions f on $\mathbb {X}_{g,n}$ with respect to the uniform measure on $\mathbb {X}_{g,n}$ . The functions on $\mathbb {X}_{g,n}$ that we consider are natural functions that arise from loops in $\Sigma _{g}$ . Given an element $\gamma \in \Gamma _{g}$ and a character $\chi $ of $S_{n}$ , we let

$$\begin{align*}\chi_{\gamma}(\phi)\stackrel{\mathrm{def}}{=}\chi(\phi(\gamma)),\quad\chi_{\gamma}:\mathbb{X}_{g,n}\to\mathbf{R}. \end{align*}$$

These functions are called Wilson loops in the physics literature [Reference LabourieLab13, Def. 6.4.1]. Our focus is on the character $\mathsf {fix}$ of $S_{n}$ which assigns to every permutation its number of fixed points.

The main motivation behind this work is its relevance to the study of random covers of the closed surface $\Sigma _{g}$ . Given some $1\ne \gamma \in \Gamma _{g}$ , consider the geodesic $C_{\gamma }$ in $\Sigma _{g}$ corresponding to the conjugacy class of $\gamma $ . For every homomorphism $\phi \in \mathbb {X}_{g,n}$ , the number of fixed points $\mathsf {fix}_{\gamma }\left (\phi \right )$ is precisely the number of lifts of $C_{\gamma }$ to a closed geodesic in the degree-n covering corresponding to $\phi $ . Indeed, the results of this paper are crucial ingredients in a subsequent work [Reference Magee, Naud and PuderMNP22] which gives new results on spectral gaps of random covers of a closed surface.

Another source of motivation is the rich theory that has been discovered around similar questions when surface groups are replaced by free groups (e.g., [Reference NicaNic94, Reference Puder and ParzanchevskiPP15, Reference Magee and PuderMP19, Reference Hanany and PuderHP22] and see $\S \S $ 1.2 below). Expanding this theory to other groups is challenging, as the presence of a relation between the generators presents a fundamental difficulty that is not present for free groups. Surface groups, among the best understood and best behaved one-relator groups, are a natural starting point for this quest. To overcome the difficulty brought up by the existence of a relation, we develop in this work new machinery, both in representation theory of $S_{n}$ and in combinatorial group theory.

Expected number of fixed points

Recall that the expectation $\mathbb {E}_{g,n}\left [\mathsf {fix}_{\gamma }\right ]$ is the average number of fixed points in $\phi \left (\gamma \right )$ where $\phi \colon \Gamma _{g}\to S_{n}$ is uniformly random. Our main results are the following two theorems.

Theorem 1.1. Fix $g\geq 2$ and $\gamma \in \Gamma _{g}$ . Then there is an infinite sequence of rational numbers

$$\begin{align*}a_{1}\left(\gamma\right),a_{0}\left(\gamma\right),a_{-1}\left(\gamma\right),a_{-2}\left(\gamma\right),\ldots \end{align*}$$

such that for any $M\in \mathbf {N}$ , as $n\to \infty $ ,

(1.2) $$ \begin{align} \mathbb{E}_{g,n}\left[\mathsf{fix}_{\gamma}\right]=a_{1}\left(\gamma\right)n+a_{0}\left(\gamma\right)+\frac{a_{-1}\left(\gamma\right)}{n}+\ldots\frac{a_{-\left(M-1\right)}\left(\gamma\right)}{n^{M-1}}+O\left(\frac{1}{n^{M}}\right). \end{align} $$

Theorem 1.2. If $\gamma \in \Gamma _{g}$ is not the identity, then, as $n\to \infty $ ,

$$\begin{align*}\mathbb{E}_{g,n}[\mathsf{fix}_{\gamma}]=O(1). \end{align*}$$

In fact, if $q\in \mathbf {N}$ is maximal such that $\gamma =\gamma _{0}^{~q}$ for some $\gamma _{0}\in \Gamma $ , then, as $n\to \infty $ ,

$$ \begin{align*} \mathbb{E}_{g,n}[\mathsf{fix}_{\gamma}] & =d(q)+O\left(\frac{1}{n}\right), \end{align*} $$

where $d(q)$ is the number of divisors function. In other words, $a_{1}(\gamma )=0$ and $a_{0}(\gamma )=d(q)$ .

For example, consider the element a in $\Gamma _{2}=\left \langle a,b,c,d\,\middle |\,\left [a,b\right ]\left [c,d\right ]\right \rangle $ . This element is not a proper power and so $\mathbb {E}_{2,n}\left [\mathsf {fix}_{a}\right ]=1+O\left (n^{-1}\right )$ by Theorem 1.2. By Theorem 1.1, this average can be approximated to any order $n^{-M}$ by a rational function in n. In this particular case, this rational function can be computed to get for, for example, $M=5$ ,

$$\begin{align*}\mathbb{E}_{2,n}\left[\mathsf{fix}_{a}\right]=1+\frac{1}{n^{2}}+\frac{2}{n^{3}}+\frac{10}{n^{4}}+O\left(\frac{1}{n^{5}}\right). \end{align*}$$

Given a finite group G, the number of homomorphisms $\Gamma _{g}\to G$ is related to the Witten zeta function of G,

$$\begin{align*}\zeta^{G}\left(s\right)\stackrel{\mathrm{def}}{=}\sum_{\chi\in\mathrm{Irr}G}\chi\left(1\right)^{-s}, \end{align*}$$

the summation being over the isomorphism classes of irreducible complex representations of G. These functions were introduced by Zagier [Reference ZagierZag94] after Witten’s work in [Reference WittenWit91]. The connection is given by

(1.3) $$ \begin{align} \left|\mathrm{Hom}\left(\Gamma_{g},G\right)\right|=\left|G\right|{}^{2g-1}\zeta^{G}\left(2g-2\right). \end{align} $$

This result goes back to Hurwitz [Reference HurwitzHur02], who gave a more general formula for arbitrary Fuchsian groups (a proof in English is given in [Reference Liebeck and ShalevLS04, Prop. 3.2]). It is also sometimes called ‘Mednykh’s formula’ in the literature after [Reference MednyhkMed78]. For the case $G=S_{n}$ , the zeta function $\zeta ^{S_{n}}$ was studied in [Reference LulovLul96, Reference Müller and PuchtaMP02, Reference Liebeck and ShalevLS04, Reference GamburdGam06]. Inter alia, these works show that, for every $s>0$ ,

$$\begin{align*}\zeta^{S_{n}}\left(s\right)\underset{n\to\infty}{\to}2. \end{align*}$$

Moreover, their results yield an asymptotic expansion in n which approximates $\zeta ^{S_{n}}\left (s\right )$ as $n\to \infty $ , in a similar manner to the one in Theorem 1.1. As such, their results can be thought of as the special case of $\gamma =1$ of a version of Theorem 1.1. We elaborate more in $\S \S $ 5.1.

Common fixed points of subgroups

Our proof also yields the following more general result that concerns not only elements of $\Gamma _{g}$ but also f.g. (finitely generated) subgroups. We write $J\le _{\mathrm {f.g.}}\Gamma _{g}$ to denote a f.g. subgroup J of $\Gamma _{g}$ . Given $J\le _{\mathrm {f.g.}}\Gamma _{g}$ and $\phi \in \mathbb {X}_{g,n}$ , we let $\mathsf {fix}_{J}\left (\phi \right )$ denote the number of elements in $1,\ldots ,n$ that are fixed by all permutations in $\phi \left (J\right )$ :

$$\begin{align*}\mathsf{fix}_{J}\left(\phi\right)\stackrel{\mathrm{def}}{=}\left|\left\{ i\in\left\{ 1,\ldots,n\right\} \,\middle|\,\sigma\left(i\right)=i~\mathrm{for~all}~\sigma\in\phi\left(J\right)\right\} \right|. \end{align*}$$

In particular, $\mathsf {fix}_{\left \langle \gamma \right \rangle }=\mathsf {fix}_{\gamma }$ for all $\gamma \in \Gamma _{g}$ . For $J\le _{\mathrm {f.g.}}\Gamma _{g}$ , we let

(1.4) $$ \begin{align} \chi_{\max}\left(J\right)\stackrel{\mathrm{def}}{=}\max\left\{ \chi\left(K\right)\,\middle|\,J\le K\le_{\mathrm{f.g.}}\Gamma_{g}\right\} \end{align} $$

denote the largest Euler characteristicFootnote 2 of a f.g. subgroup $K\le _{\mathrm {f.g.}}\Gamma _{g}$ which contains J. Note that $\chi \left (\Gamma _{g}\right )=2-2g\le \chi _{\max }\left (J\right )\le 1$ and that $\chi _{\max }\left (J\right )\ge \chi \left (J\right )$ . It is also true that $\chi _{\max }\left (J\right )=1$ if and only if $J=\left \{ 1\right \} $ , and $\chi _{\max }\left (J\right )\ge 0$ if and only if J is cyclic. In addition, we let

$$\begin{align*}\mathfrak{MOG}\left(J\right)\stackrel{\mathrm{def}}{=}\left\{ K\le_{\mathrm{f.g.}}\Gamma_{g}\,\middle|\,J\le K~\mathrm{and}~\chi\left(K\right)=\chi_{\max}\left(J\right)\right\} \end{align*}$$

denote the set of ‘maximal overgroups’ – f.g. subgroups achieving the maximum from equation (1.4). This set is always finite – see Corollary 2.16.

Theorem 1.3. Let $J\le _{\mathrm {f.g.}}\Gamma _{g}$ be a f.g. subgroup. Then

$$\begin{align*}\mathbb{E}_{g,n}\left[\mathsf{fix}_{J}\right]=\left|\mathfrak{MOG}\left(J\right)\right|\cdot n^{\chi_{\max}\left(J\right)}+O\left(n^{\chi_{\max}\left(J\right)-1}\right). \end{align*}$$

Theorem 1.3 generalizes Theorem 1.2, as for $\gamma \ne 1$ , $\chi _{\max }\left (\left \langle \gamma \right \rangle \right )=0$ and

$$\begin{align*}\mathfrak{MOG}\left(\left\langle \gamma\right\rangle \right)=\left\{ \,\left\langle \gamma_{0}^{~m}\right\rangle \big|~\,m|q\,\right\}. \end{align*}$$

The analog of Theorem 1.1 holds too for f.g. subgroups: There is an infinite sequence of rational numbers

$$\begin{align*}a_{1}\left(J\right),a_{0}\left(J\right),a_{-1}\left(J\right),\ldots \end{align*}$$

such that for any $M\in \mathbf {N}$ , as $n\to \infty $ ,

$$\begin{align*}\mathbb{E}_{g,n}\left[\mathsf{fix}_{J}\right]=\sum_{i=-(M-1)}^{1}a_{i}\left(J\right)n^{i}+O\left(n^{-M}\right), \end{align*}$$

and such that $a_{1}=a_{0}=\ldots =a_{\chi _{\max }\left (J\right )+1}=0$ and $a_{\chi _{\max }\left (J\right )}=\left |\mathfrak {MOG}\left (J\right )\right |$ .

1.1 Related works I: Mirzakhani’s integral formulas

In [Reference MirzakhaniMir07], Mirzakhani considered a similar problem to the one in this paper. Instead of integrating over the finite space $\mathrm {Hom}(\Gamma _{g},S_{n})$ , Mirzakhani obtained formulas for the integral of geometric functions over the the moduli space $\mathcal {M}_{g}$ of complete hyperbolic surfaces of genus g, with respect to the Weil–Petersson volume form $d\mathrm {Vol}_{\mathrm {wp}}$ .

The geometric functions that Mirzakhani considers are very much like our Wilson loops. Given any closed curve $\gamma \in \Sigma _{g}$ , for any complete hyperbolic metric J on $\Sigma _{g}$ there is a unique curve isotopic to $\gamma $ that is shortest with respect to J, and the length of this curve is called the length of $\gamma $ , denoted by $\ell _{J}([\gamma ])$ . Here, $[\gamma ]$ is the isotopy class of $\gamma $ .

