2.1 Ramanujan graphs and complexes

We shortly note that the results of this paper, and in particular Theorem 1.5 (or the stronger Theorem 7.4) apply to Ramanujan complexes, by which we mean here the situation when G is p-adic, $\Gamma _1$ is cocompact and no nontempered and nontrivial spherical representation appears in the decomposition of $L^{2}(\Gamma _{N}\backslash G)$ . In this case, the sequence $(\Gamma _N) $ obviously satisfies the spherical density hypothesis. The Ramanujan complexes themselves are the quotients of the Bruhat–Tits buildings of G by $\Gamma _{N}$ . Such complexes were constructed (with the same definition) by Lubotzky, Samuels and Vishne (see [Reference Lubotzky, Samuels and Vishne51, Reference Lubotzky, Samuels and Vishne50]). Similar results to Theorem 7.4 appear in [Reference Kamber38, Reference Lubetzky, Lubotzky and Parzanchevski47, Reference Chapman and Parzanchevski13].

Note that the definition we use is not the same as in the more modern approach to Ramanujan complexes, where one considers not only spherical representation but also representations with a nontrivial vector fixed by the Iwahori subgroup (see [Reference Lubotzky49, Subsection 2.3] and the references within).

The standard way of proving that a complex is a Ramanujan complex is to use results from the Langlands program (e.g., [Reference Lafforgue45]) and to eventually apply Deligne’s proof of the Weil conjectures ([Reference Deligne17]). This approach has obvious limitations, and in particular, it seems that one must assume that G is p-adic for it to succeed. We refer to [Reference Evra and Parzanchevski21] for some recent work on this subject and to an ongoing and yet unpublished work of Shai Evra.

We will avoid diving deeper into this subject, as it is based on spectral input, unlike our approach which is geometric.

2.2 Adding a parameter to the properties

It is useful to add a parameter $0<\alpha \le 1$ to the properties, with $\alpha =1$ being equivalent to the property without the parameter.

Definition. We say that the sequence $(\Gamma _N) $ satisfies the weak injective radius property with parameter $\alpha $ , if for every $0\le d_{0}\le 2\alpha \log _{q}([\Gamma _1:\Gamma _N])$ , $\epsilon>0$ ,

$$\begin{align*}\frac{1}{[\Gamma_1:\Gamma_N]}\sum_{y\in\Gamma_{N}\backslash\Gamma_{1}}\boldsymbol{N}(\Gamma_N,d_{0},y)\ll_{\epsilon}[\Gamma_1:\Gamma_N]^{\epsilon}q^{d_{0}/2}. \end{align*}$$

Definition. We say that the sequence $(\Gamma _N) $ satisfies the general density hypothesis with parameter $\alpha $ if for every precompact subset $A\subset \Pi (G)$ and every $\epsilon>0$ , N and $p\ge 2$ ,

$$\begin{align*}M(A,\Gamma,p)\ll_{\epsilon,A}[\Gamma_1:\Gamma_N]^{1-\alpha(1-2/p)+\epsilon}. \end{align*}$$

One may similarly define the spherical density hypothesis with parameter $\alpha $ and the pointwise density hypothesis with parameter $\alpha $ , by changing the exponent $2/p$ to $1-\alpha (1-2/p).$

Figure 1.1 remains true if we replace the properties with their parameterized version, for the same parameter $0<\alpha \le 1$ , with the exception that the derivation of the optimal lifting property from the spherical density hypothesis requires $\alpha =1$ .

In the work itself, we work with the parameterized version of the properties.

2.3 Congruence subgroups of arithmetic groups

Let $\Gamma _1$ be an arithmetic lattice in a semisimple noncompact Lie group G. Following [Reference Sarnak and Xue58], by arithmetic we mean that G is defined over $\mathbb {Q}$ , $\Gamma _1\subset G(\mathbb {Q})$ , there is a $\mathbb {Q}$ -embedding $\rho \colon G\to \operatorname {\mathrm {GL}}_{n}$ and $\Gamma _1$ is commensurable with $\rho ^{-1}(\operatorname {\mathrm {GL}}_{n}(\mathbb {Z}))$ .

In this case, we may define a sequence of principal congruence subgroups $(\Gamma _N) $ of $\Gamma _1$ by letting

$$\begin{align*}\Gamma_{N}=\Gamma_1(N)=\Gamma_1\cap\rho^{-1}(\left\{ A\in \operatorname{\mathrm{GL}}_{n}(\mathbb{Z}):A\equiv I\quad \mod N\right\}). \end{align*}$$

It is a well-known fact that such subgroups have an injective radius which is logarithmic in the index (e.g., [Reference Sarnak and Xue58, Lemma 1] or [Reference Guth and Lubotzky30, Proposition 16]).

Let us shortly give the argument – we assume by moving to a finite index in $\Gamma _1$ that $\Gamma _1\subset \rho ^{-1}(\operatorname {\mathrm {GL}}_{n}(\mathbb {Z}))$ . In this case, $\Gamma _{N}$ is normal in $\Gamma _1$ and it is left to verify that for every $d_{0}\le 2\alpha \ln ([\Gamma _1:\Gamma _N])$ and $\epsilon>0$ ,

$$\begin{align*}\#\left\{ \gamma\in\Gamma_{N}:l(\gamma)\le d_{0}\right\} \ll_{\epsilon}e^{d_{0}(1/2+\epsilon)}. \end{align*}$$

We note that there exists a constant C such that for $g\in G$ outside a compact set the length $l(g)$ we defined satisfies

$$\begin{align*}C^{-1}\ln({\left\Vert {\rho(g)}\right\Vert })\le l(g), \end{align*}$$

where $\|\cdot \|\colon \operatorname {\mathrm {GL}}_{n}(\mathbb {R})\to \mathbb {R}_{\ge 0}$ is the maximal absolute value of an entry of the matrix. Since each element of $\rho (\Gamma _N)$ which is not the identity has an entry of size N, We deduce that for every N large enough,

$$\begin{align*}\left\{ \gamma\in\Gamma_{N}:l(\gamma)<C\ln(N)\right\} =\{ I\}. \end{align*}$$

On the other hand, there is an injective map of $\Gamma _1/\Gamma _N$ into $\operatorname {\mathrm {GL}}_n(\mathbb {Z}/N\mathbb {Z})$ , so

$$\begin{align*}[\Gamma_1:\Gamma_N]\ll N^{n^{2}} \end{align*}$$

(and the exponent $n^2$ can usually be improved).

Combining the different estimates, for $d_{0}<2\frac {C}{2n^{2}}\ln ([\Gamma _{1}:\Gamma _N])\le 2\frac {C}{2n^{2}}N^{n^{2}})=C\ln (N)$ , it holds that for N large enough,

$$\begin{align*}\#\left\{ \gamma\in\Gamma_{N}:l(\gamma)\le d_{0}\right\} =1. \end{align*}$$

Therefore, the weak injective radius property is satisfied with parameter $\alpha =\frac {C}{2n^{2}}$ .

As a direct application of this fact, we have:

As a matter of fact, this corollary can be easily extended to principal congruence subgroups of S-arithmetic groups, once those are properly defined.

For arbitrary congruence subgroups, let us recall some of the results of [Reference Finis and Lapid22] (the same result can be deduced from [Reference Abert, Bergeron, Biringer, Gelander, Nikolov, Raimbault and Samet1, Theorem 1.11 and Theorem 5.6]).

First, for $\gamma \in \Gamma _1$ , we denote

$$\begin{align*}c_{\Gamma_{N}}(\gamma)=\left|\left\{ y\in\Gamma_{N}\backslash\Gamma_{1}:y\gamma y^{-1}\in\Gamma_N\right\} \right|, \end{align*}$$

which is the number of fixed points of the right action of $\gamma $ on $\Gamma _{N}\backslash \Gamma _1$ . Then the weak injective radius property with parameter $\alpha $ is equivalent to the fact that for every $0\le d_{0}\le 2\alpha \ln ([\Gamma _1:\Gamma _N])$ , $\epsilon>0$ ,

$$\begin{align*}\frac{1}{[\Gamma_1:\Gamma_N]}\sum_{l(\gamma)\le d_{0}}c_{\Gamma_{N}}(\gamma)\ll_{\epsilon}[\Gamma_1:\Gamma_N]^{\epsilon}e^{d_{0}(1/2+\epsilon)}. \end{align*}$$

A congruence subgroup is a subgroup of one of the groups $\Gamma _1(N)$ as above.

Theorem 2.3 (Following [Reference Finis and Lapid22, Corollary 5.9]).

Let G be Archimedean, semisimple, almost-simple and simply connected; let $\Gamma _1$ be an arithmetic lattice and let $(\Gamma _N) $ be a sequence of congruence subgroups of $\Gamma _1$ . Then there exist constants $\mu>0,c>0$ depending only on $\Gamma _1$ such that for every $d_{0}>0$ , $N\ge 1$ it holds that

(2.1) $$ \begin{align} \frac{1}{[\Gamma_1:\Gamma_N]}\sum_{\gamma\in\Gamma_1,l(\gamma)\le d_{0}}c_{\Gamma_{N}}(\gamma)\ll1+e^{cd_{0}}[\Gamma_1:\Gamma_N]^{-\mu}. \end{align} $$

In particular, the injective radius property holds with parameter $\alpha =\frac {\mu }{c}$ .

Proof. By [Reference Finis and Lapid22, Corollary 5.9], there exists a constant $c'$ such that for every $\gamma \in \Gamma _1$ which does not belong to a proper normal subgroup of G it holds that

$$\begin{align*}\frac{1}{[\Gamma_1:\Gamma_N]}c_{\Gamma_{N}}(\gamma)\ll e^{c'l(\gamma)}[\Gamma_1:\Gamma_N]^{-\mu}. \end{align*}$$

We remark that the dependence on $\gamma $ in [Reference Finis and Lapid22] is different, but it is obvious that $e^{cl(\gamma )}$ for c large enough is an upper bound on it. By our assumptions on G, the number of $\gamma \in \Gamma _1$ belonging to a proper normal subgroup is $\ll 1$ .

By summing over all $\gamma \in \Gamma _1$ with $l(\gamma )\le d_{0}$ , we get

$$ \begin{align*} \frac{1}{[\Gamma_1:\Gamma_N]}\sum_{\gamma\in\Gamma_1,l(\gamma)\le d_{0}}c_{\Gamma_{N}}(\gamma) & \ll\frac{[\Gamma_1:\Gamma_N]}{[\Gamma_1:\Gamma_N]}+e^{c'd_{0}}[\Gamma_1:\Gamma_N]^{-\mu}\sum_{\gamma\in\Gamma_1,l(\gamma)\le d_{0}}1\\ & \ll1+e^{(1+c')d_{0}}[\Gamma_1:\Gamma_N]^{-\mu}, \end{align*} $$

which implies equation (2.1) with $c=1+c'$ .

