Let \(G : = \text{PGL}(2,\mathbb{K})\) where \(\mathbb{K} = \mathbb{R}, \mathbb{C}\) and let \(X := \mathbb{H}^2, \mathbb{H}^3\), respectively. The paper under review studies the geometry and topology of locally symmetric spaces \(\Gamma \backslash X\) and the growth of multiplicities of irreducible unitary representations in \(L^2(\Gamma \backslash G)\) as \(\Gamma\) varies among the set of all arithmetic lattices in \(G\). Let us denote by \(C_c(G)\) the space of continuous compactly supported functions on \(G\). The main result of the paper is Theorem 1.1, which can be stated as follows.
Let \(\Gamma \subset G\) be an arithmetic lattice with invariant trace field \(k\) and let \(W \subset G\) be the set of regular semisimple non-torsion elements. Then there exists a constant \(1/2 > \eta > 0\) with the following property: for every \(C > 0\) and every \(f \in C_c(G)\) with \(\|f\|_{\infty}\leq 1,\) and \(\text{supp}\ f \subset \text{B}(\eta[k : \mathbb{Q}] + C)\) we have \[ \left| \underset{[\gamma]\in \Gamma \cap W} \sum \text{Vol}(\Gamma_{\gamma}\backslash G_{\gamma}) \int_{G_{\gamma}\backslash G} f(g^{-1}\gamma g)dg \right|\ll_C \text{Vol}(\Gamma \backslash G)\mathit{\Delta}_k^{-4/9}. \] Moreover, if \(\Gamma\) is a congruence lattice, then the RHS of the above inequality is \(\text{Vol}(\Gamma \backslash G)^{11/12}\mathit{\Delta}_k^{-4/9}\).
The proof of this theorem is given in Section 8.2. It is divided into two cases, congruence and non-congruence arithmetic lattices. In the previous sections a number of necessary results for the proof are obtained.
Several other results are proved using Theorem 1.1. Let \((\Gamma_n)_{n\in \mathbb{N}}\) be a sequence of pairwise non-conjugate torsion-free co-compact congruence arithmetic lattices in \(G\). Then \((\Gamma_n)_{n\in \mathbb{N}}\) has the limit multiplicity property (Theorem 1.2). In Section 9 the following result is proved (Theorem 1.3): There exists \(\eta > 0\) with the following property. Let \(\Gamma\) be a torsion-free arithmetic lattice in \(G\), with invariant trace field \(k\) and let \(R = C + \eta[k : \mathbb{Q}]\) for some constant \(C > 0\). Then \(\text{Vol}((\Gamma\backslash X)_{<R}) \ll_C \text{Vol}(\Gamma\backslash X)^{11/12}\mathit{\Delta}_k^{-4/9}\) if \(\Gamma\) is a congruence lattice, or \(\text{Vol}((\Gamma\backslash X)_{<R})) \ll_C\text{Vol}(\Gamma\backslash X)\mathit{\Delta}_k^{-4/9}\) otherwise.
As an application of Theorem 1.3, the author proves a conjecture due to Gelander on the homotopy type of arithmetic hyperbolic 3-manifolds. The proof is given in Theorem 1.5: There exist absolute positive constants \(A\), \(B\) such that every arithmetic, hyperbolic 3-manifold \(M\) is homotopy equivalent to a simplicial complex with at most \(A\text{Vol}(M)\) vertices where each vertex has degree bounded by \(B\), and if \(M\) is compact \(B = 3104\) can be taken. It is done in Section 10.1.
Two corollaries are deduced from Theorem 1.5. Corollary 1.6 assures that there exists a constant \(C > 0\) such that any torsion-free arithmetic lattice \(\Gamma\) in \(\text{PGL}(2,\mathbb{C})\) admits a presentation \(\Gamma = \langle S\ |\ \Sigma\rangle\) where \(|S|\), \(|\Sigma|\) are bounded by \(C\text{Vol}(M)\) and all relations in \(\Sigma\) are of length at most 3. In Corollary 1.7 it is proved that if \(\Gamma\) is a torsion-free, arithmetic lattice in \(\text{PGL}(2,\mathbb{C})\) then \(\log |H_1(\Gamma \backslash \mathbb{H}^3,\mathbb{Z})_{\text{tors}}| \ll \text{Vol}(\Gamma \backslash \mathbb{H}^3)\).
Let \(b_i(\Gamma \backslash X) := \text{dim}_{\mathbb{C}} H_i(\Gamma \backslash X, \mathbb{C})\) and let \(b_i^{(2)}(X)\) be the \(L^2\)-Betti numbers of \(X\). In the final Section 10.2 two results about the growth of Betti numbers are obtained. Let \((\Gamma_n)_{n\in\mathbb{N}}\) be a sequence of pairwise distinct congruence arithmetic co-compact torsion-free lattices in \(\text{PGL}(2,\mathbb{C})\). Then \(\underset{n \rightarrow \infty} \lim \frac{b_i(\Gamma_n \backslash X)}{\text{Vol}(\Gamma_n \backslash X)}=b_i^{(2)}(X)=0\) (Corollary 1.8), and Theorem 1.9: For any co-compact torsion-free arithmetic lattice \(\Gamma \subset \text{PGL}(2,\mathbb{C})\) with invariant trace field \(k\) we have \(\frac{b_1(\Gamma\backslash \mathbb{H}^3)}{\text{Vol}(\Gamma\backslash \mathbb{H}^3)} \ll [k:\mathbb{Q}]^{-1}\).