Mirzakhani requires that $\gamma $ be simple, meaning that it does not intersect itself. This condition is not present in the current paper and can be viewed as an advantage of our work. To obtain a function on $\mathcal {M}_{g}$ , given a continuous function $f:\mathbf {R}_{+}\to \mathbf {R}_{+}$ , Mirzakhani considers the averaged function

$$\begin{align*}f_{\gamma}(J)\stackrel{\mathrm{def}}{=}\sum_{[\gamma']\in\mathrm{MCG}(\Sigma_{g}).[\gamma]}f(\ell_{J}([\gamma'])), \end{align*}$$

where $\mathrm {MCG}(\Sigma _{g})$ is the mapping class group of $\Sigma _{g}$ . Because of the averaging over the mapping class group, $f_{\gamma }$ descends to a function on $\mathcal {M}_{g}$ . This type of averaging is not necessary in the current paper because $\mathbb {X}_{g,n}=\mathrm {Hom}(\Gamma _{g},S_{n})$ is already finite; here, $\mathbb {X}_{g,n}$ is playing the role of the Teichmüller space and not the moduli space. In [Reference MirzakhaniMir07, Thm. 8.1], Mirzakhani gives a formula for

$$\begin{align*}\int_{\mathcal{M}_{g}}f_{\gamma}\,d\mathrm{Vol}_{\mathrm{wp}} \end{align*}$$

in terms of integrating f against Weil–Petersson volumes of moduli spaces. The power of this formula is that in the same paper [Reference MirzakhaniMir07], Mirzakhani gives explicit recursive formulas for the calculations of Weil–Petersson volumes. For a more detailed discussion of these formulas, the reader should consult Wright’s survey of Mirzakhani’s work [Reference WrightWri20, §4].

1.2 Related works II: Free groups

Let $\mathrm {\mathbf {F}} _{r}$ denote a free group on r generators. For $\gamma \in \mathrm {\mathbf {F}} _{r}$ , the problem of integrating the Wilson loop

$$\begin{align*}\mathsf{fix}_{\gamma}(\phi)\stackrel{\mathrm{def}}{=}\mathsf{fix}(\phi(\gamma)),\quad\mathsf{fix}_{\gamma}:\mathrm{Hom}(\mathrm{\mathbf{F}} _{r},S_{n})\to\mathbf{R} \end{align*}$$

over $\mathrm {Hom}(\mathrm {\mathbf {F}} _{r},S_{n})$ with respect to the uniform probability measure is a basic problem that serves as a precursor to that of the current paper. As mentioned above, many of the considerations used with free groups no longer apply in the present paper. Indeed, $\mathrm {Hom}(\mathrm {\mathbf {F}} _{r},S_{n})$ can be identified with $S_{n}^{r}$ and hence techniques for integrating over groups are relevant in a much more direct way than in the case of $\mathrm {Hom}\left (\Gamma _{g},S_{N}\right )$ .

Despite being an easier problem, the theory is very rich. It was proved by Nica in [Reference NicaNic94] that the analog of Theorem 1.2 holds for $\mathbb {E}_{\mathrm {\mathbf {F}} _{r},n}[\mathsf {fix}_{\gamma }]$ . A significantly sharper result was given by Puder and Parzanchevski in [Reference Puder and ParzanchevskiPP15] where they proved that if $\gamma \in \mathrm {\mathbf {F}} _{r}$ , then as $n\to \infty $

$$\begin{align*}\mathbb{E}_{\mathrm{\mathbf{F}} _{r},n}\left[\mathsf{fix}_{\gamma}\right]=1+\frac{c(\gamma)}{n^{\pi(\gamma)-1}}+O\left(\frac{1}{n^{\pi(\gamma)}}\right), \end{align*}$$

where $\pi (\gamma )\in \{0,\ldots ,r\}\cup \{\infty \}$ is an algebraic invariant of $\gamma $ called the primitivity rank and $c(\gamma )\in \mathbf {N}$ is explained in terms of the enumeration of special subgroups of $\mathrm {\mathbf {F}} _{r}$ determined by $\gamma $ . Obtaining a similarly sharp result in the context of $\Gamma _{g}$ is an interesting problem that should be taken up in the future.

Similar Laurent series expansions for the expected value of $\chi _{\gamma }$ on $\mathrm {Hom}(\mathrm {\mathbf {F}} _{r},G(n))$ have been proved to exist, and studied, when $G(n)$ is one of the families of compact Lie groups $\mathrm {U}(n),\mathrm {O}(n),\mathrm {Sp}(n)$ [Reference Magee and PuderMP19, Reference Magee and PuderMP22b], when $G(n)$ is a generalized symmetric group [Reference Magee and PuderMP21], and when $G(n)=\mathrm {\mathrm {GL}}_{n}(\mathbb {F}_{q})$ , where $\mathbb {F}_{q}$ is a fixed finite field [Reference Ernst-West, Puder and SeidelEWPS21]. In all cases, $\chi $ is taken to be a natural character. For example, when $G(n)=\mathrm {U}(n)$ , one such $\chi $ is the trace of the matrix in the group. Moreover, for $G(n)=\mathrm {U}(n),\mathrm {O}(n),\mathrm {Sp}(n)$ and $\chi $ the trace, all the coefficients of the Laurent series are understood [Reference Magee and PuderMP19, Reference Magee and PuderMP22b].

In works undertaken after the completion of this paper, the first named author has obtained analogs of Theorem 1.1 and the first part of Theorem 1.2 forFootnote 3 $\mathrm {Hom}(\Gamma _{g},\mathrm {U(}n))$ and the standard matrix trace [Reference MageeMag22, Reference MageeMag21]. The methods used in (ibid.) are inspired by those of the current work.

1.3 Related works III: Noncommutative probability

Theorem 1.2 has a direct consequence in the setting of Voiculescu’s noncommutative probability theory. Following [Reference Voiculescu, Dykema and NicaVDN92, Def. 2.2.2], a $C^{*}$ -probability space is a pair $(\mathcal {B},\tau )$ , where $\mathcal {B}$ is a unital $C^{*}$ -algebra and $\tau $ is a stateFootnote 4 on $\mathcal {B}$ . We say that a sequence $\{(\mathcal {B},\tau _{n})\}_{n=1}^{\infty }$ of $C^{*}$ -probability spaces converges to $(\mathcal {B},\tau )$ if for all elements $b\in \mathcal {B}$

$$\begin{align*}\lim_{n\to\infty}\tau_{n}(b)=\tau(b). \end{align*}$$

The functions $\tau _{n}:\Gamma _{g}\to \mathbf {R}$ defined by $\tau _{n}(\gamma )\stackrel {\mathrm {def}}{=} n^{-1}\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$ extend to states on the full group $C^{*}$ -algebra $C^{*}(\Gamma {}_{g})$ of $\Gamma _{g}$ . There is also a unique state $\tau _{\mathrm {reg}}$ on $C^{*}(\Gamma _{g})$ that satisfies $\tau _{\mathrm {reg}}(g)=0$ for $g\neq 1$ ; we use the subscript $\mathrm {reg}$ because the GNS representation of $\tau _{\mathrm {reg}}$ is the left regular representation. One has the following corollary of Theorem 1.2:

Corollary 1.4. The $C^{*}$ -probability spaces $(C^{*}(\Gamma _{g}),\tau _{n})$ converge to $(C^{*}(\Gamma _{g}),\tau _{\mathrm {reg}})$ as $n\to \infty $ .

It is reasonable to hope that similar results can be obtained when $\Gamma _{g}$ is replaced by any residually finite one-relator group (cf. $\S \S $ 1.4). We view Corollary 1.4 as an important first step in this program.

1.4 Related works IV: Residual finiteness

A f.g. discrete group $\Lambda $ is residually finite if for any nonidentity $\lambda \in \Lambda $ there is a finite index subgroup $H\le \Lambda $ such that $\lambda \notin H$ . The residual finiteness of $\Gamma _{g}$ has been known for a long time [Reference BaumslagBau62, Reference HempelHem72]. More recently, various quantifications of residual finiteness and of the related property of LERFFootnote 5 have been proposed by various authors [Reference Bou-RabeeBR10, Reference Lazarovich, Levit and MinskyLLM23]. Theorem 1.2 can serve as a strengthening of the residual finiteness of $\Gamma _{g}$ , as we now explain.

Note that residual finiteness of a group $\Lambda $ is equivalent to, for all $e\neq \lambda \in \Lambda $ , the existence of $n\in \mathbf {N}$ and $\phi \in \mathrm {Hom}(\Lambda ,S_{n})$ such that $\phi (\lambda )\ne 1$ . Theorem 1.2 combined with Markov’s inequality implies the following quantitative version of residual finiteness.

Corollary 1.5. Given a nonidentity element $e\ne \gamma \in \Gamma _{g}$ , for large enough n,

(1.5) $$ \begin{align} \frac{\left|\left\{ \phi\in\mathrm{Hom}(\Gamma_{g},S_{n})\,:\,\text{\(\phi(\gamma)\neq1\)}\right\} \right|}{|\mathrm{Hom}(\Gamma_{g},S_{n})|}\ge1-\frac{d\left(q\right)}{n}-O\left(\frac{1}{n^{2}}\right), \end{align} $$

where q and $d\left (q\right )$ are as in Theorem 1.2, and the implied constant in the big-O term depends on $\gamma $ .

In fact, the techniques of this paper can be used to show that, for example, for every $m\in \mathbb {N}$ , the expected value of $\mathsf {fix}_{\gamma }^{~m}$ is of the form $c\left (q\right )+O\left (n^{-1}\right )$ , where q is as in Theorem 1.2 and $c\left (q\right )$ is a positive integer. This would yield a probability bound similar to equation (1.5) but of the form $1-\frac {c\left (q\right )}{n^{m}}+O\left (n^{-m-1}\right )$ . This is done explicitly in [Reference Puder and ZimhoniPZ22, Corollary 1.8].

1.5 Related works V: Benjamini–Schramm convergence

In [Reference Benjamini and SchrammBS01], Benjamini and Schramm introduced a notion of convergence of a sequence of finite graphs to a limiting graph, known now as Benjamini–Schramm convergence. This concept was extended to convergence of sequences of Riemannian manifolds in [Reference Abert, Bergeron, Biringer, Gelander, Nikolov, Raimbault and SametABB+11, Reference Abert, Bergeron, Biringer, Gelander, Nikolov, Raimbault and SametABB+17]. Theorem 1.2 has consequences for the Benjamini–Schramm convergence of random covers of Riemannian surfaces; there are various of these consequences but we present just one representative one here.Footnote 6

Corollary 1.6. Let X be a closed hyperbolic surface of genus $\geq 2$ . Then as $n\to \infty $ , uniformly random degree-n covering spaces of X converge in the sense of Benjamini–Schramm to the hyperbolic upper half plane $\mathbb {H}$ .

Concretely, this means that for any $L>0$ and $\varepsilon>0$ , if $X_{n}$ denotes a random degree-n cover of X (as above), then a.a.s. as $n\to \infty $ ,

$$\begin{align*}\frac{\mathrm{area}\left(X_{n}^{<L}\right)}{\mathrm{area}\left(X_{n}\right)}<\varepsilon, \end{align*}$$

where $X_{n}^{<L}$ is the points of $X_{n}$ with local injectivity radius $<L$ . To see how this follows from Theorem 1.2, viewing L as a constant, any point in $X_{n}^{<L}$ is in a neighborhood, with bounded area depending on L, of some simple closed geodesic of $X_{n}$ with length $<2L$ [Reference BuserBus10, proof of Thm. 4.1.6]. Any such geodesic covers a closed (possibly nonprimitive) geodesic in X of length $<2L$ , and these in turn correspond to a finite list of conjugacy classes in $\Gamma _{g}$ . Starting with a conjugacy class $[\gamma ]$ , the number of corresponding closed lifted geodesics in $X_{n}$ is at most $\mathsf {fix}_{\gamma }$ . Using Markov’s inequality with Theorem 1.2 gives therefore a.a.s. that the number of simple closed geodesics of $X_{n}$ with length $<2L$ is bounded (depending on L). This means $\mathrm {area}\left (X_{n}^{<L}\right )$ is bounded a.a.s. and as $\mathrm {area}\left (X_{n}\right )$ is linear in n, this completes the proof.

1.6 Structure of the proofs and the issues that arise

The reader of the paper is advised to first read this $\S $ 1.6, and then $\S $ 6, where all the ideas of the paper are brought together to give concise proofs of Theorems 1.1, 1.2 and 1.3, before reading the other sections.