When $d_{0}\le 2\frac {\mu }{c}\ln ([\Gamma _{1}:\Gamma _N])$ or $[\Gamma _1:\Gamma _N]^{-\mu }\le e^{d_{0}c/2}$ , we get

$$\begin{align*}\frac{1}{[\Gamma_1:\Gamma_N]}\sum_{\gamma\in\Gamma_1,l(\gamma)\le d_{0}}c_{\Gamma_{N}}(\gamma)\ll e^{d_{0}/2}, \end{align*}$$

which implies the weak injective radius property with parameter $\alpha =\frac {\mu }{c}$ , as needed.

In the cocompact case, Theorem 1.10 implies the pointwise multiplicity hypothesis with parameter $\alpha =\frac {\mu }{c}$ , namely

It should be compared with [Reference Abert, Bergeron, Biringer, Gelander, Nikolov, Raimbault and Samet1, Theorem 7.15], which states

$$\begin{align*}m(\pi,\Gamma_N)\ll_{\pi}[\Gamma_1:\Gamma_N]^{1-\alpha(\pi)}. \end{align*}$$

Finally, we wish to make the following conjecture:

The conjecture generalizes Conjecture $1$ and Conjecture $2$ from the work of Sarnak and Xue ([Reference Sarnak and Xue58]). Finally, a similar conjecture should also hold in the S-arithmetic setting when G is p-adic.

2.4 The Work of Sarnak and Xue and its implications

Sarnak and Xue proved the weak injective radius property with parameter $\alpha =1$ for principal congruence subgroups of cocompact arithmetic lattices in $\operatorname {\mathrm {SL}}_{2}(\mathbb {R})$ and $\operatorname {\mathrm {SL}}_{2}(\mathbb {C})$ and proved the weak injective radius property with parameter $\alpha =5/6$ for principal congruence subgroups of cocompact arithmetic lattices of $\operatorname {\mathrm {SU}}(2,1)$ .

We refer to [Reference Sarnak and Xue58, Section 3] for their calculations. Their argument actually works also for principal congruence subgroups of lattices in $\operatorname {\mathrm {SL}}_2$ over p-adic fields coming from division algebras. See, for example, [Reference Davidoff, Sarnak and Valette15, Theorem 4.4.4].

As a simple example, let us prove here the weak injective radius property for the principal congruence subgroups of $\operatorname {\mathrm {SL}}_{2}(\mathbb {Z})$ . In this case, we need to show that the number $\boldsymbol {N}(T,\Gamma (N))$ , of solutions to $ad-bc=1$ , with $a\equiv d\equiv 1\ \mod N$ , $b\equiv c\equiv 0\ \mod N$ and $\max \left \{ \left |a\right |,\left |b\right |,\left |c\right |,\left |d\right |\right \} \le T$ , for $T\le N^{3}$ , is bounded by

$$\begin{align*}\boldsymbol{N}(T,\Gamma(N))\ll_{\epsilon}T^{1+\epsilon}. \end{align*}$$

This is done as follows (see also [Reference Gamburd23, Proposition 5.3]). From the congruence condition, it follows that

$$\begin{align*}a+d-2=-(a-1)(d-1)+bc\equiv0\quad \mod N^{2}. \end{align*}$$

One may therefore choose $a+d$ in $(2T/N^{2}+1)$ ways, and choose $(a,d)$ in $(2T/N^{2}+1)(2T/N+1)$ ways.

If $ad\ne 1$ , from $bc=1-ad$ and bounds on the divisor function, there are $T^{\epsilon }$ ways of choosing $bc$ . If $ad=1$ , it is also simple to bound the number of possibilities of $b,c$ by $4(T/N+1)$ . In total, we get for $T\le N^{3}$ ,

$$\begin{align*}\boldsymbol{N}(T,\Gamma(N))\ll_{\epsilon}T^{\epsilon}(T/N^{2}+1)(T/N+1)\ll_{\epsilon}T^{1+\epsilon}. \end{align*}$$

2.5 On Some congruence subgroups of $\operatorname {\mathrm {SL}}_3(\mathbb {Z})$

Consider the following subgroups of $\Gamma _{1}=\operatorname {\mathrm {SL}}_{3}(\mathbb {Z})$ :

$$ \begin{align*} \Gamma_{0}(N)&=\left\{ \left(\begin{array}{ccc} * & * & *\\* * & * & *\\ a & b & * \end{array}\right)\in \operatorname{\mathrm{SL}}_{3}(\mathbb{Z}):a\equiv b\equiv0\quad \mod N\right\} \subset \\ \Gamma_{2}(N)&=\left\{ \left(\begin{array}{ccc} * & * & *\\* c & * & *\\ a & b & * \end{array}\right)\in \operatorname{\mathrm{SL}}_{3}(\mathbb{Z}):a\equiv b\equiv c \equiv 0\quad \mod N\right\} \subset\Gamma_{1}=\operatorname{\mathrm{SL}}_{3}(\mathbb{Z}). \end{align*} $$

In a companion paper by the second named author and Hagai Lavner ([Reference Kamber and Lavner40]), the following is proven:

We refer to [Reference Kamber and Lavner40] for an interpretation of this result in terms of the action of $\operatorname {\mathrm {SL}}_{3}(\mathbb {Z})$ on the projective plane over the field with N elements, or on the corresponding flag space.

The work [Reference Kamber and Lavner40] is strongly influenced by a deep work of Blomer, Buttcane and Maga on $\Gamma _0(N)$ ([Reference Blomer, Buttcane and Maga6]), which is based on the Kuznetsov trace formula:

Theorem 2.6 [Reference Blomer, Buttcane and Maga6, Theorem 4].

For $\pi \in \Pi (\operatorname {\mathrm {SL}}_3(\mathbb {R}))_{\operatorname {\mathrm {sph}}}$ , let $m_{\operatorname {\mathrm {cusp}}}(\pi ,\Gamma _{0}(N))$ be the multiplicity of $\pi $ in the cuspidal part of $L^{2}(\Gamma _{0}(N)\backslash \operatorname {\mathrm {SL}}_3(\mathbb {R}))$ . Then for every compact $A\subset \Pi (\operatorname {\mathrm {SL}}_3(\mathbb {R}))_{\operatorname {\mathrm {sph}}}$ , it holds that for every N prime, $p>2$ , $\epsilon>0$ ,

$$\begin{align*}\sum_{\pi\in A,p(\pi)>0}m_{\operatorname{\mathrm{cusp}}}(\pi,\Gamma_{0}(N))\ll_{A,\epsilon}[\Gamma_1:\Gamma_{0}(N)]^{1-2(1-2/p)+\epsilon}. \end{align*}$$

The result of the theorem is very similar to the spherical density hypothesis with parameter $\alpha =2$ . There are also a number of differences: first, $\operatorname {\mathrm {SL}}_{3}(\mathbb {Z})$ is not cocompact, so our discussion does not apply to it. In particular, we have to deal with the continuous spectrum if we wish to deduce the optimal lifting property. Secondly and less crucially, the dependence on the subset A is not explicit, as needed in the definition of the spherical density hypotheses, so it is hard to use it for geometric applications.

This result was recently generalized by Blomer ([Reference Blomer5]) to general $\operatorname {\mathrm {SL}}_{M}(\mathbb {Z})$ , where the subgroup $\Gamma _{0}(N)$ is similarly defined to be the set of matrices with the entries in the last row, except for the $(M,M)$ entry, equal to $0$ modulo N. We remark that Blomer uses a slightly weaker way to measure temperedness, rather than $p(\pi )$ (for $\operatorname {\mathrm {SL}}_3$ the two ways are equivalent).

In any case, it seems that the methods of [Reference Blomer, Buttcane and Maga6, Reference Blomer5] are not applicable to the subgroups of the form $\Gamma _2(N)$ .

2.6 The weak injective radius of principal congruence subgroups of $\operatorname {\mathrm {SL}}_{n}(\mathbb {Z})$

If the weak injective radius property would hold for the principal congruence subgroups of $\operatorname {\mathrm {SL}}_{n}(\mathbb {Z})$ , then it will imply that

$$\begin{align*}\#\left\{ \gamma\in \operatorname{\mathrm{SL}}_{n}(\mathbb{Z}):\gamma\equiv I\quad \mod N,{\left\Vert {\gamma}\right\Vert }\le T\right\} \ll_{\epsilon} T^{\epsilon}N^{\epsilon}\left(\frac{T^{n^{2}-n}}{N^{n^{2}-1}}+T^{(n^{2}-n)/2}\right), \end{align*}$$

where ${\left \Vert {\cdot }\right \Vert }$ is the maximal absolute value of an entry. One may also try to improve this estimate, in particular, the ‘error’ part $T^{(n^{2}-n)/2}$ .

One of the results of [Reference Katznelson42], which is also cited in [Reference Sarnak and Xue58], states that

$$\begin{align*}\#\left\{ \gamma\in \operatorname{\mathrm{SL}}_{n}(\mathbb{Z}):\gamma\equiv I\quad \mod N,{\left\Vert {\gamma}\right\Vert }\le T\right\} \ll\frac{T^{n^{2}-n}}{N^{n^{2}-1}}\log T+1. \end{align*}$$

As was pointed to us by Sarnak, the proof contains an error. As a matter of fact, this naive estimate is actually false, for simple reasons. For example, in $\operatorname {\mathrm {SL}}_{2}$ ,

$$\begin{align*}\#\left\{ \left(\begin{array}{cc} 1 & *\\ 0 & 1 \end{array}\right)\in \operatorname{\mathrm{SL}}_{2}(\mathbb{Z}):\gamma\equiv I\quad \mod N,{\left\Vert {\gamma}\right\Vert }\le T\right\} \asymp T/N+1. \end{align*}$$

and this is larger than $\frac {T^{2}}{N^{3}}\log T$ in the range $N^{1+\epsilon }<T<N^{2(1-\epsilon )}$ . The same argument works for every n, but there are $n(n-1)/2$ entries we can use. Therefore, we have the lower bound:

$$\begin{align*}\#\left\{ \gamma\in \operatorname{\mathrm{SL}}_{n}(\mathbb{Z}):\gamma\equiv I\quad \mod N,{\left\Vert {\gamma}\right\Vert }\le T\right\} \gg\left(\frac{T^{n^{2}-n}}{N^{n^{2}-1}}+\left(\frac{T}{N}\right)^{(n^{2}-n)/2}+1\right). \end{align*}$$

Up to $T^{\epsilon }$ , this is also the upper bound for $n=2$ . We conjecture that up to $T^{\epsilon }$ this is also the upper bound for larger values of n.

