Consider the following classes of pairs consisting of a group and a finite collection of subgroups:

• • $ \mathcal{C}= \left \{ (G,\mathcal{H}) \mid \text{$\mathcal{H}$ is hyperbolically embedded in $G$} \right \}$

• • $ \mathcal{D}= \left \{ (G,\mathcal{H}) \mid \text{the relative Dehn function of $(G,\mathcal{H})$ is well-defined} \right \} .$

Let $G$ be a group that splits as a finite graph of groups such that each vertex group $G_v$ is assigned a finite collection of subgroups $\mathcal{H}_v$, and each edge group $G_e$ is conjugate to a subgroup of some $H\in \mathcal{H}_v$ if $e$ is adjacent to $v$. Then there is a finite collection of subgroups $\mathcal{H}$ of $G$ such that

1. 1. If each $(G_v, \mathcal{H}_v)$ is in $\mathcal C$, then $(G,\mathcal{H})$ is in $\mathcal C$.

2. 2. If each $(G_v, \mathcal{H}_v)$ is in $\mathcal D$, then $(G,\mathcal{H})$ is in $\mathcal D$.

3. 3. For any vertex $v$ and for any $g\in G_v$, the element $g$ is conjugate to an element in some $Q\in \mathcal{H}_v$ if and only if $g$ is conjugate to an element in some $H\in \mathcal{H}$.

That edge groups are not assumed to be finitely generated and that they do not necessarily belong to a peripheral collection of subgroups of an adjacent vertex are the main differences between this work and previous results in the literature. The method of proof provides lower and upper bounds of the relative Dehn functions in terms of the relative Dehn functions of the vertex groups. These bounds generalize and improve analogous results in the literature.

We derive an upper bound on the density of Jones polynomials of knots modulo a prime number $p$, within a sufficiently large degree range: $4/p^7$. As an application, we classify knot Jones polynomials modulo two of span up to eight.

Let $p \;:\; Y \to X$ be a finite, regular cover of finite graphs with associated deck group $G$, and consider the first homology $H_1(Y;\;{\mathbb{C}})$ of the cover as a $G$-representation. The main contribution of this article is to broaden the correspondence and dictionary between the representation theory of the deck group $G$ on the one hand and topological properties of homology classes in $H_1(Y;\;{\mathbb{C}})$ on the other hand. We do so by studying certain subrepresentations in the $G$-representation $H_1(Y;\;{\mathbb{C}})$.

The homology class of a lift of a primitive element in $\pi _1(X)$ spans an induced subrepresentation in $H_1(Y;\;{\mathbb{C}})$, and we show that this property is never sufficient to characterize such homology classes if $G$ is Abelian. We study $H_1^{\textrm{comm}}(Y;\;{\mathbb{C}}) \leq H_1(Y;\;{\mathbb{C}})$—the subrepresentation spanned by homology classes of lifts of commutators of primitive elements in $\pi _1(X)$. Concretely, we prove that the span of such a homology class is isomorphic to the quotient of two induced representations. Furthermore, we construct examples of finite covers with $H_1^{\textrm{comm}}(Y;\;{\mathbb{C}}) \neq \ker\!(p_*)$.

We formalize the concepts of holomorphic affine and projective structures along the leaves of holomorphic foliations by curves on complex manifolds. We show that many foliations admit such structures, we provide local normal forms for them at singular points of the foliation, and we prove some index formulae in the case where the ambient manifold is compact. As a consequence of these, we establish that a regular foliation of general type on a compact algebraic manifold of even dimension does not admit a foliated projective structure. Finally, we classify foliated affine and projective structures along regular foliations on compact complex surfaces.

In this paper, we prove using elementary techniques that any group of diffeomorphisms acting on the 2-sphere and properly extending the conformal group of Möbius transformations must be at least 4-transitive or, more precisely, arc 4-transitive. As an important consequence, we derive that any such group must always contain an element of positive topological entropy. We also provide a self-contained characterization, in terms of transitivity, of the Möbius transformations within the full group of sphere diffeomorphisms.

We define smooth notions of concordance and sliceness for spatial graphs. We prove that sliceness of a spatial graph is equivalent to a condition on a set of linking numbers together with sliceness of a link associated with the graph. This generalizes the result of Taniyama for $\theta $ -curves.