There are two main ideas of the paper that we will discuss momentarily. Here, we give a ‘high-level’ account of the strategy of proving our main theorems. At times, we oversimplify definitions to be more instructive. Let us fix $g=2$ and discuss only Theorems 1.1 and 1.2. The extension of these results from cyclic groups to more general finitely generated subgroups is along the same lines. So fix $\gamma \in \Gamma _{2}$ .

Firstly, we view $\mathbb {X}_{n}=\mathbb {X}_{2,n}$ as a space of random coverings of a fixed genus 2 surface $\Sigma _{2}$ . By fixing an octagonal fundamental domain of $\Sigma _{2}$ , each covering of $\Sigma _{2}$ is tiled by octagons. This leads us to the notion of a tiled surface, defined precisely in Definition 2.1. A tiled surface involves not just a tiling but a labeling of the edges of the tiling by generators of the fundamental group of $\Sigma _{2}$ . Hence, all the main theorems can be reinterpreted in terms of random tiled surfaces that are called $X_{\phi }$ for $\phi \in \mathbb {X}_{n}$ .

The first observation is that $\mathbb {E}_{n}\left [\mathsf {fix}_{\gamma }\right ]=\mathbb {E}_{2,n}\left [\mathsf {fix}_{\gamma }\right ]$ , the expected number of fixed points of $\gamma $ under $\phi \in \mathbb {X}_{n}$ , is the expected number of times that we see a fixed annulus A, specified by $\gamma $ , immersed in the random tiled surface $X_{\phi }$ . This annulus A may be the ‘core surface’ corresponding to $\left \langle \gamma \right \rangle $ – see Definition 2.6, the left part of Figure 2.3 and [Reference Magee and PuderMP22a, Lem. 5.1]. However, A needs not be embedded in $X_{\phi }$ . On the other hand, it is possible to produce a finite collection $\mathcal {R}$ of tiled surfaces, each of which has an immersed copy of A, such that

(1.6) $$ \begin{align} \mathbb{E}_{n}\left[\mathsf{fix}_{\gamma}\right]=\sum_{Y\in\mathcal{R}}\mathbb{E}_{n}^{\mathrm{emb}}(Y), \end{align} $$

where $\mathbb {E}_{n}^{\mathrm {emb}}(Y)$ is the expected number of times that Y is embedded in the random $X_{\phi }$ .

We formalize types of collections $\mathcal {R}$ that have the above property in Definition 2.8; we call them resolutions (of A). Of course, there is a great deal of flexibility in how $\mathcal {R}$ is chosen; we will come back to this point shortly. The benefit to having equation (1.6) brings us to the first main idea of the paper:

We have a new method of calculating $\mathbb {E}_{n}^{\mathrm {emb}}(Y)$ , using the representation theory of symmetric groups $S_{n}$ and more specifically, the approach to the representation theory of $S_{n}$ developed by Vershik and Okounkov in [Reference Vershik and OkounkovVO96].

This methodology is developed in $\S $ 5. The necessary background on representation theory is given in $\S $ 3, and in $\S $ 4 we prove some preliminary representation theoretic results needed for $\S $ 5. The reader may be interested to see that Theorem 1.1 has, at its source, Proposition 4.6. See also the overview of $\S $ 5 in $\S \S $ 5.1.

This new methodology to calculate $\mathbb {E}_{n}^{\mathrm {emb}}(Y)$ is sufficient to prove Theorem 1.1. However, in the proof of Theorem 1.2, a critical issue now intervenes. We expect, based on experience with similar projects (e.g., [Reference Puder and ParzanchevskiPP15, Reference Magee and PuderMP19]) that

(1.7) $$ \begin{align} \mathbb{E}_{n}^{\mathrm{emb}}(Y)\approx n^{\chi(Y)} \end{align} $$

as $n\to \infty $ . However, this cannot always be the case. For example, if, roughly speaking, it is possible to glue some octagons to Y to increase the Euler characteristic, forming $Y'$ , then the observation that $\mathbb {E}_{n}^{\mathrm {emb}}(Y)\geq \mathbb {E}_{n}^{\mathrm {emb}}(Y')$ breaks equation (1.7). Then it is not unsurprising that the bounds we obtain for $\mathbb {E}_{n}^{\mathrm {emb}}(Y)$ do not always agree with equation (1.7).

On the other hand, if Y has special properties that we call boundary reduced and strongly boundary reduced, then we can get appropriate bounds on $\mathbb {E}_{n}^{\mathrm {emb}}(Y)$ . We give the precise definitions of these properties in Definitions 2.4 and 2.5. They involve forbidding certain constellations from appearing in the boundary of Y. Even though these constellations are dictated by representation theory, forbidding them remarkably relates to natural geometric properties of Y. For example, if Y is not boundary reduced, then it is possible to add octagons to Y to decrease the number of edges in its boundary. To give some more intuition, being boundary reduced can be viewed as a discrete analog of a hyperbolic surface having geodesic boundary. This means that these properties are closely related with the problem of finding shortest representatives (with respect to word length) of elements of $\Gamma _{g}$ that is addressed by Dehn’s algorithm [Reference DehnDeh12].

If Y is boundary reduced, then we can prove (Theorem 5.10 and Proposition 5.25)

$$\begin{align*}\mathbb{E}_{n}^{\mathrm{emb}}(Y)=O\left(n^{\chi(Y)}\right), \end{align*}$$

and if Y is strongly boundary reduced, we can prove (Theorem 5.10 and Proposition 5.26)

$$\begin{align*}\mathbb{E}_{n}^{\mathrm{emb}}(Y)=n^{\chi\left(Y\right)}\left(1+O\left(n^{-1}\right)\right) \end{align*}$$

(see, again, Section 5.1 for a more detailed overview).Footnote 7 Therefore, to prove Theorem 1.2, it suffices to produce resolutions of the annulus A where we can control which elements are (strongly) boundary reduced, control their Euler characteristics and count the number of elements with maximal Euler characteristic. The design of these resolutions is the second main theme of the paper.

For any tiled surface Z, we describe an algorithm to produce finite resolutions of Z with careful control on their properties as above. This is the main topic of $\S $ 2. Precisely defining the annulus A that should be used as input, as well as its generalization for noncyclic subgroups $J\leq \Gamma $ , and counting the outputs of our algorithm, requires introducing the concept of a core surface of a subgroup $J\leq \Gamma $ . For example, above, A should be taken to be the core surface of $\langle \gamma \rangle $ . The theory of core surfaces that we develop in a companion paper [Reference Magee and PuderMP22a] is analogous to that of Stallings’ core graphs for subgroups of free groups due to Stallings [Reference StallingsSta83], and we hope that the results therein may be of independent interest.

Remark 1.7. Another perspective on the value of $\mathbb {E}_{n}^{\mathrm {emb}}(Y)$ for arbitrary tiles surfaces is given in [Reference Puder and ZimhoniPZ22, Thm. 2.6]. Let Y be an arbitrary tiled surface, $p\colon Y\to \Sigma _{g}$ the restricted covering map, and $\chi ^{\mathrm {grp}}\left (Y\right )$ the Euler characteristic of the subgroup $p_{*}\left (\pi _{1}\left (Y\right )\right )\le \pi _{1}\left (\Sigma _{g}\right )=\Gamma _{g}$ . Then

$$\begin{align*}\mathbb{E}_{n}^{\mathrm{emb}}(Y)=n^{\chi^{\mathrm{grp}}\left(Y\right)}\left(a_{0}+O\left(n^{-1}\right)\right), \end{align*}$$

where $a_{0}$ is some positive integer. This theorem heavily relies on the results of the current paper.

1.7 Notation

Write $\mathbf {N}$ for the natural numbers $1,2,\ldots $ and so on. For $n\in \mathbf {N}$ , we use the notation $[n]$ for the set $\{1,\ldots ,n\}$ . For $m\leq n$ , $m,n\in \mathbf {N}$ we write $[m,n]$ for the set $\{m,m+1,\ldots ,n\}$ . If A and B are sets, we write $A\backslash B$ for the elements of A that are not in B. We write $(n)_{\ell }$ for the Pochhammer symbol

$$\begin{align*}(n)_{\ell}\stackrel{\mathrm{def}}{=} n(n-1)\ldots(n-\ell+1). \end{align*}$$

If V is a vector space, we write $\mathrm {End}(V)$ for the linear endomorphisms of V. If V is a unitary representation of some group, we write $\check {V}$ for the dual representation. If $P_{1},\ldots ,P_{k}$ are a series of expressions we write $\mathbf {1}_{\left \{ P_{1},\ldots ,P_{k}\right \} }$ for a value which is 1 if all the statements $P_{i}$ are true and $0$ else. If V is a vector space, we write $\mathrm {Id}_{V}$ for the identity operator on that space. All integrals over finite sets are with respect to the uniform probability measure on the set. If X is a CW-complex, we write $X^{(i)}$ for its i-skeleton. If we use the symbol $\pm $ more than once in the same expression or equation, we mean that the same sign is chosen each time. If implied constants in big-O notation depend on other constants, we indicate this by adding the constants as a subscript to the O, for example, $O_{\varepsilon }(f(n))$ means the implied constant depends on $\varepsilon $ . We use Vinogradov notation $f(n)\ll g(n)$ to mean that there are constants $n_{0}\ge 0$ and $C_{0}>0$ such that for $n>n_{0}$ , $|f(n)|\leq C_{0}g(n)$ . We add subscripts to indicate dependence of the implied constants on other quantities or objects. If $a,b$ are elements of the same group, we write $[a,b]\stackrel {\mathrm {def}}{=} aba^{-1}b^{-1}$ for their commutator.

2 Resolutions of core surfaces

2.1 Tiled surfaces and core surfaces

In this $\S \S $ 2.1, we summarize some definitions and results from [Reference Magee and PuderMP22a].Footnote 8

2.1.1 Tiled surfaces

Consider the construction of the surface $\Sigma _{g}$ from a $4g$ -gon by identifying its edges in pairs according to the pattern $a_{1}b_{1}a_{1}^{-1}b_{1}^{-1}\ldots a_{g}b_{g}a_{g}^{-1}b_{g}^{-1}$ . This gives rise to a CW-structure on $\Sigma _{g}$ consisting of one vertex (denoted o), $2g$ oriented $1-$ cells (denoted $a_{1},b_{1},\ldots ,a_{g},b_{g}$ ) and one $2$ -cell which is the $4g$ -gon glued along $4g 1$ -cells.Footnote 9 See Figure 2.1 (in our running examples with $g=2$ , we denote the generators of $\Gamma _{2}$ by $a,b,c,d$ instead of $a_{1},b_{1},a_{2},b_{2}$ ). We identify $\Gamma _{g}$ with $\pi _{1}\left (\Sigma _{g},o\right )$ so that in the presentation (1.1), words in the generators $a_{1},\ldots ,b_{g}$ correspond to the homotopy class of the corresponding closed paths based at o along the $1$ -skeleton of $\Sigma _{g}$ .

Figure 2.1 The fixed CW-structure on $\Sigma _{2}$ .

Note that every covering space $p\colon \Upsilon \to \Sigma _{g}$ inherits a CW-structure from $\Sigma _{g}$ : The vertices are the preimages of o, and the open $1$ -cells (2-cells) are the connected components of the preimages of the open 1-cells (2-cells, respectively) in $\Sigma _{g}$ . In particular, this is true for the universal covering space $\widetilde {\Sigma _{g}}$ of $\Sigma _{g}$ , which we can now think of as a CW-complex. A subcomplex of a CW-complex is a subspace consisting of cells such that if some cell belongs to the subcomplex, then so do the cells of smaller dimension at its boundary.

Definition 2.1 (Tiled surface).

[Reference Magee and PuderMP22a, Def. 3.1] A tiled surface Y is a subcomplex of a (not-necessarily-connected) covering space of $\Sigma _{g}$ . In particular, a tiled surface is equipped with the restricted covering map $p\colon Y\to \Sigma _{g}$ which is an immersion. We write $\mathfrak {v}\left (Y\right )$ for the number of vertices of Y, $\mathfrak {e}\left (Y\right )$ for the number of edges and $\mathfrak {f}\left (Y\right )$ for the number of $4g$ -gons.

Alternatively, instead of considering a tiled surface Y to be a complex equipped with a restricted covering map, one may consider Y to be a complex as above with directed and labeled edges: The directions and labels ( $a_{1},b_{1},\ldots ,a_{g},b_{g}$ ) are pulled back from $\Sigma _{g}$ via p. These labels uniquely determine p as a combinatorial map between complexes. Figures 2.1 and 2.3 feature examples of tiled surfaces.