4.1 Distances and length of elements

Besides our definition of length, the following is standard; see, for example, [Reference Ghosh, Gorodnik and Nevo26, Section 3]. We mainly follow [Reference Knapp43] when G is Archimedean and [Reference Cartier10] when G is non-Archimedean.

Let k be $\mathbb {R}$ or a p-adic field, and $\left |\cdot \right |{}_{k}\colon k\to \mathbb {R}_{+}$ its standard nontrivial valuation. Let G be a semisimple noncompact algebraic group over k, of k-rank r. Let $T\cong G_{m}^{r}\subset G$ be a maximal k-split torus. The choice of T determines the set of weights $X^{\ast }(T)$ , that is, of rational characters of T. Let $\Phi (T,G)\subset X^{\ast }(T)$ be the set of roots of G with respect to T.

In the Archimedean case, if $T_{0}\cong \left \{ \pm 1\right \} ^{r}$ is the maximal compact subgroup of T, the connected component $A\cong T/T_{0}$ of the identity of T is the Lie group of a real Cartan subalgebra $\mathfrak {a}$ of $\mathfrak {g}$ , and we define $\nu \colon T\to A\to \mathfrak {a}\cong \mathbb {R}^{r}$ by the logarithm map.

In the non-Archimedean case, let $T_{0}$ be a maximal compact subgroup of T. Then $T/T_{0}\cong \mathbb {Z}^{r}$ , this identification defines $\nu \colon T\to \mathbb {Z}^{r}\subset \mathbb {R}^{r}$ , and we identify $\mathbb {R}^{r}$ with $\mathfrak {a}$ .

Let $X^{\ast }(T)_{\mathbb {R}}\cong X^{\ast }(T)\otimes \mathbb {R}$ be the weight space. For an element $\alpha \in X^{\ast }(T)$ , we let $\chi _{\alpha }\in \mathfrak {a}^{\ast }$ be the linear functional defined such that for $t\in T \left |\alpha (t)\right |{}_{k}=q^{\chi _{\alpha }(\nu (t))}$ , where $q=e$ in the Archimedean case and otherwise the size of the quotient field of k. For $\alpha \in X^{\ast }(T)_{\mathbb {R}}$ , we define $\chi _{\alpha }\in \mathfrak {a}^{\ast }$ by extension of the action above. This isomorphism (as linear spaces) between $X^{\ast }(T)_{\mathbb {R}}$ and $\mathfrak {a}^{*}$ defines an isomorphism between the coweight space $(X^{*}(T)_{\mathbb {R}})^{*}$ and $\mathfrak {a}$ .

Choose an ordering on the root system which defines the positive roots $\Phi _{+}\subset \Phi (T,G)$ and let $\Delta =\left \{ \alpha _1,...,\alpha _{r}\right \} \subset X^{*}(T)_{\mathbb {R}}\cong \mathfrak {a}^{*}$ be the simple roots of $\Phi $ with respect to this ordering. Let $\left \{ \omega _1,...,\omega _{r}\right \} \subset \mathfrak {a}$ be the set of simple coweights, that is, the dual basis to $\Delta $ . The set $\left \{ \sum _{i=1}^{r}x_{i}\omega _{i}:x_{i}\ge 0\right \} \subset \mathfrak {a}$ is called the dominant sector, or the positive Weyl chamber. It is isomorphic to $\mathfrak {a}/W$ , where W is the Weyl group of the root system.

Let $\Phi _{+}^{\vee }\subset \mathfrak {a}$ be the corresponding coroot system, let $\Delta ^{\vee }=\left \{ \alpha _{1}^{\vee },...,\alpha _{r}^{\vee }\right \} $ be the dual basis of simple coroots and let $\left \{ \omega _{1}^{\vee },...,\omega _{r}^{\vee }\right \} \subset \mathfrak {a}^{*}$ be the set of simple weights. We define a partial ordering on $\mathfrak {a}$ by $\alpha \ge _{\mathfrak {a}}\alpha '$ if and only if $\omega _{i}^{\vee }(\alpha )\ge \omega _{i}^{\vee }(\alpha )$ for every simple weight, or alternatively $\alpha -\alpha '$ is a nonnegative sum of elements of $\Delta ^{\vee }$ .

Let P be the Borel subgroup with respect to the set of positive roots. It holds that $P=MN$ , where M is the centralizer of T in G and N is the unipotent radical of P ([Reference Cartier10, p. 134]). Let K be a maximal special compact open subgroup (i.e., in the p-adic case we choose it to be ‘good’ in the sense of Bruhat and Tits [Reference Bruhat and Tits8]). The Iwasawa decomposition $G=KP$ holds ([Reference Knapp43, Proposition 1.2],[Reference Cartier10, p. 140]). It holds that $M=(M\cap K)\cdot T$ , and we extend $\nu \colon M\to \mathbb {R}^{r}$ by $\nu (k)=1$ for $k\in M\cap K$ .

Let us recall the Cartan decomposition $G=KA_{+}K$ :

• • In the real case, following [Reference Knapp43, Theorem 5.20], $A\cong T/T_{0}$ is the Lie group of the Cartan subalgebra $\mathfrak {a}$ of $\mathfrak {g}$ . $A_{+}$ is the exponent of the closure of the dominant sector in $\mathfrak {a}$ , that is, the set of elements $A_{+}=\left \{ t\in A:\forall \alpha \in \Phi _{+},\alpha (t)\ge 1\right \} $ . It is isomorphic (as a set, using the exponential map) to the dominant sector in $\mathfrak {a}$ .

• • In the p-adic case, following [Reference Cartier10, p. 140], let $\Lambda \cong M/M^{0}$ , where $M^{0}$ is the maximal compact subgroup of M. We identify elements of $\Lambda $ with elements of $M\subset P$ . Elements of the weight space are unramified characters of M, and therefore are characters of $\Lambda $ as well. We let $A_{+}=\left \{ \lambda \in \Lambda :\forall \alpha \in \Phi _{+},\alpha (\lambda )\ge 1\right \}$ . The action $\nu \colon T/T_{0}\to \mathfrak {a}$ extends to $\nu \colon M/M^{0}\to \mathfrak {a}$ . Then $A_{+}$ is isomorphic as a set with the intersection of $\nu (\Lambda )$ with the dominant sector in $\mathfrak {a}$ . It is also isomorphic to a subset of the special vertices in the dominant sector in an apartment in the Bruhat–Tits building of G.

In both cases, there exists a map $\nu \colon A_{+}\to \mathfrak {a}$ . It is also useful to extend it to a map $H\colon G\to \mathfrak {a}$ , using the Iwasawa decomposition $G=KMN$ , by $H(kmn)=\nu (m)$ .

We will need the following fundamental technical lemma:

Lemma 4.1. For $a\in A_{+}$ and $k\in K$

$$\begin{align*}H(ak)\le_{\mathfrak{a}}H(a). \end{align*}$$

Proof. We first notice the following property of the H-function: for $k\in K$ , $g\in G$ , $m\in M$ , $n\in N$

$$ \begin{align*} H(kgmn) & =H(g)+H(m) \end{align*} $$

Now, if $KaKa'K\cap Ka"K\ne \phi $ , then $a"=k_{0}ak_{1}a'k_{2}$ . Applying the Iwasawa decomposition to $a'k_{2}$ we have $a'k_{2}=k_{2}^{\prime }mn$ with $H(m)\le _{\mathfrak {a}}H(a')$ by Lemma 4.1. Applying Lemma 4.1 again, we have

$$ \begin{align*} \nu(a") & =H(k_{0}ak_{1}k_{2}^{\prime}mn)=H(k_{0}ak_{1}k_{2}^{\prime})+H(m)\\ & \le_{\mathfrak{a}}H(a)+H(a')=\nu(a)+\nu(a').\\[-38pt] \end{align*} $$

Corollary 4.2 in the non-Archimedean case is [Reference Bruhat and Tits8, Proposition 4.4.4(iii)], and is deduced from Lemma 4.1 in the same way.

Let $\delta (\tilde {p})$ be the left modular character of P, that is, if $d\tilde {p}$ is a left Haar measure on P, then $\delta (\tilde {p})d\tilde {p}$ is a right Haar measure. Normalize the measures so that for $f\in C_c(G)$ , $\int _{G}f(g)dg=\int _{K}\int _{P}f(k\tilde {p})\delta (\tilde {p})d\tilde {p}dk=\int _{K}\int _{P}f(\tilde {p}k)d\tilde {p}dk$ , $\int _{K}dk=1$ .

It can also be defined as follows: M acts by conjugation on the Lie algebra $\mathfrak {n}$ of N. Then for $m\in M$ , $\delta (m)=\left |\operatorname {\mathrm {Det}}\operatorname {\mathrm {Ad}}_{\mathfrak {n}}(m)\right |{}_{k}$ ([Reference Cartier10, p. 135], [Reference Knapp43, Proposition 5.25]). Unwinding the definitions, $\delta (m)=q^{2\rho (\nu (m))}$ , where $\rho =\frac {1}{2}\sum _{\alpha \in \Phi _{+}}(\dim \mathfrak {g}_{\alpha })\chi _{\alpha }$ . Here, $\mathfrak {g}_{\alpha }$ is the root space of $\alpha $ in the Lie algebra $\mathfrak {g}$ of G. Also, recall that $q=e$ for $k=\mathbb {R}$ and otherwise is the size of the quotient field of k. As an example, for $G=\operatorname {\mathrm {SL}}_{n}(\mathbb {R})$ , and the matrix $a=\operatorname {\mathrm {diag}}(a_{0},\dots ,a_{n-1})$ , $\delta (a)=\prod a_{i}^{n-1-2i}$ .

We associate with each element $a\in A_{+}\subset G$ a length $l\colon A_{+}\to \mathbb {R}_{\ge 0}$ by $l(a)=\log _{q}\delta (a)=2\rho (a)$ . We extend $l\colon G\to \mathbb {R}_{\ge 0}$ by $l(kak')=l(a)$ . By definition, l is left and right K-invariant.

For $a\in T$ , we may identify $l(a)$ by the entropy (taken with logarithm in base q) of the dynamical system of translation of $\Gamma \backslash G$ by a, with respect to the Haar measure ( $\Gamma $ here is an arbitrary lattice; see [Reference Einsiedler and Lindenstrauss20, Theorem 7.9]). Using this fact, we have for $a\in A$ , $l(a)=l(a^{-1})$ and therefore for every $g\in G$ , $l(g)=l(g^{-1})$ . The same fact can be proven directly.