We introduce “braided” versions of self-similar groups and Röver–Nekrashevych groups, and study their finiteness properties. This generalizes work of Aroca and Cumplido, and the first author and Wu, who considered the case when the self-similar groups are what we call “self-identical.” In particular, we use a braided version of the Grigorchuk group to construct a new group called the “braided Röver group,” which we prove is of type $\operatorname {\mathrm {F}}_\infty $ . Our techniques involve using so-called d-ary cloning systems to construct the groups, and analyzing certain complexes of embedded disks in a surface to understand their finiteness properties.

The equivariant Heegaard genus of a 3-manifold W with the action of a finite group G of diffeomorphisms is the smallest genus of an equivariant Heegaard splitting for W. Although a Heegaard splitting of a reducible manifold is reducible and although if W is reducible, there is an equivariant essential sphere, we show that equivariant Heegaard genus may be super-additive, additive, or sub-additive under equivariant connected sum. Using a thin position theory for 3-dimensional orbifolds, we establish sharp bounds on the equivariant Heegaard genus of reducible manifolds, similar to those known for tunnel number. Along the way, we make use of a new invariant for W which is much better behaved under equivariant sums.

We initiate the study of outer automorphism groups of special groups $G$, in the Haglund–Wise sense. We show that $\operatorname {Out}(G)$ is infinite if and only if $G$ splits over a co-abelian subgroup of a centraliser and there exists an infinite-order ‘generalised Dehn twist’. Similarly, the coarse-median preserving subgroup $\operatorname {Out}_{\rm cmp}(G)$ is infinite if and only if $G$ splits over an actual centraliser and there exists an infinite-order coarse-median-preserving generalised Dehn twist. The proof is based on constructing and analysing non-small, stable $G$-actions on $\mathbb {R}$-trees whose arc-stabilisers are centralisers or closely related subgroups. Interestingly, tripod-stabilisers can be arbitrary centralisers, and thus are large subgroups of $G$. As a result of independent interest, we determine when generalised Dehn twists associated to splittings of $G$ preserve the coarse median structure.

Residual torsion-free nilpotence has proved to be an important property for knot groups with applications to bi-orderability and ribbon concordance. Mayland proposed a strategy to show that a two-bridge knot group has a commutator subgroup which is a union of an ascending chain of para-free groups. This paper proves Mayland’s assertion and expands the result to the subgroups of two-bridge link groups that correspond to the kernels of maps to $\mathbb{Z}$ . We call these kernels the Alexander subgroups of the links. As a result, we show the bi-orderability of a large family of two-bridge link groups. This proof makes use of a modified version of a graph-theoretic construction of Hirasawa and Murasugi in order to understand the structure of the Alexander subgroup for a two-bridge link group.

The space of Fredholm operators of fixed index is stratified by submanifolds according to the dimension of the kernel. Geometric considerations often lead to questions about the intersections of concrete families of elliptic operators with these submanifolds: Are the intersections nonempty? Are they smooth? What are their codimensions? The purpose of this article is to develop tools to address these questions in equivariant situations. An important motivation for this work are transversality questions for multiple covers of J-holomorphic maps. As an application, we use our framework to give a concise exposition of Wendl’s proof of the superrigidity conjecture.

Given a finitely generated free group $ {\mathbb {F} }$ of $\mathsf {rank}( {\mathbb {F} } )\geq 3$, we show that the mapping torus of $\phi$ is (strongly) relatively hyperbolic if $\phi$ is exponentially growing. As a corollary of our work, we give a new proof of Brinkmann's theorem which proves that the mapping torus of an atoroidal outer automorphism is hyperbolic. We also give a new proof of the Bridson–Groves theorem that the mapping torus of a free group automorphism satisfies the quadratic isoperimetric inequality. Our work also solves a problem posed by Minasyan and Osin: the mapping torus of an outer automorphism is not virtually acylindrically hyperbolic if and only if $\phi$ has finite order.

We show that any set of distinct homotopy classes of simple closed curves on the torus that pairwise intersect at most k times has size $k+O(\sqrt k \log k)$ . Prior to this work, a lemma of Agol, together with the state of the art bounds for the size of prime gaps, implied the error term $O(k^{21/40})$ , and in fact the assumption of the Riemann hypothesis improved this error term to the one we obtain $O(\sqrt k\log k)$ . By contrast, our methods are elementary, combinatorial, and geometric.

The main result includes as special cases on the one hand, the Gerstenhaber–Rothaus theorem (1962) and its generalisation due to Nitsche and Thom (2022) and, on the other hand, the Brodskii–Howie–Short theorem (1980–1984) generalising Magnus’s Freiheitssatz (1930).