Note that a tiled surface is not always a surface: It may also contain vertices or edges with no $2$ -cells incident to them. However, as Y is a subcomplex of a covering space of $\Sigma _{g}$ , namely, of a surface, any neighborhood of Y inside the covering is a surface, and it is sometimes beneficial to think of Y as such.

Definition 2.2 (Thick version of a tiled surface).

[Reference Magee and PuderMP22a, Def. 3.2] Given a tiled surface Y which is a subcomplex of the covering space $\Upsilon $ of $\Sigma _{g}$ , adjoin to Y a small, closed, tubular neighborhood in $\Upsilon $ around every edge and a small closed disc in $\Upsilon $ around every vertex. The resulting closed surface, possibly with boundary, is referred to as the thick version of Y.

We let $\partial Y$ denote the boundary of the thick version of Y and $\mathfrak {d}\left (Y\right )$ denote the number of edges along $\partial Y$ (so if an edge of Y does not border any $4g$ -gon, it is counted twice).

In particular, $\mathfrak {d}\left (Y\right )=2\mathfrak {e}\left (Y\right )-4g\mathfrak {f}\left (Y\right )$ . We stress that we do not think of Y as a subcomplex but rather as a complex for its own sake, which happens to have the capacity to be realized as a subcomplex of a covering space of $\Sigma _{g}$ . See [Reference Magee and PuderMP22a, §3] for a more detailed discussion.

It is occasionally useful, for example in Section 5, to augment the tiled surface Y by adding some new half-edges. Here, formally, a half-edge is a copy of the interval $[0,\frac {1}{2})$ which is an (open) half of an edge of a covering space of $\Sigma _{g}$ .

Definition 2.3 (Tiled surface with hanging half-edges).

[Reference Magee and PuderMP22a, §§3.2] Let Y be a tiled surface which is a subcomplex of the covering space $p\colon \Upsilon \to \Sigma _{g}$ . We denote by $Y_{+}$ the tiled surface Y together with half-edges of $\Upsilon $ which do not belong to Y but are incident to vertices of Y. Every half-edge of $Y_{+}$ added to Y in this manner is called a hanging half-edge. The thick version of $Y_{+}$ is, as above, $Y_{+}$ together with a small, closed, tubular neighborhood in $\Upsilon $ around every edge or hanging half-edge, and a small closed disc in $\Upsilon $ around every vertex. We denote by $\partial Y_{+}$ the boundary of the think version of $Y_{+}$ .

Note that there are exactly $4g$ half-edges incident to every vertex in $Y_{+}$ : Some of them originate from edges in Y and some are hanging half-edges.

Morphisms of tiled surfaces

If $Y_{1}$ and $Y_{2}$ are tiled surfaces, a morphism from $Y_{1}$ to $Y_{2}$ is a map of $CW$ -complexes which maps i-cells to i-cells for $i=0,1,2$ and respects the directions and labels of edges. Equivalently, this is a morphism of CW-complexes which commutes with the restricted covering maps $p_{j}\colon Y_{j}\to \Sigma _{g}$ ( $j=1,2$ ). In particular, the restricted covering map from a tiled surface to $\Sigma _{g}$ is itself a morphism of tiled surfaces. It is an easy observation that every morphism of tiled surfaces is an immersion (locally injective).

2.1.2 Blocks and chains

Some of the notions we use below are taken from [Reference Birman and SeriesBS87]. See [Reference Magee and PuderMP22a, §§3.2] for a more detailed account.

Given a covering space $\Upsilon $ of $\Sigma _{g}$ , every path in the $1$ -skeleton $\Upsilon ^{\left (1\right )}$ corresponds to a word in $\left \{ a_{1}^{\pm 1},\ldots ,b_{g}^{\pm 1}\right \} $ . A path that follows a (part of the) boundary of a $4g$ -gon is called a block . If it has length at least $2g+1$ it is called a long block , and if it has length exactly $2g$ , it is called a half-block . If a (noncyclic) block of length b sits along the boundary of a $4g$ -gon O, the complement of the block is the block of length $4g-b$ consisting of the complement set of edges along O, so the block and its complement share the same starting point and the same terminal point.

Figure 2.2 A long chain of total length $17$ (blocks of sizes $4,3,3,3,4$ , in blue) and its complement of length $15$ (in red).

A chain is a path in $\Upsilon ^{\left (1\right )}$ that consists of a sequence of blocks $b_{1},\ldots ,b_{r}$ such that if the last vertex of $b_{i}$ and the first vertex of $b_{i+1}$ is v, there is exactly one edge incident to v between the last edge of $b_{i}$ and the first edges of $b_{i+1}$ . In other words, if the $4g$ -gons corresponding to the blocks $b_{1},\ldots ,b_{r}$ are $O_{1},\ldots ,O_{r}$ , then $O_{i}$ and $O_{i+1}$ share an edge e with an endpoint v, and $b_{i}$ ends at v and $b_{i+1}$ starts at v. See Figure 2.2. A long chain is a chain with corresponding blocks of lengths

$$\begin{align*}2g,2g-1,2g-1,\ldots,2g-1,2g. \end{align*}$$

A half-chain is a cyclic chain (so the corresponding path is closed) consisting of blocks each of which is of length $2g-1$ . The complement of a long chain is the chain with blocks of lengths $2g-1,2g-1,\ldots ,2g-1$ which sits along the other side of the $4g$ -gons bordering the long chain and with the same starting point and endpoint. Note that the complement of a long chain is shorter by two edges from the long chain (see Figure 2.2). The complement of a half-chain is defined as follows. If the half-chain sits along the boundary of the $4g$ -gons $O_{1},\ldots ,O_{r}$ , its complement is the half-chain sitting along the other sides of these $4g$ -gons: A block (of length $2g-1$ ) of the half-chain along $O_{i}$ is replaced by the path of length $2g-1$ along $O_{i}$ , with starting and terminal points one edge away from the starting and terminal points, respectively, of the block. The complement of a half-chain has the same length as the original half-chain. The left part of Figure 2.3 illustrates two complementing half-chains of length $6$ each (with two octagons in between).

A boundary cycle of Y is a cycle in $Y^{\left (1\right )}$ corresponding to an oriented boundary component of the thick version of Y (see Definition 2.2). We always choose the orientation so that there are no $4g$ -gons to the immediate left of the boundary component as it is traversed. Therefore, boundary components of Y correspond to unique cycles. Note that $\mathfrak {d}\left (Y\right )$ is equal to the sum over boundary cycles of Y of the number of edges in each such cycle.

2.1.3 Boundary reduced and strongly boundary reduced tiled surfaces

The following definitions came up from our results in representation theory in $\S $ 5, but they fit perfectly with classical results in combinatorial group theory [Reference DehnDeh12] and in particular with [Reference Birman and SeriesBS87].

Definition 2.4 (Boundary reduced).

A tiled surface Y is boundary reduced if no boundary cycle of Y contains a long block or a long chain.

In particular, if Y is boundary reduced, then every path that reads $\left [a_{1},b_{1}\right ]\ldots \left [a_{g},b_{g}\right ]$ is not only closed, but there is also a $4g$ -gon attached to it. We also need a stronger version of this property.

Definition 2.5 (Strongly boundary reduced).

A tiled surface Y is strongly boundary reduced if no boundary cycle of Y contains a half-block or is a half-chain.

Because a long block contains (at least two) half-blocks and a long chain contains (two) half-blocks, a strongly boundary reduced tiled surface is in particular boundary reduced. The relevance of the notions of being (strongly) boundary reduced is that our techniques for estimating $\mathbb {E}_{n}^{\mathrm {emb}}\left (Y\right )$ for a tiled surface Y only give the right type of estimates when Y is boundary reduced – see Proposition 5.25. If Y is strongly boundary reduced we get even better estimates – see Proposition 5.26.

Figure 2.3 Fix $g=2$ , and let $\Gamma _{2}=\left \langle a,b,c,d\,\middle |\,\left [a,b\right ]\left [c,d\right ]\right \rangle $ . On the left is the core surface $ \mathrm {Core}\left (\left \langle aba^{-2}b^{-1}c\right \rangle \right )$ . It consists of $12$ vertices, $14$ edges and two octagons and topologically it is an annulus. On the right is the core surface $ \mathrm {Core}\left (\left \langle a,b\right \rangle \right )$ . It consists of four vertices, six edges and one octagon and topologically it is a genus-1 torus with one boundary component.

Let Y be a compact tiled surface which is a subcomplex of the covering space $\Upsilon $ of $\Sigma _{g}$ . In [Reference Magee and PuderMP22a, §4], we describe the ‘boundary reduced closure’ $\mathsf {BR}\left (Y\hookrightarrow \Upsilon \right )$ of Y in $\Upsilon $ which is the smallest intermediate tiled surface which is boundary reduced. By (ibid., Proposition 4.6), $\mathsf {BR}\left (Y\hookrightarrow \Upsilon \right )$ is compact too. Likewise, $\mathsf {SBR}\left (Y\hookrightarrow \Upsilon \right )$ , the strongly boundary reduced closure, is the smallest intermediate tiled surface which is strongly boundary reduced, but this one is not always compact. Our resolutions in Section 2.3 are based on a ‘compromise’ between these two types of closures.

2.1.4 Core surfaces

Finally, let us define the main object which was introduced and analyzed in [Reference Magee and PuderMP22a], with motivation coming from the current paper. In analogy to Stallings core graphs and their role in the study of free groups and their subgroups, we introduced the notion of core surfaces which relates to subgroups of $\Gamma _{g}$ :

Definition 2.6 (Core surfaces).

[Reference Magee and PuderMP22a, Def. 1.1] Given a subgroup $1\ne J\le \Gamma _{g}=\pi _{1}\left (\Sigma _{g},o\right )$ , consider the covering space $p\colon \Upsilon \to \Sigma _{g}$ corresponding to J, so $\Upsilon =J\backslash \widetilde {\Sigma _{g}}$ . Define the core surface of J, denoted $\mathrm {Core}\left (J\right )$ , as the tiled surface which is a subcomplex of $\Upsilon $ as follows: $\left (i\right )$ take the union of all shortest-representative cycles in the 1-skeleton $\Upsilon ^{\left (1\right )}$ of every free-homotopy class of essential closed curve in $\Upsilon $ , and $\left (ii\right )$ add every connected component of the complement which contains finitely many $4g$ -gons.

For completeness define the core surface of the trivial subgroup to be the zero-dimensional tiled surface consisting of a single vertex.

Note that the quotient $\Upsilon =J\backslash \widetilde {\Sigma _{g}}$ is invariant under conjugation of J, so $\mathrm {Core}\left (J\right )$ depends only on the conjugacy class of J in $\Gamma _{g}$ . Figure 2.3 gives two examples of core surfaces. As another example, if J is of finite index in $\Gamma $ , $\mathrm {Core}\left (J\right )$ is identical to $J\backslash \widetilde {\Sigma _{g}}$ and is a compact closed surface.

In [Reference Magee and PuderMP22a], we give an intrinsic definition of a core surface and show there is one-to-one correspondence between core surfaces (labeled by $a_{1},\ldots ,b_{g}$ ) and conjugacy classes of subgroups of $\Gamma _{g}$ , we provide a ‘folding process’ to construct $\mathrm {Core}\left (J\right )$ from a finite generating set of J (provided, of course, that J is f.g.) and prove basic properties of core surfaces. In particular, $\mathrm {Core}\left (J\right )$ is connected and strongly boundary reduced (ibid., Proposition 5.3), and whenever J is f.g., $\mathrm {Core}\left (J\right )$ is compact (ibid, Proposition 5.8). We also show that whenever $H\le J\le \Gamma _{g}$ , the natural morphism between the corresponding covering spaces $H\backslash \widetilde {\Sigma _{g}}\to J\backslash \widetilde {\Sigma _{g}}$ , restricts to a morphism $\mathrm {Core}\left (H\right )\to \mathrm {Core}\left (J\right )$ .