For $a\in A_{+}$ , define $S(a)=\int \delta (ak)dk$ . For $f\in C_c(G)$ , we have

$$\begin{align*}\int f(g)dg=\int_{K}\int_{K}\int_{A_{+}}f(kgk')S(a)dkdk'da. \end{align*}$$

We interpret $S(a)$ as the measure of the ‘circle’ $KaK$ . In the non-Archimedean case, $S(a)\asymp \delta (a)$ ([Reference Cartier10, p. 141]). In the Archimedean case, by [Reference Knapp43, Proposition 5.28],

$$\begin{align*}S(a)=\prod_{\alpha\in\Phi_{+}}(\sinh(\chi_{\alpha}(a)))^{\dim\mathfrak{g}_{\alpha}}. \end{align*}$$

Since $\sinh (x)\asymp _{\beta }e^{x}$ for $x>\beta $ , for $a\in A_{+}$ ‘far from the walls’, that is, with $\chi _{\alpha }(a)>\beta $ for every $\alpha \in \Delta $ (and therefore $\chi _{\alpha }(a)>\beta $ for every $\alpha \in \Phi _{+}$ ), we have $S(a)\asymp _{\beta }\delta (a)=e^{l(a)}$ . Near the walls where $\sinh x\approx x$ this approximation fails, but we still have $S(a)\ll \delta (a)$ . In any case, if we choose some norm ${\left \Vert {\cdot }\right \Vert }_{\mathfrak {a}}$ on $A_{+}$ , then for every $\tau>0$ , $a\in A_{+}$ , $\mu (\left \{ g:{\left \Vert {a_{g}-a}\right \Vert }_{\mathfrak {a}}\le \tau \right \} )\asymp _{\tau }q^{l(a)}$ , since the set $\left \{ a'\in A_{+}:{\left \Vert {a'-a}\right \Vert }_{\mathfrak {a}}\le \tau \right \} $ contains elements that are far enough (with respect to $\tau $ ) from the walls.

We deduce that the size of balls for $l\gg 1$ ,

(4.1) $$ \begin{align} q^{l}\ll\mu(\cup_{a':l(a')\le l}Ka'K)=\intop_{a\in A_{+}:l(a)\le l}S(a)da\ll p(l)q^{l}\ll_{\epsilon}q^{l(1+\epsilon)}, \end{align} $$

for some polynomial p.

Remark 4.4. The literature has two popular choices of ‘distance’ or ‘length’ on $G/K$ or G.

1. 1. For $G\subset \operatorname {\mathrm {SL}}_{n}$ , where $K=G\cap K'$ for $K'$ maximal compact in $\operatorname {\mathrm {SL}}_{n}$ , one defines $\tilde {l}(g)=\log {\left \Vert g\right \Vert }$ , where ${\left \Vert v\right \Vert }$ is some matrix norm on $\operatorname {\mathrm {GL}}_{n}$ . Recall that we are mainly interested in distances as $g\to \infty $ , so the specific choice of matrix norm does not matter. Such choice (without calling it a distance) is studied in [Reference Sarnak and Xue58, Reference Duke, Rudnick and Sarnak19].

2. 2. For G Archimedean, let $\mathfrak {g},\mathfrak {t}$ be the Lie algebras of G and K, and let $B\colon \mathfrak {g}\times \mathfrak {g}\to \mathbb {C}$ the Killing form of G. Let $\mathfrak {p}=\left \{ X\in \mathfrak {g}:B(X,Y)=0\,\forall Y\in \mathfrak {t}\right \} $ . Then $B|_{\mathfrak {p}\mathfrak {\times }\mathfrak {p}}$ is positive definite. $\mathfrak {p}$ can be identified with the tangent space of $G/K$ at the identity, and it defines a natural Riemannian structure on $G/K$ , with length $\hat {l}(g)=\hat {d}(g,1)$ . See, for example, [Reference DeGeorge and Wallach16, Section 2], [Reference Sarnak and Xue58]. A similar distance is used in the p-adic case in [Reference Tits62, 2.3].

Since we mostly care about far distances in the group, and since K is compact, by the Cartan decomposition it suffices to compare l to other distances on $A_{+}$ . In general, l and $\tilde {l}$ look like an $L^{1}$ -norm on $A_{+}$ , and $\hat {l}$ looks like an $L^{2}$ -norm.

Let us concentrate on $G=\operatorname {\mathrm {SL}}_{n}(\mathbb {R})$ and $\tilde {l}(g)=\ln ({\left \Vert g\right \Vert } _{2})$ , ${\left \Vert g\right \Vert }_{2}^{2}=\operatorname {\mathrm {tr}} (gg^{t})$ . For $G=\operatorname {\mathrm {SL}}_{2}(\mathbb {R})$ , its symmetric space is the hyperbolic plane with the standard metric of curvature $-1$ , and l we defined above coincides with the hyperbolic metric. For example, consider the matrix $g=\left (\begin {array}{cc} e^{\frac {t}{2}} & 0\\ 0 & e^{-\frac {t}{2}} \end {array}\right )$ . Then $l(g)=t$ , and $\tilde {l}(g)=\frac {1}{2}\ln (e^{t}+e^{-t})\approx \frac {t}{2}$ . In fact, for every $g\in \operatorname {\mathrm {SL}}_{2}(\mathbb {R})$ , $l(g)-2\tilde {l}(g)=O(1)$ , and l and $\tilde {l}$ are equal up to an additive constant.

For $G=\operatorname {\mathrm {SL}}_{3}(\mathbb {R})$ , it is no longer true. For

$$\begin{align*}g_{1}=\left(\begin{array}{ccc} e^{t/3}\\ & e^{t/3}\\ & & e^{-2t/3} \end{array}\right), g_{2}=\left(\begin{array}{ccc} e^{2t/3}\\ & e^{-t/3}\\ & & e^{-t/3} \end{array}\right), \end{align*}$$

it holds that $l(g_{1})=l(g_{2})=2t$ , while $\tilde {l}(g_{1})\approx t/3$ , $\tilde {l}(g_{2})\approx 2t/3$ . So the two distances $l,\tilde {l}$ are not equal up to an additive constant, but are only Lipschitz-equivalent, with $\frac {3}{2}\tilde {l}\le l+O(1)\le 3\tilde {l}$ . However, if we chose $\tilde {\tilde {l}}(g)=2(\tilde {l}(g)+\tilde {l}(g^{-1}))$ , then $l(g)-2\tilde {\tilde {l}}(g)=O(1)$ . This solution no longer works for $\operatorname {\mathrm {SL}}_{4}(\mathbb {R})$ .

4.2 Growth of matrix coefficients and Harish-Chandra’s bounds

For $2\le p\le \infty $ , let be the p-th version of Harish-Chandra’s function. Let $\Xi (g)=\Xi _{2}(g)$ be the standard Harish-Chandra’s function. Note that since $\Xi (g)$ is left and right K-invariant it only depends on $a\in A_{+}$ from the Cartan decomposition of g.

An explicit upper bound on Harish-Chandra’s function is given by the following theorem:

Theorem 4.5. There is a constant C such that for every $g\in G$ and $\epsilon>0$ ,

A representation $(\pi ,V)$ of G is called tempered if $p(\pi )\le 2$ . The following theorem is the standard reference for upper bounds on matrix coefficients:

The work [Reference Cowling, Haagerup and Howe14] also provides a bound when $p(\pi )>2$ :

This theorem is not sufficient for this work since we need more precise bounds when $p(\pi )\notin 2\mathbb {N}$ . The following theorem contains a general upper bound that is good enough for all the applications of this paper.

Theorem 4.8. Let $(\pi ,V)$ be a unitary irreducible representation of G with $p(\pi )\le p$ , $p\ge 2$ and let $v_1,v_{2}\in V$ be two K-finite vectors such that $\dim \operatorname {\mathrm {span}} Kv_{1}=d_{1}$ , $\dim \operatorname {\mathrm {span}} Kv_{2}=d_{2}$ . Then

$$\begin{align*}\left|\left\langle v_1,gv_{2}\right\rangle \right|\le\sqrt{d_{1}d_{2}}{\left\Vert {v_{1}}\right\Vert }{\left\Vert {v_{2}}\right\Vert }\Xi_{p}(g). \end{align*}$$

Let us discuss older similar results, slightly weaker than Theorem 4.8.

We first consider bounds on matrix coefficients of a single representation $\pi $ . In the Archimedean case, we may consider the theory of leading exponents ([Reference Knapp43, Chapter VIII]), which we describe in Subsection 9.1. In the non-Archimedean case, analogous results hold ([Reference Casselman11, Section 4]).

Theorem 4.9 (See Theorems 9.1,9.2).

Let $(\pi ,V)$ be a unitary irreducible representation of G with $p(\pi )\le p$ , $p\ge 2$ , and let $v_1,v_{2}\in V$ be two K-finite vectors. Then for every $\epsilon>0$ ,

Next, we consider bounds on matrix coefficients when $v_1,v_{2}$ are K-fixed. In such case (if $v_1,v_{2}\ne 0$ ), the representation is spherical and well understood, at least in the Archimedean case (see Subsection 4.4 below). From those bounds, we have:

Theorem 4.10 (See [Reference Ghosh, Gorodnik and Nevo26, Section 3]).

Let G be Archimedean, and let $(\pi ,V)$ be a unitary irreducible representation of G with $p(\pi )\le p$ , and let $v_1,v_{2}\in V$ be two K-fixed vectors. Then for every $\epsilon>0$ ,

Let us provide a proof of Theorem 4.8 when $v_1,v_{2}$ are K-fixed. A bit more work can give bounds for K-finite vectors as well, using the ideas of [Reference Cowling, Haagerup and Howe14]. Let us first prove the following lemma, which is based on the proof of [Reference Cowling, Haagerup and Howe14, Theorem 2]. For $f\in L^{p}(G)$ and $g\in G$ , we let $gf\in L^{p}(G)$ be $gf(g')=f(g^{-1}g')$ .

Proof. To avoid integrability questions, we assume that $f_1,f_2\in C_c(G)$ and deduce the theorem by density. Denote $p'=\frac {p}{p-1}$ . For $f\in C_c(G)$ , it holds that

$$\begin{align*}\intop_{G}f(x)dx=\intop_{K}\intop_{P}f(k\tilde{p})\delta(\tilde{p})d\tilde{p}dk, \end{align*}$$

so

Since $f_{1}$ is K-fixed, we have for every $k\in K$ ,

write $g^{-1}k=k_{0}\tilde {p}_{0}$ . Then

Since $\delta $ is left K-fixed and $\tilde {p}_{0}=k_{0}^{-1}g^{-1}k$

The lemma has a nice corollary: For $a\in A_{+}$ let $A_{a}$ be the operator $A_{a}\colon C(K\backslash G)\to C(K\backslash G)$ defined by $A_{a}f(g)=\intop _{K}f(a^{-1}k^{-1}g)dk=\intop _{K}\intop _{K}f(k^{\prime -1}a^{-1}k^{-1}g)dkdk'$ . We may define $A_{g}$ for $g\in G$ , but it only depends on the $A_{+}$ component of g from the Cartan decomposition. Since $A_{a}$ is a translation followed by an average, its $L^{p}$ -norm is bounded by $1$ on $L^{p}(G)\cap C(K\backslash G)$ , and therefore it defines an operator $A_{a}\colon L^{p}(K\backslash G)\to L^{p}(K\backslash G)$ .