This paper provides a convenient and practical method to compute the homology and intersection pairing of a branched double cover of the 4-ball.

To projections of links in the 3-ball, and to projections of surfaces in the 4-ball into the boundary sphere, we associate a sequence of homology groups, called the disoriented homology. We show that the disoriented homology is isomorphic to the homology of the double branched cover of the link or surface. We define a pairing on the first disoriented homology group of a surface and show that this is equal to the intersection pairing of the branched cover. These results generalize work of Gordon and Litherland, for embedded surfaces in the 3-sphere, to arbitrary surfaces in the 4-ball. We also give a generalization of the signature formula of Gordon–Litherland to the general setting.

Our results are underpinned by a theorem describing a handle decomposition of the branched double cover of a codimension-2 submanifold in the n-ball, which generalizes previous results of Akbulut–Kirby and others.

For a pseudo-Anosov flow $\varphi $ without perfect fits on a closed $3$-manifold, Agol–Guéritaud produce a veering triangulation $\tau $ on the manifold M obtained by deleting the singular orbits of $\varphi $. We show that $\tau $ can be realized in M so that its 2-skeleton is positively transverse to $\varphi $, and that the combinatorially defined flow graph $\Phi $ embedded in M uniformly codes the orbits of $\varphi $ in a precise sense. Together with these facts, we use a modified version of the veering polynomial, previously introduced by the authors, to compute the growth rates of the closed orbits of $\varphi $ after cutting M along certain transverse surfaces, thereby generalizing the work of McMullen in the fibered setting. These results are new even in the case where the transverse surface represents a class in the boundary of a fibered cone of M. Our work can be used to study the flow $\varphi $ on the original closed manifold. Applications include counting growth rates of closed orbits after cutting along closed transverse surfaces, defining a continuous, convex entropy function on the ‘positive’ cone in $H^1$ of the cut-open manifold, and answering a question of Leininger about the closure of the set of all stretch factors arising as monodromies within a single fibered cone of a $3$-manifold. This last application connects to the study of endperiodic automorphisms of infinite-type surfaces and the growth rates of their periodic points.

We establish central limit theorems for an action of a group $G$ on a hyperbolic space $X$ with respect to the counting measure on a Cayley graph of $G$. Our techniques allow us to remove the usual assumptions of properness and smoothness of the space, or cocompactness of the action. We provide several applications which require our general framework, including to lengths of geodesics in geometrically finite manifolds and to intersection numbers with submanifolds.

In this paper, we focus on dynamical properties of (real) convex projective surfaces. Our main theorem provides an asymptotic formula for the number of free homotopy classes with roughly the same renormalized Hilbert length for two distinct convex real projective structures. The correlation number in this asymptotic formula is characterized in terms of their Manhattan curve. We show that the correlation number is not uniformly bounded away from zero on the space of pairs of hyperbolic surfaces, answering a question of Schwartz and Sharp. In contrast, we provide examples of diverging sequences, defined via cubic rays, along which the correlation number stays larger than a uniform strictly positive constant. In the last section, we extend the correlation theorem to Hitchin representations.

Recently, Gross, Mansour and Tucker introduced the partial-dual polynomial of a ribbon graph as a generating function that enumerates all partial duals of the ribbon graph by Euler genus. It is analogous to the extensively studied polynomial in topological graph theory that enumerates by Euler genus all embeddings of a given graph. To investigate the partial-dual polynomial, one only needs to focus on bouquets: that is, ribbon graphs with exactly one vertex. In this paper, we shall further show that the partial-dual polynomial of a bouquet essentially depends on the signed intersection graph of the bouquet rather than on the bouquet itself. That is to say, two bouquets with the same signed intersection graph have the same partial-dual polynomial. We then give a characterisation of when a bouquet has a planar partial dual in terms of its signed intersection graph. Finally, we consider a conjecture posed by Gross, Mansour and Tucker that there is no orientable ribbon graph whose partial-dual polynomial has only one nonconstant term; this conjecture is false, and we give a characterisation of when all partial duals of a bouquet have the same Euler genus.

We consider free symmetries on cobordisms between knots, which is equivalent to cobordisms between knots in lens spaces. We classify which freely periodic knots bound equivariant surfaces in the 4-ball in terms of corresponding homology classes in lens spaces. We give a numerical condition determining the free periods for which torus knots bound equivariant surfaces in the 4-ball.