2.2 Expectations and probabilities of tiled surfaces

Correspondence between $ \mathrm {Hom}\left (\Gamma _{g},S_{n}\right )$ and n-sheeted covering spaces of $\Sigma _{g}$

Let M be a connected topological space with basepoint m. Consider n-sheeted covering spaces of M with the fiber above m labeled by $\left [n\right ]$ so that every point in the fiber has a different label. If M is ‘nice enough’, in particular if M is a surface, there is a one-to-one correspondence between these labeled n-sheeted covering spaces and the set of homomorphisms $\mathrm {Hom}\left (\pi _{1}\left (M,m\right ),S_{n}\right )$ (see, for instance, [Reference HatcherHat05, pp. 68-70]). If $\hat {M}$ is a labeled n-sheeted covering space and $p\colon \hat {M}\to M$ the covering map, the corresponding homomorphism $\theta \colon \pi _{1}\left (M,m\right )\to S_{n}$ is given as follows: For $h\in \pi _{1}\left (M,m\right )$ , consider $\gamma $ , a closed path in M, based at m, which represents h. Then $\theta \left (h\right )\left (i\right )=j$ if and only if the lift of $\gamma $ at the point i ends at the point j.Footnote 10

In our case, this translates to a one-to-one correspondence between the representation space $\mathbb {X}_{g,n}=\mathrm {Hom}\left (\Gamma _{g},S_{n}\right )$ and labeled n-sheeted covering spaces of $\Sigma _{g}$ (pointed at o). For $\phi \colon \Gamma _{g}\to S_{n}$ , denote the corresponding covering space by $p_{\phi }\colon X_{\phi }\to \Sigma _{g}$ . As explained above, $X_{\phi }$ inherits a CW-structure from $\Sigma _{g}$ and is, therefore, a tiled surface. The fiber above o is precisely the vertices of $X_{\phi }$ , and they are labeled by $\left [n\right ]$ in this construction.

Expected number of fixed points as expected number of lifts

Given a compact tiled surface Y, we are interested in the expected number of morphisms from Y to a random n-covering of $\Sigma _{g}$ , namely, in

$$\begin{align*}\mathbb{E}_{n}\left(Y\right)\stackrel{\mathrm{def}}{=}\mathbb{E}_{\phi\in\mathbb{X}_{g,n}}\left[\#\left\{ \mathrm{morphisms}~Y\to X_{\phi}\right\} \right], \end{align*}$$

where $\phi $ is sampled uniformly at random from $\mathbb {X}_{g,n}$ . Equivalently, this is the expected number of lifts of the restricted covering map $p:Y\to \Sigma _{g}$ to the random n-covering $X_{\phi }$ :

Note that if Y is connected and $\phi \in \mathbb {X}_{g,n}$ , the number of morphisms $Y\to X_{\phi }$ is at most n, as any vertex of Y can be lifted to one of the n vertices of $X_{\phi }$ , and each such lift can be extended in at most one way to a lift of the whole of Y. For suitable choices of Y, $\mathbb {E}_{n}\left (Y\right )$ is equal to the quantities $\mathbb {E}_{g,n}\left [\mathsf {fix}_{\gamma }\right ]$ and $\mathbb {E}_{g,n}\left [\mathsf {fix}_{J}\right ]$ that feature in our main theorems (Theorems 1.1, 1.2 and 1.3):

Lemma 2.7. Let Y be a connected tiled surface and $p\colon Y\to \Sigma _{g}$ the restricted covering map. For arbitrary vertex $y\in Y$ , assume that $p_{*}\left (\pi _{1}\left (Y,y\right )\right )$ is conjugate to $J\le _{\mathrm {f.g.}}\Gamma _{g}$ (as a subgroup of $\pi _{1}\left (\Sigma _{g},o\right )=\Gamma _{g}$ ). Then for all $n\in \mathbb {N}$ ,

$$\begin{align*}\mathbb{E}_{n}\left(Y\right)=\mathbb{E}_{g,n}\left[\mathsf{fix}_{J}\right]. \end{align*}$$

In particular, for $J\le _{\mathrm {f.g.}}\Gamma _{g}$ ,

$$\begin{align*}\mathbb{E}_{n}\left(\mathrm{Core}\left(J\right)\right)=\mathbb{E}_{g,n}\left[\mathsf{fix}_{J}\right]. \end{align*}$$

Proof. In fact, the equality holds at the level of the individual representation $\phi \in \mathbb {X}_{n}=\mathrm {Hom}\left (\Gamma _{g},S_{n}\right )$ : The number of morphisms $Y\to X_{\phi }$ is equal to the number of common fixed points $\mathsf {fix}_{J}\left (\phi \right )$ . Indeed, because the number of common fixed points of $\phi \left (J\right )$ is the same as the number of fixed points of any conjugate, we may assume without loss of generality that $p_{*}\left (\pi _{1}\left (Y,y\right )\right )=J$ . Now, $i\in \left [n\right ]$ is a common fixed point of $\phi \left (J\right )$ if and only if $J\le \pi _{1}\left (X_{\phi },v_{i}\right )$ , where $v_{i}$ is the vertex of $X_{\phi }$ labeled i, and $\pi _{1}\left (X_{\phi },v_{i}\right )$ is identified with the subgroup

$$\begin{align*}\left(p_{\phi}\right)_{*}\left(\pi_{1}\left(X_{\phi},v_{i}\right)\right)\le\Gamma_{g}. \end{align*}$$

By standard facts from the theory of covering spaces [Reference HatcherHat05, Prop. 1.33 and 1.34], there is a lift of p to $X_{\phi }$ mapping the vertex y to $v_{i}$ if and only if (the image in $\Gamma _{g}$ of) $\pi _{1}\left (Y,y\right )$ is contained in (the image in $\Gamma _{g}$ of) $\pi _{1}\left (X_{\phi },v_{i}\right )$ , and this lift, if exists, is unique.

The statement about core surfaces follows from the fact that (the image in $\Gamma _{g}$ of) $\pi _{1}\left (\mathrm {Core}\left (J\right )\right )$ is conjugate to J [Reference Magee and PuderMP22a, Prop. 5.3].

Another type of expectation will also feature in this work. Given a compact tiled surface Y, denote

$$\begin{align*}\mathbb{E}_{n}^{\mathrm{emb}}\left(Y\right)\stackrel{\mathrm{def}}{=}\mathbb{E}_{\phi\in\mathbb{X}_{g,n}}\left[\#\left\{ \mathrm{injective~morphisms}~Y \to X_{\phi} \right\} \right], \end{align*}$$

where the expectation is over a uniformly random $\phi \in \mathbb {X}_{g,n}$ .

2.3 Resolutions

Definition 2.8 (Resolutions).

A resolution $\mathcal {\mathcal {R}}$ of a tiled surface Y is a collection of morphisms of tiled surfaces

$$\begin{align*}\mathcal{R}=\left\{ f\colon Y\to W_{f}\right\} , \end{align*}$$

such that every morphism $h\colon Y\to Z$ of Y into a tiled surface Z with no boundary decomposes uniquely as $Y\stackrel {f}{\to }W_{f}\stackrel {\overline {h}}{\hookrightarrow }Z$ , where $f\in \mathcal {R}$ and $\overline {h}$ is an embedding.

The purpose of introducing resolutions is the following obvious lemma. Recall the notation $\mathbb {E}_{n}\left (Y\right )$ and $\mathbb {E}_{n}^{\mathrm {emb}}\left (Y\right )$ from Section 2.2.

Lemma 2.9. If Y is a compact tiled surface and $\mathcal {R}$ is a finite resolution of Y, then

(2.1) $$ \begin{align} \mathbb{E}_{n}\left(Y\right)=\sum_{f\in\mathcal{R}}\mathbb{E}_{n}^{\mathrm{emb}}\left(W_{f}\right). \end{align} $$

Our main goal in the rest of this subsection is to prove the existence of a finite resolution whenever we are given a compact tiled surface Y – this is the content of Theorem 2.14 below. This resolution will consist strictly of boundary reduced tiled surfaces $W_{f}$ , and some of these will even be strongly boundary reduced. We shall make use of Theorem 2.14 mainly for Y a core surface of a finitely generated subgroup of $\Gamma $ . In this case, the resolution we construct has even nicer properties – see Proposition 2.15.

Ideally, we would have liked to get a resolution where all the elements are strongly boundary reduced. Unfortunately, such a resolution does not always exist. For example, when $g=2$ and $\Gamma _{2}=\left \langle a,b,c,d\,\middle |\,\left [a,b\right ]\left [c,d\right ]\right \rangle $ , the core surface $Y=\mathrm {Core}\left (\left \langle \left [a,b\right ]\right \rangle \right )$ does not admit such a resolution as can be inferred from [Reference Magee and PuderMP22a, Fig. 4.2].

To prove the existence of a resolution with nice properties, we first define a process which outputs a ‘compromise’ between the $\mathsf {BR}$ -closure of a tiled surface and the $\mathsf {SBR}$ -closure, introduced in [Reference Magee and PuderMP22a, §4].

Definition 2.10. Fix $\chi _{0}\in \mathbb {Z}$ . Assume that $h\colon Y\to Z$ is a morphism between tiled surfaces where Y is compact and Z has no boundary. Let $W_{0}$ denote the h-image of Y in Z. Set $i=0$ . Perform the following algorithm we call the growing process:

  1. 1. If one of the following conditions holds:

    1. (a) $W_{i}$ is strongly boundary reduced, or

    2. (b) $W_{i}$ is boundary reduced and $\chi \left (W_{i}\right )<\chi _{0}$ ,

    terminate and return $h\colon Y\to W_{i}$ .

  2. 2. Obtain $W_{i+1}$ from $W_{i}$ by adding to $W_{i}$ (the closure of) every $4g$ -gon in $Z\setminus W_{i}$ which touches along its boundary an edge of $\partial W_{i}$ which is part of a half-block (this includes the case of a long block), a long chain or a half-chain. Set $i:=i+1$ and return to item $1$ .

It is clear that every step of this process is deterministic. Note that if the process ends when $W_{i}$ is strongly boundary reduced, then $W_{i}$ is the unique smallest strongly boundary reduced tiled surface inside Z containing $W_{0}$ , denoted $\mathsf {SBR}(Y\hookrightarrow Z)$ [Reference Magee and PuderMP22a, §4]. (In general, $\mathsf {SBR}(Y\hookrightarrow Z)$ is not always compact, but in this case it is.) The growing process always terminates after finitely many steps.

Lemma 2.11. The process described in Definition 2.10 always terminates.

Proof. Let $\mathfrak {he}\left (W_{i}\right )$ denote the number of hanging half-edges along the boundary of $\left (W_{i}\right )_{+}$ , and consider the triple

(2.2) $$ \begin{align} \left(\mathfrak{d}\left(W_{i}\right),\chi\left(W_{i}\right),-\mathfrak{he}\left(W_{i}\right)\right). \end{align} $$

For every i, $W_{i}$ is a compact subsurface of Z, and so the three quantities are well-defined integers. We claim that at every step in the growing process, the triple (2.2) strictly reduces with respect to the lexicographic order.

Indeed, assume we do not halt after i steps, and let $O_{1},\ldots ,O_{k}$ be the list of $4g$ -gons in $Z\setminus W_{i}$ which are added to $W_{i}$ in order to obtain $W_{i+1}$ . By the choice of $4g$ -gons, it is clear that $\mathfrak {d}\left (W_{i+1}\right )\le \mathfrak {d}\left (W_{i}\right )$ . If the inequality is strict, we are done. So assume $\mathfrak {d}\left (W_{i+1}\right )=\mathfrak {d}\left (W_{i}\right )$ . This means that $\partial \left (W_{i}\right )$ contains no long blocks nor long chains, so it is boundary reduced, and that the edges in the complements in Z of the half-blocks and half-chains at $\partial \left (W_{i}\right )$ all belong to $\partial \left (W_{i+1}\right )$ . In other words, let $p_{1},\ldots ,p_{m}$ be these complements in Z: So $p_{j}$ is either a half-block or a half-chain. The equality $\mathfrak {d}\left (W_{i+1}\right )=\mathfrak {d}\left (W_{i}\right )$ means that all the edges in $p_{1},\ldots ,p_{m}$ belong to $\partial W_{i+1}$ .

It is easy to see that in this case $\chi \left (W_{i+1}\right )\le \chi \left (W_{i}\right )$ : The number of new $4g$ -gons and vertices in $W_{i+1}$ at most balances the number of new edges. Let V denote the set of internal vertices in $p_{1},\ldots ,p_{m}$ (so not at their endpoints). We have strict inequality $\chi \left (W_{i+1}\right )<\chi \left (W_{i}\right )$ if and only if some $v\in V$ belongs to $W_{i}$ or to two different complements from $p_{1},\ldots ,p_{m}$ .

Now, assume that $\mathfrak {d}\left (W_{i+1}\right )=\mathfrak {d}\left (W_{i}\right )$ and $\chi \left (W_{i+1}\right )=\chi \left (W_{i}\right )$ . Then $W_{i}$ is boundary reduced and each of the complements $p_{1},\ldots ,p_{m}$ is a connected piece of $\partial W_{i+1}$ . If $O_{j}$ touches a half-block of $\partial W_{i}$ , its annexation adds a net of $\left (2g-1\right )\left (4g-2\right )-2=8g\left (g-1\right )$ hanging half-edges. Every $4g$ -gon along a half-chain of $\partial W_{i}$ also adds on average a net of $8g\left (g-1\right )$ hanging half-edges. So if we add at least one $4g$ -gon at the $\left (i+1\right )$ st step, $-\mathfrak {he}$ strictly decreases. So indeed the triple (2.2) strictly decreases lexicographically in every step.

Finally, there are at most finitely many steps in which $\mathfrak {d}\left (W_{i}\right )$ decreases because this is a nonnegative integer. So it is enough to show there cannot be infinitely many steps in which $\mathfrak {d}\left (W_{i}\right )$ is constant. If $\mathfrak {d}\left (W_{i+1}\right )=\mathfrak {d}\left (W_{i}\right )$ , then $W_{i}$ is boundary reduced. If $\chi \left (W_{i}\right )$ keeps decreasing, then eventually we hit the bound $\chi \left (W_{i}\right )<\chi _{0}$ and halt. If $\mathfrak {d}\left (W_{i}\right )$ and $\chi \left (W_{i}\right )$ are constant, then $\mathfrak {he}\left (W_{i}\right )$ increases constantly, but in every tiled surface W, $\mathfrak {he}\left (W\right )\le 4g\mathfrak {d}\left (W\right )$ , so there cannot be infinitely many steps of this type too. This proves the lemma.

Lemma 2.12. There is a bound $B=B\left (Y\right )$ , independent of $h\colon Y\to Z$ , such that in the entire growing process, at most $B=B\left (Y\right )$ $4g$ -gons are added to $W_{0}$ .

Proof. Note that every boundary edge of $W_{0}$ is necessarily an h-image of a boundary edge of Y so that $\mathfrak {d}\left (W_{i}\right )\le \mathfrak {d}\left (W_{0}\right )\le \mathfrak {d}\left (Y\right )$ . In every step, we add at most $\frac {\mathfrak {d}\left (W_{i}\right )}{2g-1}\le \frac {\mathfrak {d}\left (Y\right )}{2g-1}$ $4g$ -gons. So it is enough to bound the number of steps performed in the growing process until it terminates. There are at most $\mathfrak {d}\left (Y\right )$ steps in which $\mathfrak {d}\left (W_{i}\right )$ strictly decreasesFootnote 11 ( $\mathfrak {d}\left (W_{i+1}\right )<\mathfrak {d}\left (W_{i}\right )$ ), so there are at most $\mathfrak {d}\left (Y\right )+1$ possible values of $\mathfrak {d}\left (W_{i}\right )$ . In steps where $\mathfrak {d}\left (W_{i}\right )$ is unchanged, $W_{i}$ is boundary reduced, so by definition $\chi \left (W_{i}\right )\ge \chi _{0}$ (otherwise, the process terminates). Let $\pi _{0}\left (Y\right )$ denote the number of connected components of Y. For all i, $W_{i}$ is a subsurface of Z with at most $\pi _{0}\left (Y\right )$ connected components, and by the classification of surfaces, $\chi \left (W_{i}\right )\le 2\pi _{0}\left (Y\right )$ . There are at most $2\pi _{0}\left (Y\right )-\chi _{0}$ steps with $\mathfrak {d}\left (W_{i}\right )$ fixed and $\chi \left (W_{i}\right )$ strictly decreasing. Finally, when $\mathfrak {d}\left (W_{i}\right )$ is constant there are at most $2\pi _{0}\left (Y\right )-\chi _{0}+1$ possible values of $\chi \left (W_{i}\right )$ , and if $\mathfrak {d}\left (W_{i+1}\right )=\mathfrak {d}\left (W_{i}\right )$ and $\chi \left (W_{i+1}\right )=\chi \left (W_{i}\right )$ , then $\mathfrak {he}\left (W_{i+1}\right )\ge \mathfrak {he}\left (W_{i}\right )+8g\left (g-1\right )$ and $\mathfrak {he}\left (W_{i}\right )\le 4g\mathfrak {d}\left (W_{i}\right )\le 4g\mathfrak {d}\left (Y\right )$ , so there are at most $\frac {\mathfrak {d}\left (Y\right )}{2\left (g-1\right )}$ steps with the same value of $\mathfrak {d}\left (W_{i}\right )$ and $\chi \left (W_{i}\right )$ . Overall there are at most

(2.3) $$ \begin{align} \mathfrak{d}\left(Y\right)+\left(\mathfrak{d}\left(Y\right)+1\right)\left[\left(2\pi_{0}\left(Y\right)-\chi_{0}\right)+\left(2\pi_{0}\left(Y\right)-\chi_{0}+1\right)\cdot\frac{\mathfrak{d}\left(Y\right)}{2\left(g-1\right)}\right] \end{align} $$

steps in the growing process. Define $B\left (Y\right )$ to be $\frac {\mathfrak {d}\left (Y\right )}{2g-1}$ times equation (2.3).

We can now define the sought-after resolution for compact tiled surfaces.

Definition 2.13. Suppose that Y is a compact tiled surface and $\chi _{0}\in \mathbb {Z}$ a fixed integer. Define the $\chi _{0}$ -resolution of Y to be the collection

$$\begin{align*}\mathcal{R}=\mathcal{R}\left(Y,\chi_{0}\right)=\left\{ f\colon Y\to W_{f}\right\} \end{align*}$$

obtained from all possible morphisms $h\colon Y\to Z$ from Y to a tiled surface Z with no boundary via the growing process (the process applied with the parameter $\chi _{0}$ ).

Theorem 2.14. Suppose Y is a compact tiled surface and $\chi _{0}\in \mathbb {Z}$ a fixed integer. The collection $\mathcal {R}=\mathcal {R}\left (Y,\chi _{0}\right )$ from Definition 2.13 is a finite resolution of Y which satisfies further

  1. R1 for every $f\in \mathcal {R}$ , the tiled surface $W_{f}$ is compact and boundary reduced, and

  2. R2 for every $f\in \mathcal {R}$ with $\chi \left (W_{f}\right )\ge \chi _{0}$ , the tiled surface $W_{f}$ is strongly boundary reduced.

Proof. By Lemma 2.11 and the halting conditions of the growing process, it is clear that every such morphism in $\mathcal {R}$ satisfies R1 and R2. Given a morphism $h\colon Y\to Z$ as in Definition 2.13, $h\left (Y\right )$ (also named $W_{0}$ ) is a quotient of Y and therefore the number of cells in $h\left (Y\right )$ is bounded. From Lemma 2.12, we now conclude that that there is a bound on the number of cells in any $W_{f}$ with $f\in \mathcal {R}$ . This shows that $\mathcal {R}$ is finite as there are finitely many tiled surfaces with given bounds on the number of cells and finitely many morphisms between two given compact tiled surfaces.

It remains to show that $\mathcal {R}$ is a resolution. By the way it was constructed, it is clear that every morphism $h\colon Y\to Z$ with $\partial Z=\emptyset $ decomposes as

(2.4) $$ \begin{align} Y\stackrel{f}{\to}W_{f}\hookrightarrow Z \end{align} $$

and that $f\in \mathcal {R}$ . To show uniqueness, assume that h decomposes in an additional way

(2.5) $$ \begin{align} Y\stackrel{\varphi}{\to}W_{\varphi}\hookrightarrow Z, \end{align} $$

where $W_{\varphi }$ is the result of the growing process for some $h'\colon Y\to Z'$ with $\partial Z'=\emptyset $ . We claim that equations (2.4) and (2.5) are precisely the same decompositions of h. Indeed, the growing process defined by $h'\colon Y\to Z'$ takes place entirely inside $W_{\varphi }$ and does not depend on the structure of $Z'\backslash W_{\varphi }$ : In the $\left (i+1\right )$ st step of the growing process, the decision whether or not to annex more $4g$ -gons and where depends only on the structure and boundary of $W_{i}$ . Consequently, the growing process defined by the morphism $h'\colon Y\to Z'$ has the exact same output, in terms of the resulting element we add to $\mathcal {R}$ , as the growing process defined by the composition $Y\stackrel {\varphi }{\to }W_{\varphi }\hookrightarrow Z$ . But because the growing process is deterministic, the latter is identical to the growing process defined by $h\colon Y\to Z$ .

As mentioned above, we will use Theorem 2.14 mainly with Y being a core surface. In this case, the theorem can be strengthened as follows. Recall from Section 1 that, given $J\le _{\mathrm {f.g.}}\Gamma _{g}$ , we denote by $\mathfrak {MOG}\left (J\right )$ the set of f.g. overgroups of J with maximal Euler characteristic, and by $\chi _{\max }\left (J\right )$ this maximal Euler characteristic.

Proposition 2.15 (Addendum to Theorem 2.14).

Let $J\le _{\mathrm {f.g.}}\Gamma _{g}$ , and let $\chi _{0}\in \mathbb {Z}$ . Let $\mathcal {R}_{J,\chi _{0}}=\left \{ f\colon \mathrm {Core}\left (J\right )\to W_{f}\right \} $ be the resolution $\mathcal {R}\left (\mathrm {Core}\left (J\right ),\chi _{0}\right )$ from Definition 2.13. Then $\mathcal {R}_{J,\chi _{0}}$ satisfies further the following two properties.

  1. R2 For every $f\in \mathcal {R}_{J,\chi _{0}}$ with $\chi \left (W_{f}\right )\ge \chi _{0}$ , the tiled surface $W_{f}$ is the core surface of some $K\le _{\mathrm {f.g.}}\Gamma _{g}$ with $J\le K$ and f is the natural morphism between the two core surfaces (the restriction of $J\backslash \widetilde {\Sigma _{g}}\to K\backslash \widetilde {\Sigma _{g}}$ ).

  2. R4 Assume that $\chi _{0}\le \chi _{\max }\left (J\right )$ . Then for every $K\in \mathfrak {MOG}\left (J\right )$ , the natural morphism $\mathrm {Core}\left (J\right )\to \mathrm {Core}\left (K\right )$ belongs to $\mathcal {R}_{J,\chi _{0}}$ .

For $K\le _{\mathrm {f.g.}}\Gamma _{g}$ , we have $\chi \left (K\right )=\chi \left (\mathrm {Core}\left (K\right )\right )$ [Reference Magee and PuderMP22a, Prop. 5.3]. Proposition 2.15 thus shows that, as long as $\chi _{0}\le \chi _{\max }\left (J\right )$ , there is a bijection between the elements of $\mathfrak {MOG}\left (J\right )$ and the elements in the resolution with maximal Euler characteristic.

Corollary 2.16. For every $J\le _{\mathrm {f.g.}}\Gamma _{g}$ , the set $\mathfrak {MOG}\left (J\right )$ of f.g. overgroups of maximal Euler characteristic is finite.

Proof of Proposition 2.15.

Suppose that $f\colon \mathrm {Core}\left (J\right )\to W_{f}$ satisfies $\chi \left (W_{f}\right )\ge \chi _{0}$ . In particular, $W_{f}$ is strongly boundary reduced by R2. Let $j\in \mathrm {Core}\left (J\right )$ be a vertex and assume without loss of generality that $J=p_{*}\left (\pi _{1}\left (\mathrm {Core}\left (J\right ),j\right )\right )$ (the fact that $p_{*}\left (\pi _{1}\left (\mathrm {Core}\left (J\right ),j\right )\right )\le \pi _{1}\left (\Sigma _{g},o\right )=\Gamma _{g}$ is conjugate to J follows from [Reference Magee and PuderMP22a, Prop. 5.3 and Cor. 4.11]). Define $K\stackrel {\mathrm {def}}{=}\pi _{1}\left (W_{f},f\left (j\right )\right )$ . As $W_{f}$ is compact, K is finitely generated. Let $p_{K}\colon \left (K\backslash \widetilde {\Sigma _{g}},k\right )\to \Sigma _{g}$ be the pointed covering space with $\pi _{1}\left (K\backslash \widetilde {\Sigma _{g}},k\right )=K$ . By the unique lifting property from the theory of covering spaces [Reference HatcherHat05, Prop. 1.33 and 1.34], as $W_{f}$ is connected, there is a unique lift $\alpha $ of the restricted covering map $p_{W_{f}}\colon W_{f}\to \Sigma _{g}$ to $K\backslash \widetilde {\Sigma _{g}}$ , which maps $f\left (j\right )$ to k. We will show that $\alpha $ gives an isomorphism between $W_{f}$ and $\mathrm {Core}\left (K\right )$ .

First, we show that $\alpha \left (W_{f}\right )\subseteq \mathrm {Core}\left (K\right )$ . Recall that $W_{f}$ is the result of the growing process for some $h\colon \mathrm {Core}\left (J\right )\to Z$ with $\partial Z=\emptyset $ . Consider $W_{0}\stackrel {\mathrm {def}}{=} h\left (\mathrm {Core}\left (J\right )\right )\subseteq Z$ . Recall that $W_{f}=\mathsf {SBR}\left (W_{0}\hookrightarrow Z\right )$ . By the unique lifting property, $\alpha \circ f$ is the natural morphism $\mathrm {Core}\left (J\right )\to K\backslash \widetilde {\Sigma _{g}}$ , which, by [Reference Magee and PuderMP22a, Lem. 5.4], has an image contained in $\mathrm {Core}\left (K\right )$ . Hence, $\alpha \left (W_{0}\right )\subseteq \mathrm {Core}\left (K\right )$ . As $\mathrm {Core}\left (K\right )$ is strongly boundary reduced [Reference Magee and PuderMP22a, Prop. 5.3], we have that $\mathsf {SBR}\left (\alpha \left (W_{0}\right )\hookrightarrow K\backslash \widetilde {\Sigma _{g}}\right )$ is contained in $\mathrm {Core}\left (K\right )$ . By [Reference Magee and PuderMP22a, Lem. 4.7],

$$\begin{align*}\alpha\left(W_{f}\right)=\alpha\left(\mathsf{SBR}\left(W_{0}\hookrightarrow Z\right)\right)\subseteq\mathsf{SBR}\left(\alpha\left(W_{0}\right)\hookrightarrow K\backslash\widetilde{\Sigma_{g}}\right)\subseteq\mathrm{Core}\left(K\right). \end{align*}$$

Now, Z is a covering space of $\Sigma _{g}$ and we may assume it is connected (because $\mathrm {Core}\left (J\right )$ is). Thus, Z is identical to $L\backslash \widetilde {\Sigma _{g}}$ for some $L=\pi _{1}\left (Z,h\left (j\right )\right )$ . By property R2, $W_{f}$ is strongly boundary reduced, and so by [Reference Magee and PuderMP22a, Cor. 4.11] its embedding in Z is $\pi _{1}$ -injective. In other words, $K\le L$ and, therefore, there is a morphism $m\colon \left (K\backslash \widetilde {\Sigma _{g}},k\right )\to \left (Z,f\left (j\right )\right )$ . By the unique lifting property, the composition $m\circ \alpha \colon \left (W_{f},f\left (j\right )\right )\to \left (Z,f\left (j\right )\right )$ must be identical to the embedding $\left (W_{f},f\left (j\right )\right )\hookrightarrow \left (Z,f\left (j\right )\right )$ and therefore $\alpha $ is injective. So $\alpha \left (W_{f}\right )$ is a strongly boundary reduced subtiled surface of $\mathrm {Core}\left (K\right )$ with fundamental group K. By [Reference Magee and PuderMP22a, Lem. 5.7], it follows that $\alpha \left (W_{f}\right )\supseteq \mathrm {Core}\left (K\right )$ . We conclude that $\alpha \colon W_{f}\to \mathrm {Core}\left (K\right )$ is an isomorphism, and R3 is proven.

To prove R4, suppose that $K\in \mathfrak {MOG}\left (J\right )$ . Let $h\colon \mathrm {Core}\left (J\right )\to K\backslash \widetilde {\Sigma _{g}}$ be the natural morphism. By the definition of the resolution $\mathcal {R}_{J}$ , the morphism h factors as $\mathrm {Core}\left (J\right )\stackrel {f}{\to }W_{f}\stackrel {}{\hookrightarrow }K\backslash \widetilde {\Sigma _{g}}$ for some $f\in \mathcal {R}_{J}$ . Because $h\left (\mathrm {Core}\left (J\right )\right )\subseteq \mathrm {Core}\left (K\right )$ (by [Reference Magee and PuderMP22a, Lem. 5.4]) and because $\mathrm {Core}\left (K\right )$ is strongly boundary reduced, we have $W_{f}\subseteq \mathrm {Core}\left (K\right )$ .

Let C be a connected component of the difference between the thick version of $\mathrm {Core}\left (K\right )$ and the thick version of $W_{f}$ . As $\mathrm {Core}\left (K\right )$ is compact, $\overline {C}$ is compact. As $\mathrm {Core}\left (K\right )$ is connected, $\overline {C}$ must intersect $\partial W_{f}$ and in particular has at least one boundary component. Since $W_{f}$ is boundary reduced, $\overline {C}$ is not homeomorphic to a disc, and so $\chi \left (\overline {C}\right )\le 0$ . Now,

$$\begin{align*}\chi\left(K\right)=\chi\left(\mathrm{Core}\left(K\right)\right)=\chi\left(W_{f}\right)+\sum_{C}\chi\left(\overline{C}\right), \end{align*}$$

the sum being over all connected components as above. We conclude that $\chi \left (W_{f}\right )\ge \chi \left (K\right )= \chi _{\max }\left (J\right )\ge \chi _{0}$ . By R2, $W_{f}$ is strongly boundary reduced and by R3, $W_{f}=\mathrm {Core}\left (M\right )$ for some subgroup M. But then $M\in \mathfrak {MOG}\left (J\right )$ , $\chi \left (\mathrm {Core}\left (M\right )\right )=\chi \left (K\right )$ , and every connected component C as above satisfies $\chi \left (\overline {C}\right )=0$ . As $\overline {C}$ has at least one boundary component, it must be an annulus. But then $\mathrm {Core}\left (M\right )$ is a deformation retract of $\mathrm {Core}\left (K\right )$ , so $M=K$ up to conjugation and so $W_{f}=\mathrm {Core}\left (M\right )=\mathrm {Core}\left (K\right )$ .

Corollary 2.17. Suppose $1\ne \gamma \in \Gamma _{g}$ is a nontrivial element. Let q be the maximal natural number such that $\gamma =\gamma _{0}^{~q}$ for some $\gamma _{0}\in \Gamma _{g}$ , and $d(q)$ the number of positive divisors of q. Then $\mathrm {Core}\left (\left \langle \gamma \right \rangle \right )$ has a finite resolution $\mathcal {R}_{\gamma }=\left \{ f\colon \mathrm {Core}\left (\left \langle \gamma \right \rangle \right )\to W_{f}\right \} $ with $W_{f}$ boundary reduced for every $f\in R_{\gamma }$ and with exactly $d\left (q\right )$ elements $f\in \mathcal {R}_{\gamma }$ with $\chi \left (W_{f}\right )\ge 0$ . Moreover, these $d\left (q\right )$ elements are precisely

(2.6) $$ \begin{align} f_{m}\colon\mathrm{Core}\left(\left\langle \gamma\right\rangle \right)\to\mathrm{Core}\left(\left\langle \gamma_{0}^{~m}\right\rangle \right) \end{align} $$

for $m\mid q$ , where $f_{m}$ is the natural morphism between the core surfaces.

Proof. Construct $\mathcal {R}_{\gamma }=\left \{ f\colon \mathrm {Core}\left (\left \langle \gamma \right \rangle \right )\to W_{f}\right \} $ as $\mathcal {R}\left (\mathrm {Core}\left (\left \langle \gamma \right \rangle \right ),0\right )$ from Definition 2.13. By Theorem 2.14 and Proposition 2.15, the elements in $\mathcal {R}_{\gamma }$ with $\chi \left (W_{f}\right )=0$ are precisely the core surfaces of the subgroups in $\mathfrak {MOG}\left (\left \langle \gamma \right \rangle \right )$ . So it only remains to show that $\mathfrak {MOG}\left (\left \langle \gamma \right \rangle \right )$ are precisely $\left \langle \gamma _{0}^{~m}\right \rangle $ with $m\mid q$ .

But f.g. subgroups $K\le \Gamma _{g}$ with $\chi \left (K\right )=0$ are necessarily cyclic. Assume $K=\left \langle \delta \right \rangle \in \mathfrak {MOG}\left (\left \langle \gamma \right \rangle \right )$ , so $\gamma \in \left \langle \delta \right \rangle $ , and we may assume that $\gamma $ is a positive power of $\delta $ (otherwise switch to $\delta ^{-1}$ ). Every finitely generated subgroup of $\Gamma $ of infinite index is free (e.g., [Reference ScottSco78]), and a subgroup of finite index of $\Gamma _{g}$ is isomorphic to $\Gamma _{h}$ for some $h\ge g$ and so cannot be generated by less then $2g$ elements. We conclude that the subgroup $\left \langle \delta ,\gamma _{0}\right \rangle \le \Gamma $ is free. Because there is a relation $\gamma _{0}^{~q}=\delta ^{k}$ for some $k\in \mathbb {N}$ , it must be a cyclic subgroup. By definition, $\gamma _{0}$ is not a proper power, and so $\delta $ must be a positive power of $\gamma _{0}$ , and hence $\delta =\gamma _{0}^{~m}$ for some $m\mid q$ .

3 Background: representation theory of the symmetric group

In this section, we give background on the complex representation theory of $S_{n}$ that will be used in the sequel. We follow the Vershik–Okounkov approach to the representation theory of $S_{n}$ developed in [Reference Vershik and OkounkovVO96].

3.1 Young diagrams

A partition is a sequence $\lambda =(\lambda _{1},\lambda _{2},\ldots ,\lambda _{\ell })$ with each $\lambda _{i}\in \mathbf {N}$ and $\lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{\ell }$ . If $\sum \lambda _{i}=n$ , we write this as $\lambda \vdash n$ . Such partitions are in one-to-one correspondence with Young diagrams (YD). A YD consists of a collection of left-aligned rows of identical square boxes, where the number of boxes in each row is nonincreasing from top to bottom. Given a partition $\lambda =(\lambda _{1},\lambda _{2},\ldots ,\lambda _{\ell })$ , the corresponding YD has $\ell $ rows and $\lambda _{i}$ boxes in the i th row, where i increases from top to bottom. We think of partitions as YDs, and vice versa, freely throughout the sequel. If $\lambda $ and $\mu $ are two YDs, we say $\mu \subset \lambda $ if every box of $\mu $ is a box of $\lambda $ . We say $\mu \subset _{k}\lambda $ if $\mu \subset \lambda $ and $\mu $ and $\lambda $ differ by k boxes.

A skew YD (SYD) is formally a pair of YDs $\mu $ and $\lambda $ with $\mu \subset \lambda $ and is denoted by $\lambda /\mu $ . We also think of $\lambda /\mu $ as a diagram consisting of the boxes of $\lambda $ that are not in $\mu $ . We can think of a YD $\lambda $ also as a skew diagram $\lambda =\lambda /\emptyset $ , where $\emptyset $ is the empty diagram with no boxes. Therefore, statements that we make about SYDs apply in this way also to YDs.

The size $|\lambda /\mu |$ of an SYD $\lambda /\mu $ is the number of boxes that it contains, or $\sum \lambda _{i}-\sum \mu _{i}$ . If $\square $ is a particular box appearing in an SYD, we let $i(\square )$ be the row number (starting at 1, counting from top to bottom) of the box and $j(\square )$ the column number (starting at 1, counting from left to right) of the box. The content of a box $\square $ in an SYD is

$$\begin{align*}c(\square)\stackrel{\mathrm{def}}{=} j(\square)-i(\square). \end{align*}$$

If $\square _{1}$ and $\square _{2}$ are two boxes in an SYD, we let

$$\begin{align*}\mathrm{ax}(\square_{1},\square_{2})\stackrel{\mathrm{def}}{=} c(\square_{1})-c(\square_{2}); \end{align*}$$

this is called the axial distance between $\square _{1}$ and $\square _{2}$ .