Proof. Let . Let $f\in L^{p}(K\backslash G)$ . Let $f_{1}\in L^{p'}(K\backslash G)$ with ${\left \Vert {f_{1}}\right \Vert }_{p'}=1$ be such that $\left \langle f_1,A_{a}f\right \rangle ={\left \Vert {A_{a}f}\right \Vert }_{p}$ . Then since $f_{1}$ is left K-invariant $\left \langle f_1,A_{a}f\right \rangle =\left \langle f_1,af\right \rangle $ . Applying Lemma 4.11, we have

$$\begin{align*}{\left\Vert {A_{a}f}\right\Vert }_{p}\le\Xi_{p}(a){\left\Vert f\right\Vert }_{p}, \end{align*}$$

as needed.

Note that if $(\pi ,V)$ is a unitary representation of G, we may define $\pi (A_{a})\colon V^{K}\to V^{K}$ by the same arguments as above, as

(4.2) $$ \begin{align} \pi(A_{a})v=\intop_{K}\pi(ka)v\,dk=\intop_{K}\intop_{K}\pi(kak')v\,dkdk', \end{align} $$

and ${\left \Vert {A_{a}}\right \Vert }_{V}\le 1$ . By a standard argument, this operation commutes with taking matrix coefficients – denote for $v_1,v_{2}\in V^{K}$ , $\varphi _{v_1,v_{2}}(g)=\left \langle v_1,\pi (g)v_{2}\right \rangle $ , $\varphi _{v_1,v_{2}}(g)\in L^{\infty }(G)$ and then $A_{a}\varphi _{v_1,v_{2}}(g)=\varphi _{A_{a}v_1,v_{2}}(g)$ .

Corollary 4.13. Let $(\pi ,V)$ be a unitary irreducible representation of G with $p(\pi )\le p$ , and let $v_1,v_{2}\in V$ be two K-fixed vectors. Then

$$\begin{align*}\left|\left\langle \pi(g)v_1,v_{2}\right\rangle \right|\le{\left\Vert {v_{1}}\right\Vert }{\left\Vert {v_{2}}\right\Vert }\Xi_{p}(g). \end{align*}$$

4.3 Upper bounds on operators

Let $(\pi ,V)$ be a unitary representation of G. For $a\in A_{+}$ , let $\pi (A_{a})\colon V^{K}\to V^{K}$ be from equation (4.2). Similarly, for $d_{0}\in \mathbb {R}_{\ge 0}$ , let $\chi _{d_0},\psi _{d_0}\in C_{c}^{\infty }(G)$ be as in Definition 3.1. Recall that $\chi _{d_0}$ is a smooth approximation for the characteristic function of $\left \{ g:l(g)\le d_{0}\right \} $ and $\psi _{d_0}$ is a smooth approximation for $q^{(d_{0}-l(g))/2}\chi _{d_0}$ .

For $h\in C_c(G)$ we define as usual $\pi (h)v=\intop _{G}h(g)\pi (g)v\,dg$ .

Corollary 4.14. Let $(\pi ,V)$ be a unitary irreducible representation of G with $p(\pi )\le p$ , and let $v\in V$ be a K-fixed vector. Then

Proof. The bound on $\pi (A_{a})$ follows from Corollary 4.13, using the explicit bounds of Theorem 4.5.

We will only prove the estimate for $\psi _{d_0}$ since the proof for $\chi _{d_0}$ is similar and a little easier.

In the last line, we used the fact that $\intop _{l(a)\le d_{0}+1}da\ll _{\epsilon }q^{d_{0}\epsilon }$ .

4.4 Spherical Functions

Let us recall the definition of spherical functions. For $\lambda \in \mathfrak {a}_{\mathbb {C}}^{\ast }=\mathfrak {a}^{\ast }\otimes \mathbb {C}$ dominant, we let $\lambda _{P}\colon P\to \mathbb {C}^{\times }$ be $\lambda _{P}(mn)=q^{\lambda (\nu (m))}\delta ^{-1/2}(m)=q^{(\lambda -\rho )(\nu (m))}$ . Extend it to $\tilde {\lambda }\colon G\to \mathbb {C}$ by $\tilde {\lambda }(kp)=\lambda _{P}(p)$ . Finally, define the spherical function

$$\begin{align*}\varphi_{\lambda}(g)=\intop_{K}\tilde{\lambda}(gk)dk. \end{align*}$$

Note that $\varphi _{0}=\Xi $ and $\varphi _{(1-2/p)\rho }=\Xi _{p}$ for $2\le p\le \infty $ .

The theory of spherical functions was developed by Harish-Chandra in the Archimedean case and by Satake in the non-Archimedean case (see [Reference Ghosh, Gorodnik and Nevo26, Subsection 3.2] and the reference therein). We will need some very basic properties of spherical functions.

Let $(\pi ,V)\in \Pi (G)_{\operatorname {\mathrm {sph}}}$ be a unitary spherical representation, that is, a unitary irreducible representation with a nontrivial K-fixed vector. If $v_1,v_{2}$ are K-fixed, then there exists a dominant $\lambda \in \mathfrak {a}_{\mathbb {C}}^{\ast }$ such that

$$\begin{align*}\left\langle v_1,\pi(g)v_{2}\right\rangle =\left\langle v_1,v_{2}\right\rangle \varphi_{\lambda}(g). \end{align*}$$

Upper bounds on spherical functions are given as follows. By [Reference Ghosh, Gorodnik and Nevo26, Lemma 3.3], it holds that for $a\in A_{+}$ , $k,k'\in K$ and $\epsilon>0$ ,

$$\begin{align*}\varphi_{\lambda}(kak')=\varphi_{\lambda}(a)\ll q^{\operatorname{\mathrm{Re}}\lambda(\nu(a))}\Xi(a)\ll_{\epsilon}q^{\operatorname{\mathrm{Re}}\lambda(\nu(a))}q^{-l(a)(1/2-\epsilon)}. \end{align*}$$

Let $\omega _1,...,\omega _{r}\in \mathfrak {a}$ be the fundamental coweights. Since $\lambda $ is dominant $\operatorname {\mathrm {Re}}\lambda (\omega _{i})\ge 0$ for $1\le i\le r$ . Using the above bounds, we deduce that

(4.3)

See also [Reference Ghosh, Gorodnik and Nevo26, Lemma 3.2], which has a small misprint.

In the Archimedean case, the other direction of the implication of equation (4.3) is also true ([Reference Knapp43, Theorem 8.48]), which gives a proof of Theorem 4.10 in this case.

However, the other direction is no longer true for the general non-Archimedean case. The simplest example is when the Bruhat–Tits building of G is a bipartite $(d_0,d_1)$ -regular tree, $d_0\ne d_1$ , where for one of the choices of a maximal compact subgroup, there is a spherical function that is square integrable but $\operatorname {\mathrm {Re}}\lambda (\omega )>0$ (this result was first observed by [Reference Matsumoto54]). In general, the other direction is true in the standard case according to MacDonald ([Reference Macdonald52, Chapter V]), which excludes the case above. In particular, every group has a standard maximal compact subgroup for which equation (4.3) is an equivalence.

Lower bounds on spherical functions can be given in general if $\operatorname {\mathrm {Re}}\lambda =\lambda $ . By Lemma 4.1, if $\operatorname {\mathrm {Re}}\lambda =\lambda $ and $\operatorname {\mathrm {Re}}\lambda (\omega _{i})\le \rho (\omega _{i})$ , then for $a\in A_{+}$ , $k,k'\in K$

(4.4) $$ \begin{align} \varphi_{\lambda}(kak')=\varphi_{\lambda}(a)=\intop_{K}\tilde{\lambda}(ak)dk\ge\intop_{K}\tilde{\lambda}(a)dk=\lambda_{P}(a). \end{align} $$

In the case when G is Archimedean, it is well known that the dominant $\lambda \in \mathfrak {a}_{\mathbb {C}}^{\ast }$ which occur this way for unitary spherical representations must satisfy $-\bar {\lambda }=w\lambda $ for some $w\in W$ ([Reference Kostant44]).

If we moreover assume that G is of rank $1$ , then $\mathfrak {a}_{\mathbb {C}}^{*}\cong \mathbb {C}$ and $W=\left \{ 1,s\right \} $ acts by $s\lambda =-\lambda $ . Therefore, the only dominant $\lambda \in \mathfrak {a}_{\mathbb {C}}^{*}$ which may occur satisfy either $\operatorname {\mathrm {Re}}\lambda =0$ or $\operatorname {\mathrm {Re}}\lambda =\lambda $ . In the case $\operatorname {\mathrm {Re}}\lambda =0$ , the corresponding unitary representation satisfies $p(\pi )=2$ . If $\operatorname {\mathrm {Re}}\lambda =\lambda $ , then $\lambda =\alpha \rho $ with $\alpha \ge 0$ . By equation (4.3), $\alpha \le 1$ . Write $\alpha =1-1/p$ , $2\le p\le \infty $ . Using equation (4.3) and the definition of $\Sigma _{p}(g)$ , we conclude that for $p>2$ , there is at most a single unitary irreducible representation $(\pi ,V)$ with a nontrivial K-fixed vector and $p(\pi )=p$ . The corresponding spherical function is $\Xi _{p}(g)$ . We conclude:

Proposition 4.15. Let G be Archimedean of rank $1$ , $p>2$ and $(\pi ,V)$ a unitary irreducible representation of G with $p(\pi )=p$ , having a non-trivial K-fixed vector v. Then $\pi (A_{a})v=\lambda _{1}v$ , $\pi (\chi _{d_0})v=\lambda _{2}v$ , with

Moreover, if $h\in C_{c}(K\backslash G/K)$ satisfies $h(g)\ge \chi _{d_0}(g)$ for every $g\in G$ , then $\pi (h)v=\lambda _{h}v$ , with $\lambda _{h}\ge \lambda _{2}$ .

Proof. It is well known that in this case the set of all K-fixed vectors in V is one-dimensional and equals $\operatorname {\mathrm {span}}\left \{ v\right \}$ ([Reference Cartier10, Theorem 4.3]). Since $\pi (A_{a})v$ , $\pi (\chi _{d_0})v$ , $\pi (h)v$ are K-fixed, we get that $\pi (A_{a})v=\lambda _{1}v$ , $\pi (\chi _{d_0})v=\lambda _{2}v$ and $\pi (h)v=\lambda _{h}v$ . Applying matrix coefficients, we see that

$$ \begin{align*} \lambda_{1} & =\Xi_{p}(a)\\ \lambda_{2} & =\intop_{G}\chi_{d_0}(g)\Xi_{p}(g)dg\\ \lambda_{h} & =\intop_{G}h(g)\Xi_{p}(g)dg\ge\intop_{G}\chi_{d_0}(g)\Xi_{p}(g)dg=\lambda_{2}. \end{align*} $$

Using equation (4.4), we get

$$\begin{align*}\Xi_{p}(g)\ge e^{-\frac{2}{p}\rho(\nu(a))}=e^{-l(a)/p}. \end{align*}$$

And therefore $\lambda _{1}\ge e^{-l(a)/p}$ and for $d_{0}\ge 1$ ,

$$ \begin{align*} \lambda_{2} & =\intop_{G}\chi_{d_0}(g)\Xi_{p}(g)dg\ge\\ & \ge\intop_{0}^{d_0}\sinh(t)e^{-t/p}dt\gg e^{(1-1/p)d_{0}}.\\[-49pt] \end{align*} $$

A similar analysis can be done for the non-Archimedean rank $1$ case, whenever the Bruhat–Tits tree of G is regular. However, when the Bruhat–Tits tree of G is not regular, the description of the spherical unitary dual is more complicated, and in particular, it is different for the two nonconjugate maximal compact subgroups. The analysis of this can be done by analyzing the slightly more general case of regular and biregular trees. See [Reference Hashimoto33] or [Reference Kamber39] for an analysis of this case.

Since the calculations are a bit long and are not the main focus of this work, we skip them and state the final result. Recall that we have a map $\nu \colon M/M^{0}\to \mathfrak {a}$ with a discrete image. We identify its image with $\mathbb {Z}$ .

Proposition 4.16. Let G be non-Archimedean of rank $1$ , $p>2$ and $(\pi ,V)$ a unitary irreducible representation of G with $p(\pi )=p$ , having a nontrivial K-fixed vector v. Then for $a\in A_{+}$ with $\nu (a)\in \mathbb {Z}$ even, we have $\pi (A_{a})v=\lambda _{1}v$ , with

$$ \begin{align*} q^{-l(a)/p} & \ll\lambda_{1}. \end{align*} $$

We can finally conclude:

4.5 Convolution of Operators

In this section, we analyze the function $\chi _{d_0}\ast \chi _{d_0}(g)$ . Similar analysis can be found for rank $1$ in [Reference Sarnak and Xue58, Lemma 3.1]. Our analysis is less accurate but is more abstract and works for every rank.

Lemma 4.18 (Convolution Lemma).

It holds that $c_{d_0}=\chi _{d_0}\ast \chi _{d_0}\in C_{c}^{\infty }(K\backslash G/K)$ and satisfies the inequality

$$\begin{align*}c_{d_0}(g)\ll_{\epsilon}q^{d_{0}\epsilon}\psi_{2d_{0}+2}(g). \end{align*}$$

The same bound holds for the convolution of every $f_1,f_{2}\in C_{c}(K\backslash G/K)$ such that $f_{1}(g),f_{2}(g)\ll _{\epsilon }q^{d_{0}\epsilon }\chi _{d_0}(g)$ .

Proof. The second statement is a consequence of the first, so we only need to prove the first statement.

The idea is to look at the action (by right convolution) of $\chi _{d_0}$ on $L^{2}(G)$ . By Lemma 4.11 and the same arguments as in Corollary 4.12, the norm of $\chi _{d_0}$ on $L^{2}(G)$ is bounded by . Therefore, the norm of $c_{d_0}$ on $L^{2}(G)$ is bounded by $\ll _{\epsilon }q^{d_{0}(1+\epsilon )}$ . Now, we need to use some continuity arguments to deduce pointwise bounds.

Notice that since $\chi _{d_0}\in C_{c}^{\infty }(K\backslash G/K)$ , the same is true for $c_{d_0}$ . In the non-Archimedean case, the arguments are simpler: We look at the action of $c_{d_0}$ on the characteristic function $\boldsymbol {1}_{K}$ of K. Then

$$\begin{align*}{\left\Vert {\boldsymbol{1}_{K}\ast c_{d_0}}\right\Vert }_{L^{2}(G)}^{2}\ll_{\epsilon}q^{2d_{0}(1+\epsilon)}{\left\Vert {\boldsymbol{1}_{K}}\right\Vert }_{L^{2}(G)}=q^{2d_{0}(1+\epsilon)}. \end{align*}$$

But if $c_{d_0}(g)=R$ then $\boldsymbol {1}_{K}\ast c_{d_0}(g)=R$ , so

$$\begin{align*}{\left\Vert {\boldsymbol{1}_{K}\ast c_{d_0}}\right\Vert }_{L^{2}(G)}^{2}\ge\mu(KgK)R^{2}\gg q^{l(g)}R^{2}. \end{align*}$$

Therefore, as needed.

In the Archimedean case, assume that $c_{d_0}(g)=R$ . Then $c_{d_{0}+2}(g')\ge R$ for every $g'\in G$ with $\left |l(g)-l(g^{\prime })\right |\le 1$ . We consider $\boldsymbol {1}_{B_{1}}$ , where $B_{1}$ is the ball of radius $1$ around the identity. It holds that

$$\begin{align*}{\left\Vert {\boldsymbol{1}_{B_{1}}\ast c_{d_{0}+2}}\right\Vert }_{L^{2}(G)}^{2}\ll_{\epsilon}q^{2d_{0}(1+\epsilon)}{\left\Vert {\boldsymbol{1}_{B_{1}}}\right\Vert }_{L^{2}(G)}\ll q^{2d_{0}(1+\epsilon)}. \end{align*}$$

It also holds that $\boldsymbol {1}_{B_{1}}\ast c_{d_{0}+2}(g)\gg R$ , for $g'$ satisfying $\left |l(g)-l(g^{\prime })\right |\le 1$ , so

$$\begin{align*}{\left\Vert {\boldsymbol{1}_{K}\ast c_{d_{0}+2}}\right\Vert }_{L^{2}(G)}^{2}\gg\mu(KB_{1}gK)R^{2}\gg q^{l(g)}R^{2}, \end{align*}$$

and as needed.

4.6 Traces of operators on irreducible unitary representations

Our goal in this section is to relate the lower and upper bounds of the previous sections to lower and upper bounds on traces.

Let us recall how to define traces of an operator $h\in C_c(G)$ on a unitary irreducible representation $(\pi ,V)$ . We will only consider the case when h is left and right K-finite, which simplifies the theory. In such case, there is an orthogonal projection $e_{h}\colon C(K)\to C(K)$ , with a finite-dimensional image, such that $h=e_{h}*h*e_{h}$ (see, e.g., [Reference Cowling, Haagerup and Howe14, proof of Theorem 2]).

Since $\pi $ is admissible, the image of $\pi (h)$ is finite-dimensional and therefore is trace class ([Reference Knapp43, Chapter X]), with trace given by

$$\begin{align*}\operatorname{\mathrm{tr}} \pi(h)=\sum_{i}\left\langle u_{i},\pi(h)u_{i}\right\rangle , \end{align*}$$

where $\left \{ u_{i}\right \} \subset V$ is an orthonormal basis.

By uniform admissibility ([Reference Knapp43, Theorem 10.2], [Reference Bernstein3]) and the fact the image of $\pi (h)$ is supported on a finite number of K-types, the image of $\pi (h)$ is of bounded dimension, depending only on the projection $e_{h}$ .

Finally, recall that the norm of a finite-dimensional operator is larger than the largest absolute value of an eigenvalue. We conclude:

Proposition 4.19. Assume that $h\in C_c(G)$ is left and right K-finite and $(\pi ,V)\in \Pi (G)$ . Then

$$\begin{align*}\left|\operatorname{\mathrm{tr}} \pi(h)\right|\ll_{e_{h}}{\left\Vert {\pi(h)}\right\Vert }, \end{align*}$$

the bound depending only on the projection $e_{h}$ such that $e_{h}*h*e_{h}$ .

As an example, if $h\in C_{c}(K\backslash G/K)$ is left and right K-invariant, the image of $\pi (h)$ is of dimension $1$ or $0$ . If the dimension is 0, obviously $\operatorname {\mathrm {tr}} \pi (h)=\pi (h)=0$ . If the dimension is $1$ , V has a K-invariant vector $v\in V$ , ${\left \Vert v\right \Vert }=1$ and

$$\begin{align*}\left|\operatorname{\mathrm{tr}} \pi(h)\right|={\left\Vert {\pi(h)}\right\Vert }=\left|\left\langle v,\pi(h)v\right\rangle \right|. \end{align*}$$

One may also deduce lower bounds on traces of nonnegative self-adjoint operators. It follows from the same considerations as in the finite-dimensional case.

Proposition 4.20. Assume that $h\in C_c(G)$ is left and right K-finite and $(\pi ,V)\in \Pi (G)$ . Moreover, assume that $\pi (h)$ is self-adjoint and nonnegative. Then

$$\begin{align*}{\left\Vert {\pi(h)}\right\Vert }=\sup_{v:{\left\Vert v\right\Vert }=1}\left\langle v,\pi(h)v\right\rangle \le\operatorname{\mathrm{tr}} \pi(h). \end{align*}$$

4.7 The pretrace formula

Let us recall the pretrace formula ([Reference Gelfand, Graev and Piatetski-Shapiro25, Chapter 1]). Let $h\in C_c(G)$ , and let $\Gamma \subset G$ be a cocompact lattice. Denote by $\hat {h}\in C_c(G)$ the function $\hat {h}(g)=h(g^{-1})$ . Then we have an operator $h\colon L^{2}(\Gamma \backslash G)\to L^{2}(\Gamma \backslash G)$ , acting on $f\in L^{2}(\Gamma \backslash G)$ by

$$ \begin{align*} (hf)(x)=f\ast\hat{h}(x) & =\intop_{G}f(xg)h(g)dg=\intop_{G}f(y)h(x^{-1}y)dy\\ & =\intop_{\Gamma\backslash G}f(y)(\sum_{\gamma\in\Gamma}h(x^{-1}\gamma y))dy=\intop_{\Gamma\backslash G}K(x,y)f(y)dy, \end{align*} $$

for $K(x,y)=\sum _{\gamma \in \Gamma }h(x^{-1}\gamma y)$ . By [Reference Gelfand, Graev and Piatetski-Shapiro25, Chapter 1], if h is also self-adjoint, then it is trace-class on $L^{2}(\Gamma \backslash G)$ and

$$\begin{align*}\operatorname{\mathrm{tr}} h|_{L^{2}(\Gamma\backslash G)}=\intop_{\Gamma\backslash G}\sum_{\gamma\in\Gamma}h(x^{-1}\gamma x)dx. \end{align*}$$

Moreover, if $h\in C_{c}^{\infty }(G)$ and $L^{2}(\Gamma \backslash G)\cong \oplus _{\pi \in \Pi (G)}m(\pi ,\Gamma )$ is the decomposition into irreducible representations, then we have the pretrace formula:

$$ \begin{align*} \operatorname{\mathrm{tr}} h|_{L^{2}(\Gamma\backslash G)} & =\intop_{\Gamma\backslash G}\sum_{\gamma\in\Gamma}h(x^{-1}\gamma x)dx\\ & =\sum_{\pi\in\Pi(G)}m(\pi,\Gamma)\operatorname{\mathrm{tr}} \pi(h), \end{align*} $$

where $\operatorname {\mathrm {tr}} (\pi (h))$ is the usual trace on the representation space $(\pi ,V)$ .