If $\lambda $ is a YD, we write $\check {\lambda }$ for the YD obtained from $\lambda $ by swapping rows and columns, namely, by transposing. This $\check {\lambda }$ is called the conjugate of $\lambda $ .

3.2 Young tableaux

Let $\lambda /\mu $ be an SYD with $\lambda \vdash n$ and $\mu \vdash m$ . A standard Young tableau of shape $\lambda /\mu $ is a filling of the boxes of $\lambda /\mu $ with the numbers $m+1,\ldots ,n$ such that

  • each number appears in exactly one box of $\lambda /\mu $ , and

  • the numbers in the boxes are strictly increasing from left to right and from top to bottom.

In the sequel, we will refer to standard Young tableaux simply as tableaux. For $\lambda /\mu $ an SYD, we write $\mathrm {Tab}(\lambda /\mu )$ for the collection of tableaux of shape $\lambda /\mu $ . If $\lambda \vdash n$ , $T\in \mathrm {Tab}(\lambda )$ and $m\in \left [n\right ]$ , we write $\mu _{m}\left (T\right )$ for the YD obtained by deleting the boxes containing $m+1,\ldots ,n$ from T, so $\mu _{m}\left (T\right )\vdash m$ . We also write $T\lvert _{\le m}\in \mathrm {Tab}\left (\mu _{m}\left (T\right )\right )$ for the tableau formed by the numbers-in-boxes of T that are $\leq m$ , and $T\lvert _{>m}$ for the tableau formed by the numbers-in-boxes of T that are $>m$ . In general, the shape of $T\lvert _{>m}$ will be an SYD. If T is a tableau of shape $\lambda /\mu $ , where $\lambda \vdash n$ and $\mu \vdash m$ , and $m<i\leq n$ , we write ${\fbox {\(i\)}}_{T}$ for the box containing i in T.

If $\lambda \vdash n$ and $\mu \subset \lambda $ , then we have a concatenation between $\mathrm {Tab}(\mu )$ and $\mathrm {Tab}(\lambda /\mu )$ : If $T\in \mathrm {Tab}(\mu )$ and $R\in \mathrm {Tab}(\lambda /\mu )$ , let $T\sqcup R$ be the tableau obtained by adjoining R to T.

3.3 Representations of symmetric groups

The irreducible unitary representations of $S_{n}$ are parameterized, up to unitary equivalence, by YDs of size n. This correspondence between YDs and representations is denoted by

$$\begin{align*}\lambda\mapsto V^{\lambda}. \end{align*}$$

Each $V^{\lambda }$ is a finite-dimensional complex vector space with a unitary action of $S_{n}$ and is also a module for the group algebra $\mathbf {C}[S_{n}]$ . Let $d_{\lambda }\stackrel {\mathrm {def}}{=}\dim V^{\lambda }$ . It is known that $d_{\lambda }=\left |\mathrm {Tab}\left (\lambda \right )\right |$ .

We now follow Vershik–Okounkov [Reference Vershik and OkounkovVO96]. The natural ordering of $[n]$ induces a filtration

$$\begin{align*}S_{1}\subset S_{2}\subset\cdots\subset S_{n-1}\subset S_{n} \end{align*}$$

of $S_{n}$ , where $S_{m}$ is the subgroup of $S_{n}$ fixing each of the numbers in $\left [m+1,n\right ]$ . If W is any unitary representation of $S_{n}$ , for $m\in [n]$ and $\mu $ a YD of size m, we write $W_{\mu }$ for the linear span in W of all elements in the image of $\mathrm {Hom}_{S_{m}}(V^{\mu },W)$ . In other words, $W_{\mu }$ is the span of copies of $V^{\mu }$ in the restriction of W to $S_{m}$ . This $W_{\mu }$ is called the $\mu $ -isotypic subspace of W.

Vershik and Okounkov describe a specific orthonormal basis of $V^{\lambda }$ , called a Gelfand–Tsetlin basis, that will be useful to us here. The basis is indexed by $T\in \mathrm {Tab}(\lambda )$ ; each such T gives a basis vector $v_{T}$ . The vectors $v_{T}$ can be characterized up to multiplication by complex scalars of modulus $1$ in the following way. The intersection of subspaces

$$\begin{align*}\left(V^{\lambda}\right)_{\mu_{1}(T)}\cap\left(V^{\lambda}\right)_{\mu_{2}(T)}\cap\cdots\cap\left(V^{\lambda}\right)_{\mu_{n-1}(T)} \end{align*}$$

is one-dimensional and contains the unit vector $v_{T}$ [Reference Vershik and OkounkovVO96, $\S$ 1]. One important corollary of this is that if $\mu \vdash m\in [n]$ , then $\left (V^{\lambda }\right )_{\mu }\neq \{0\}$ if and only if $\mu \subset \lambda $ . Also, note that if $\mu _{1},\mu _{2}\subset \lambda , \mu _{1},\mu _{2}\vdash m\in [n]$ , and $\mu _{1}\neq \mu _{2}$ , then $\left (V^{\lambda }\right )_{\mu _{1}}$ is orthogonal to $\left (V^{\lambda }\right )_{\mu _{2}}$ .

More generally, if $\lambda /\mu $ is an SYD with $\lambda \vdash n$ and $\mu \vdash m$ , then there is a skew module $V^{\lambda /\mu }$ that is a unitary representation of $S^{\prime }_{n-m}$ where we write $S^{\prime }_{n-m}$ for the copy of $S_{n-m}$ in $S_{n}$ that fixes the elements $[m]$ . Formally,

$$\begin{align*}V^{\lambda/\mu}\stackrel{\mathrm{def}}{=}\mathrm{Hom}_{S_{m}}\left(V^{\mu},V^{\lambda}\right), \end{align*}$$

where the action of $S^{\prime }_{n-m}$ is by left multiplication: for $\varphi \in \mathrm {Hom}_{S_{m}}\left (V^{\mu },V^{\lambda }\right )$ , $\tau \in S^{\prime }_{n-m}$ and $v\in V^{\mu }$ , $\left (\tau .\varphi \right )\left (v\right )\stackrel {\mathrm {def}}{=}\tau .\left (\varphi \left (v\right )\right )$ . This action preserves $V^{\lambda /\mu }$ as $S^{\prime }_{n-m}$ is in the centralizer of $S_{m}$ in $\mathbf {C}[S_{n}]$ . We write $d_{\lambda /\mu }$ for the dimension of $V^{\lambda /\mu }$ . Since $d_{\lambda /\mu }$ is the multiplicity of $V^{\mu }$ in the restriction of $V^{\lambda }$ to $S_{m}$ , by Frobenius reciprocity, it is also the multiplicity of $V^{\lambda }$ in the induced representation $\mathrm {Ind}_{S_{m}}^{S_{n}}V^{\mu }$ . By calculating the dimension of $\mathrm {Ind}_{S_{m}}^{S_{n}}V^{\mu }$ in two ways, we obtain the following result that will be useful later.

Lemma 3.1. Let $n\in \mathbf {N}$ , $m\in \left [n\right ]$ and $\mu \vdash m$ . Then,

$$\begin{align*}\sum_{\lambda\vdash n\colon\mu\subset\lambda}d_{\lambda/\mu}d_{\lambda}=\frac{n!}{m!}d_{\mu}. \end{align*}$$

The module $V^{\lambda /\mu }$ has an orthonormal basis $w_{T}$ indexed by $T\in \mathrm {Tab}(\lambda /\mu )$ [Reference Vershik and OkounkovVO96, §7]. One also has the following property that we will use later [Reference Ceccherini-Silberstein, Scarabotti and TolliCSST10, Eq. (3.65)].

Lemma 3.2. Let $n\in \mathbf {N}$ , $m\in \left [n\right ]$ , $\lambda \vdash n$ and $\mu \vdash m$ and assume that $\mu \subset \lambda $ . Then the map

$$\begin{align*}v_{T}\otimes w_{R}\mapsto v_{T\sqcup R},\quad T\in\mathrm{Tab}(\mu),\,R\in\mathrm{Tab}(\lambda/\mu) \end{align*}$$

linearly extends to an isomorphism of unitary $\left (S_{m}\times S^{\prime }_{n-m}\right )$ -representations $V^{\mu }\otimes V^{\lambda /\mu }\cong \left (V^{\lambda }\right )_{\mu }$ .

There is also an explicit formula for the action of $S^{\prime }_{n-m}$ on $V^{\lambda /\mu }$ . A full exposition of this formula can be found in [Reference Vershik and OkounkovVO96, §6]. Recall that $S^{\prime }_{n-m}$ is generated by the Coxeter generators

$$\begin{align*}s_{i}\stackrel{\mathrm{def}}{=}(i~\ i+1) \end{align*}$$

for $m<i<n$ , where $\left (i~~i+1\right )$ is our notation for a transposition switching i and $i+1$ . Therefore, it is sufficient to describe how the $s_{i}$ act on $V^{\lambda /\mu }$ . Say that T is admissible for $s_{i}$ if the boxes containing i and $i+1$ in T are neither in the same row nor the same column.

For $T\in \mathrm {Tab}(\lambda /\mu )$ , let

$$ \begin{align*} s_{i}T & =\begin{cases} T & \text{if \(T\) is not admissible for \(s_{i}\)}\\ T' & \text{if \(T\) is admissible for \(s_{i}\)}, \end{cases} \end{align*} $$

where $T'$ is the tableaux obtained from T by swapping i and $i+1$ . The admissibility condition ensures $T'$ is a valid standard Young tableau. Then one has Young’s orthogonal form

(3.1) $$ \begin{align} s_{i}w_{T}=\frac{1}{\mathrm{ax}({\fbox{\(i+1\)}}_{T},{\fbox{\(i\)}}_{T})}w_{T}+\sqrt{1-\frac{1}{\mathrm{ax}({\fbox{\(i+1\)}}_{T},{\fbox{\(i\)}}_{T})^{2}}}w_{s_{i}T}. \end{align} $$

Note that as a special case of this formula, if T is not admissible for $s_{i}$ , then $\mathrm {ax}({\fbox {\(i+1\)}}_{T},{\fbox {\(i\)}}_{T})=\pm 1$ and

(3.2) $$ \begin{align} s_{i}w_{T}=\frac{1}{\mathrm{ax}({\fbox{\(i+1\)}}_{T},{\fbox{\(i\)}}_{T})}w_{T}=\begin{cases} w_{T} & \mathrm{if}~i~\mathrm{and}~i+1~\mathrm{are~in~the~same~row},\\ -w_{T} & \mathrm{if}~i~\mathrm{and}~i+1~\mathrm{are~in~the~same~column.} \end{cases} \end{align} $$

Remark 3.3. For completeness of some of our statements, we need to define the notions above also for $S_{0}$ , the symmetric group of the empty set. This is the trivial group. Whenever $\mu =\lambda $ , we have $\mathrm {Tab}\left (\lambda /\mu \right )=\left \{ \emptyset \right \} $ , and the representation $V^{\lambda /\mu }$ is one-dimensional with basis $w_{T}$ , for T the empty tableau.

4 Preliminary representation theoretic results

In this section, we give some preliminary results on representation theory that will be used in the rest of the paper. Although some results here seem to be novel (in particular Proposition 4.4), this section plays only a supporting role in the paper.

4.1 Commutants

Recall that if V is a finite-dimensional vector space and $\mathcal {A}$ is a subalgebra of $\mathrm {End}(V)$ , then the commutant of $\mathcal {A}$ in $\mathrm {End}(V)$ is the algebra of elements $b\in \mathrm {End}(V)$ such that

$$\begin{align*}ba=ab \end{align*}$$

for all $a\in \mathcal {A}$ . For $m\in [n]$ and $\lambda \vdash n$ , let $Z(\lambda ,m,n)$ denote the commutant of the image of $\mathbf {C}[S_{m}]$ in $\mathrm {End}(V^{\lambda }).$ We identify

(4.1) $$ \begin{align} \mathrm{End}(V^{\lambda}) & \cong V^{\lambda}\otimes\check{V^{\lambda}} \end{align} $$

and give $\mathrm {End}(V^{\lambda })$ the Hermitian inner product induced from $V^{\lambda }.$

Lemma 4.1. Let