The following lemma is immediate from the pretrace formula but essential for our work.

4.8 Spectral decomposition of a characteristic function of a small ball

For $x\in \Gamma _{N}\backslash \Gamma _1\subset X_{N}=\Gamma _{N}\backslash G/K$ , let $b_{x,\delta }\in L^{2}(X_N)$ be defined as follows:

• • In the non-Archimedean case, choose $h_{\delta }\in C_{c}(K\backslash G/K)$ to be the characteristic function of K.

• • In the Archimedean case, choose $0<\delta < 1/4$ such that $l(\gamma )>\delta $ for every $\gamma \in \Gamma _1$ with $l(\gamma )>0$ . Choose a function $h_{\delta }\in C_{c}^{\infty }(K\backslash G/K)$ such that:

  1. – $0\le h_{\delta }(g)\le \frac {2}{\mu (B_{\delta }(e))}$ for all $g\in G$ , where $B_{\delta }(e)$ is the ball of radius $\delta $ around the identity $e\in G$ .

  2. – $h_{\delta }(g)$ is supported on $\{g\in G:l(g)\le \delta \}$ .

  3. – $\intop _{G}h_{\delta }(g)dg=1$

  4. – $h(g)=h(g^{-1})$ , that is, $h=\hat {h}$ .

Finally, let $b_{x,\delta }\in L^{2}(X_N)$ be

$$\begin{align*}b_{x,\delta}(y)=\sum_{\gamma\in\Gamma_{N}}h_{\delta}(x^{-1}\gamma y). \end{align*}$$

We fix $\delta>0$ once and for all (depending on $\Gamma _1$ ) and suppress the dependence on it from now on. We notice that

$$\begin{align*}\intop_{X_N}b_{x,\delta}(y)dy=1. \end{align*}$$

By the properties of $\delta $ , the sum defining it is over at most $\left |\Gamma _1\cap K\right |$ elements, and therefore (recall that we suppress the dependence on $\Gamma _1$ from our notations),

$$\begin{align*}{\left\Vert {b_{x,\delta}}\right\Vert }_{\infty}\ll 1, \ \ {\left\Vert {b_{x,\delta}}\right\Vert }_{2}\ll 1. \end{align*}$$

Let us remark that for our uses in the Archimedean rank $1$ case one can simply choose instead

$$\begin{align*}h_{\delta}(g)=\begin{cases} \frac{1}{\mu(B_{\delta}(e))} & l(g)\le\delta\\ 0 & \text{else} \end{cases}, \end{align*}$$

and for higher rank, we make this choice so that one can apply the Paley–Wiener theorem for spherical functions due to Harish-Chandra, which is used as follows:

Lemma 4.22. Let $f\in L^{2}(X_N)$ . Then,

$$\begin{align*}\sum_{x\in\Gamma_{N}\backslash\Gamma_{1}}\left|\left\langle b_{x,\delta},f\right\rangle \right|{}^{2}\ll{\left\Vert f\right\Vert }_{2}^{2}. \end{align*}$$

Moreover, in the Archimedean case, if $f\in L^{2}(X_N)$ is the K-fixed function of some irreducible representation $\pi \subset L^{2}(\Gamma _{N}\backslash G)$ with $\lambda (\pi )=\lambda $ , then for every $L'>0$ ,

$$\begin{align*}\sum_{x\in\Gamma_{N}\backslash\Gamma_{1}}\left|\left\langle b_{x,\delta},f\right\rangle \right|{}^{2}\ll_{L'}(1+\lambda)^{-L'}{\left\Vert f\right\Vert }_{2}^{2}. \end{align*}$$

(The last result will only be used in the Archimedean rank $\ge 2$ case.)

Proof. By our assumption on $\delta $ , the balls $B_{\delta }(x)$ for $x\in \Gamma _{N}\backslash \Gamma _1$ are all either equal (with multiplicity at most $\left |\Gamma _1\cap K\right |$ ) or distinct, so by Cauchy–Schwartz

$$ \begin{align*} \sum_{x\in\Gamma_{N}\backslash\Gamma_{1}}\left|\left\langle f,b_{x,\delta}\right\rangle \right|{}^{2} & \le\sum_{x\in\Gamma_{N}\backslash\Gamma_{1}}\left\Vert f|_{B_{\delta}(x)}\right\Vert _{2}^{2}\left\Vert b_{x,\delta}\right\Vert _{2}^{2}\le\left|\Gamma_1\cap K\right|{\left\Vert f\right\Vert }_{2}^{2}\max_{x\in\Gamma_{N}\backslash\Gamma_{1}}{\left\Vert {b_{x}}\right\Vert }_{2}^{2}\\ & \ll_{\delta}{\left\Vert f\right\Vert }_{2}^{2}. \end{align*} $$

For the moreover part, note that

$$\begin{align*}\left\langle f,b_{x,\delta}\right\rangle =f(x)\operatorname{\mathrm{tr}} \pi(h_{\delta}). \end{align*}$$

It is well known (see [Reference Seeger and Sogge60] for an exact statement) that there is a constant $M>0$ such that

$$\begin{align*}\left|f(x)\right|\ll(1+\lambda)^{M}\left\Vert f|_{B_{\delta}(x)}\right\Vert _{2}^{2}. \end{align*}$$

By the Paley–Wiener theorem for spherical functions ([Reference Duistermaat, Kolk and Varadarajan18, Subsection 3.4]),

$$\begin{align*}\operatorname{\mathrm{tr}} \pi(h_{\delta})\ll_{L'}(1+\lambda)^{-L'-M}. \end{align*}$$

Combining both estimates we get the required inequality.

6.1 Reduction to a spectral argument

Recall that assuming the weak injective radius property and spectral gap, we should prove that for every $\epsilon>0$ , for every $a\in A_{+}$ with $l(a)\ge (1+\epsilon )\log _{q}(\mu (X_N))$ ,

(6.1) $$ \begin{align} \#\left\{ (x,y)\in(\Gamma_1/\Gamma_N)^{2}:\exists\gamma\in\Gamma_1\text{ s.t. }\pi_{N}(\gamma)x=y,{\left\Vert {a_{\gamma}-a}\right\Vert }_{\mathfrak{a}}<\epsilon{\left\Vert a\right\Vert }_{\mathfrak{a}}\right\} =(1-o_{\epsilon}(1))[\Gamma_1:\Gamma_N]^{2}. \end{align} $$

For $(x,y)\in (\Gamma _1/\Gamma _N)^{2}$ , $a\in A_{+}$ and $\epsilon>0$ , we say that $\gamma \in \Gamma _1$ is good for $(x,y,a,\epsilon )$ if $\pi _{N}(\gamma )x=y$ and ${\left \Vert {a_{\gamma }-a}\right \Vert }_{\mathfrak {a}}<\epsilon {\left \Vert a\right \Vert }_{\mathfrak {a}}$ .

Lemma 6.1. Let $(x,y)\in (\Gamma _1/\Gamma _N)^{2}$ , and assume that there is no good $\gamma \in \Gamma _1$ for $(x,y,a,\epsilon )$ . Identify $x,y$ with elements $x,y\in X_{N}=\Gamma _{N}\backslash G/K$ . Let $f_{a}\in L^{1}(G)$ be a function supported on the set $\left \{ g\in G:{\left \Vert {a_{g}-a}\right \Vert }_{\mathfrak {a}}<\epsilon /2{\left \Vert a\right \Vert }_{\mathfrak {a}}\right \} $ , and for $\delta $ small enough with respect to $\epsilon {\left \Vert {a}\right \Vert }_{\mathfrak {a}}$ let $b_{x,\delta }$ as in Subsection 4.8. Then

$$\begin{align*}f_{a}b_{x,\delta}(y)=0. \end{align*}$$

Moreover, for every $y'\in B_{\delta }(y)$ it holds that

$$\begin{align*}f_{a}b_{x,\delta}(y')=0. \end{align*}$$

Proof. We think of $f_{a}b_{x,\delta }$ as a left $\Gamma _{N}$ -invariant function on G, and identify $x,y$ with some lifts of them in $\Gamma _1 \subset G$ . The support of $f_{a}b_{x,\delta }= b_{x,\delta }*\hat {f}_a$ is contained in the set

$$\begin{align*}\left\{ y'\in G:\exists\gamma\in\Gamma_N,x'\in G,\,f_{a}(y^{\prime-1}\gamma x')>0,\,d(x,x')\le\delta\right\}. \end{align*}$$

Assume by contradiction that $y'\in B_{\delta }(y)$ is in the support of $f_{a}b_{x,\delta }$ , and let $\gamma \in \Gamma _{N}$ , $x'\in G$ be such that $f_{a}(y^{\prime -1}\gamma x')>0,\,d(x,x')\le \delta $ . By the assumption on the support of $f_{a}$ ,

$$\begin{align*}{\left\Vert {a_{y^{\prime-1}\gamma x'}-a}\right\Vert }_{\mathfrak{a}}<\epsilon/2{\left\Vert {a}\right\Vert }_{\mathfrak{a}}. \end{align*}$$

Look at

$$\begin{align*}\gamma'=y^{-1}\gamma x\in\Gamma_1. \end{align*}$$

Then $a_{\gamma '}=a_{y^{-1}\gamma x}$ , and if $\delta $ is small enough with respect to $\epsilon {\left \Vert {a}\right \Vert }_{\mathfrak {a}}$ , then ${\left \Vert {a_{y^{-1}\gamma x}-a_{y^{\prime -1}\gamma x'}}\right \Vert }_{\mathfrak {a}}<\epsilon /2{\left \Vert a\right \Vert }_{\mathfrak {a}}$ . Therefore,

$$\begin{align*}{\left\Vert {a_{\gamma'}-a}\right\Vert }_{\mathfrak{a}}<\epsilon{\left\Vert a\right\Vert }_{\mathfrak{a}}, \end{align*}$$

and

$$\begin{align*}\pi_N(\gamma')x = x\gamma^{\prime^-1} = \gamma y. \end{align*}$$

Since $\gamma \in \Gamma _N$ , it says that $\gamma '$ sends $x\in \Gamma _{N}\backslash \Gamma _1$ to $y\in \Gamma _{N}\backslash \Gamma _1$ , as needed.

Let $\pi \in L^{2}(X_N)$ be the uniform probability distribution, that is, $\pi (x)=\frac {1}{\mu (x_N)}$ .

Lemma 6.2. Equation (6.1) holds if for every $\epsilon>0$ , for every $a\in A_{+}$ with $l(a)>(1+\epsilon )\log _{q}([\Gamma _1:\Gamma _N])$ , there exists a probability function $f_{a}\in C_{c}^{\infty }(G)$ supported on $\left \{ g\in G:{\left \Vert {a_{g}-a}\right \Vert }_{\mathfrak {a}}<\epsilon {\left \Vert a\right \Vert }_{\mathfrak {a}}\right \} $ such that

(6.2) $$ \begin{align} \sum_{x\in\Gamma_1/\Gamma_{N}}{\left\Vert {f_{a}b_{x,\delta}-\pi}\right\Vert }_{2}^{2}=o_{\epsilon}(1). \end{align} $$

Proof. We assume that equation (6.2) holds and want to prove that equation (6.1) holds.

Let $\epsilon>0$ . For $x,y\in (\Gamma _1/\Gamma _N)^{2}$ , by Lemma 6.1, if there is no good $\gamma $ for $(x,y,a,\epsilon /2)$ , then for $y'$ in the $\delta $ -neighborhood of y it holds that $\left |(f_{a}b_{x,\delta }-\pi )(y')\right |=\pi (y')=\frac {1}{\mu (X_N)}\asymp [\Gamma _1:\Gamma _N]^{-1}$ . Therefore, for a fixed $x\in \Gamma _1/\Gamma _{N}$ each y without good $\gamma $ contributes $\gg _{\delta }[\Gamma _1:\Gamma _N]^{-2}$ to ${\left \Vert {f_{a}b_{x,\delta }-\pi }\right \Vert }_{2}^{2}$ . Moreover, the contributions are distinct for $y,y'$ whose image in $X_N =\Gamma _N \backslash G / K$ is different.

Therefore,

$$ \begin{align*} & \#\left\{ (x,y)\in(\Gamma_1/\Gamma_N)^{2}:\text{There is no good }\gamma\text{ for }(x,y,a,\epsilon/2)\right\} \\ & \ll\sum_{x\in\Gamma_1/\Gamma_{N}}{\left\Vert {f_{a}b_{x,\delta}-\pi}\right\Vert }_{2}^{2}[\Gamma_1:\Gamma_N]^{2}\\ & =[\Gamma_1:\Gamma_N]^{2}\sum_{x\in\Gamma_1/\Gamma_{N}}{\left\Vert {f_{a}b_{x,\delta}-\pi}\right\Vert }_{2}^{2}\\ & =o([\Gamma_1:\Gamma_N]^{2}), \end{align*} $$

where we used equation (6.2) in the last step. This implies equation (6.1) for $\epsilon /2$ .

The following lemma explains where spectral gap is used.

Proof. First, we note that ${\left \Vert {\pi }\right \Vert }_{2}^{2}=\intop _{X_{N}}\mu (X_N)^{-2}dx=\mu (X_N)^{-1}\asymp [\Gamma _1:\Gamma _N]^{-1}$ . Therefore, if equation (6.3) holds, then also

$$\begin{align*}\sum_{x\in\Gamma_1/\Gamma_{N}}{\left\Vert {f_{a}b_{x,\delta}-\pi}\right\Vert }_{2}^{2}\le\sum_{x\in\Gamma_1/\Gamma_{N}}{\left\Vert {f_{a}b_{x,\delta}}\right\Vert }_{2}^{2}+\sum_{x\in\Gamma_1/\Gamma_{N}}{\left\Vert {\pi}\right\Vert }_{2}^{2}\ll_{\epsilon,\epsilon_{1}}q^{\epsilon_{1}l(a)}, \end{align*}$$

which is similar to equation (6.2), but $o(1)$ is replaced with $O_{\epsilon ,\epsilon _{1}}(q^{\epsilon _{1}l(a)})$ .

Let $\epsilon '>0$ , and let $a\in A_+$ be such that $l(a')=\epsilon 'l(a)$ . Let $f_{a}^{\prime }=A_{a'}*f_{a}$ . Assuming $\epsilon '$ is small enough, $f_{a}^{\prime }$ is supported on

$$\begin{align*}\left\{ g\in G:g\in G:{\left\Vert {a_{g}-a}\right\Vert }_{\mathfrak{a}}<2\epsilon{\left\Vert a\right\Vert }_{\mathfrak{a}}\right\}. \end{align*}$$

We will show that equation (6.2) holds for $f_{a}^{\prime }$ .

Notice that $A_{a'}\pi =\pi $ and $f_{a}b_{x,\delta }-\pi \perp \pi $ . By the spectral gap assumption and Corollary 4.14, for some $p'<\infty $ ,

$$ \begin{align*} {\left\Vert {f_{a}^{\prime}b_{x,\delta}-\pi}\right\Vert }_{2} & ={\left\Vert {A_{a'}(f_{a}b_{x,\delta}-\pi)}\right\Vert }_{2}\\[4pt] & \ll q^{-\epsilon'l(a)/p'}{\left\Vert {f_{a}b_{x,\delta}-\pi}\right\Vert }_{2}. \end{align*} $$

Therefore,

$$ \begin{align*} \sum_{x\in\Gamma_1/\Gamma_{N}}{\left\Vert {f_{a'}b_{x,\delta}-\pi}\right\Vert }_{2}^{2} & \ll q^{-\epsilon'l(a)/p'} \sum_{x\in\Gamma_1/\Gamma_{N}}{\left\Vert {f_{a}b_{x,\delta}-\pi}\right\Vert }_{2}^{2}\\[4pt] & \ll_{\epsilon,\epsilon_{1}}q^{-\epsilon'l(a)/p'}q^{\epsilon_1l(a)}. \end{align*} $$

If we choose $\epsilon _{1}$ small enough, this is $o(1)$ and equation (6.2) holds. Applying Lemma 6.2, we get that equation (6.1) holds as well.

6.2 Completing the proof of Theorem 1.5

Recall that Theorem 1.5 states that spectral gap and the weak injective radius property imply the optimal lifting property. In Lemma 6.3, we reduced it to some spectral statement, equation (6.3). We now claim:

Lemma 6.4. Assume that the weak injective radius property holds for a sequence $(\Gamma _n)$ . Then for every $\epsilon>0$ sufficiently small, for some $\delta>0$ , for every $a\in A_{+}$ with $l(a)\ge \log _{q}([\Gamma _1:\Gamma _N])$ , there is a probability function $f_{a}\in C_{c}^{\infty }(G)$ supported on $\left \{ g\in G:{\left \Vert {a_{g}-a}\right \Vert }_{\mathfrak {a}}<\epsilon \right \}$ such that

$$\begin{align*}\sum_{x\in\Gamma_1/\Gamma_{N}}{\left\Vert {f_{a}b_{x,\delta}}\right\Vert }_{2}^{2}\ll_{\epsilon,\epsilon_1}q^{\epsilon_{1}l(a)}. \end{align*}$$

Notice the function $f_a$ in Lemma 6.4 satisfies slightly stronger conditions than required by Lemma 6.3, so together they imply Theorem 1.5.

Proof. Let $\epsilon>0$ be sufficiently small. By a standard argument, there is a smooth probability function $f_{a}\in C_{c}^{\infty }(G)$ supported on $\left \{ g\in G:{\left \Vert {a_{g}-a}\right \Vert }_{\mathfrak {a}}<\epsilon \right \}$ and satisfies $f_{a}(x)\ll q^{-l(a)}\chi _{l(a)+1}(x)$ .

Therefore, by the Convolution Lemma 4.18 and Lemma 5.1,

$$ \begin{align*} {\left\Vert {f_{a}b_{x,\delta}}\right\Vert }_{2}^{2} & \ll_{\epsilon}q^{-2l(a)}{\left\Vert {\chi_{l(a)+1}b_{x,\delta}}\right\Vert }_{2}^{2}=q^{-2l(a)}\left\langle \chi_{l(a) +1 }b_{x,\delta},\chi_{l(a)+1}b_{x,\delta}\right\rangle \\ & =q^{-2l(a)}\left\langle \chi_{l(a)+1}*\chi_{l(a)+1}b_{x,\delta},b_{x,\delta}\right\rangle \\ & \ll_{\epsilon_{1}}q^{-2(1-\epsilon_{1})l(a)}\left\langle \psi_{2(l(a)+2)}b_{x,\delta},b_{x,\delta}\right\rangle \\ & \ll q^{-2(1-\epsilon_{1})l(a)}\intop_{0}^{2(l(a)+2)}q^{(2l(a)-d_{0})/2}\left\langle \chi_{d_0}b_{x,\delta},b_{x,\delta}\right\rangle dd_{0}.\\ & \ll q^{-2(1-\epsilon_{1})l(a)}\intop_{0}^{2(l(a)+2)}q^{(2l(a)-d_{0})/2}N(x,\Gamma_N,d_{0}+2)dd_{0}. \end{align*} $$

Therefore, applying Proposition 5.3, we get

$$ \begin{align*} \sum_{x\in\Gamma_1/\Gamma_{N}}{\left\Vert {f_{a}b_{x,\delta}}\right\Vert }_{2}^{2} & \ll_{\epsilon_{1}}q^{-2(1-\epsilon_{1})l(a)}\intop_{0}^{2(l(a)+2)}q^{(2l(a)-d_{0})/2}(q^{d_0}+q^{d_{0}/2}[\Gamma_1:\Gamma_N])dd_{0}\\ & \ll_{\epsilon_{1}}q^{l(a)\epsilon_{1}}(1+q^{-l(a)}[\Gamma_1:\Gamma_N]). \end{align*} $$

Since $l(a)\ge \log _{q}([\Gamma _1:\Gamma _N])$ , this is $O(q^{l(a)\epsilon _{1}})$ as needed.

7.1 Some technical calculations

For ease of reference, we give here a couple of technical bounds.

Proof. The fact that (2) implies (1) is simple and is left to the reader.

The fact that (1) implies (2) follows from a standard trick of integration by parts ([Reference Hardy and Wright31, Theorem 421]):

For the higher-rank Archimedean case we have: