Strong Limit Multiplicity for arithmetic hyperbolic surfaces and $3$-manifolds

TL;DR

This paper proves a strong form of the Limit Multiplicity property for sequences of arithmetic hyperbolic surfaces and 3-manifolds, with applications to their geometric structure and homotopy types.

Contribution

It establishes a quantitative Limit Multiplicity property for arithmetic lattices and confirms Gelander's conjecture on the homotopy types of arithmetic hyperbolic 3-manifolds.

Findings

Volume of the thin part grows at most as Vol(M)^{11/12}

Every arithmetic hyperbolic 3-manifold is homotopy equivalent to a bounded-degree simplicial complex

Strong Limit Multiplicity property holds for sequences of congruence lattices

Abstract

We show that every sequence of torsion-free arithmetic congruence lattices in $\mathrm{PGL}(2,\mathbb R)$ or $\mathrm{PGL}(2,\mathbb C)$ satisfies a strong quantitative version of the Limit Multiplicity property. We deduce that for $R>0$ in certain range, growing linearly in the degree of the invariant trace field, the volume of the $R$-thin part of any congruence arithmetic hyperbolic surface or congruence arithmetic hyperbolic $3$-manifold $M$ is of order at most $\mathrm{Vol}(M)^{11/12}$. As an application we prove Gelander's conjecture on homotopy type of arithmetic hyperbolic $3$-manifolds: We show that there are constants $A,B$ such that every such manifold $M$ is homotopy equivalent to a simplicial complex with at most $A\mathrm{Vol}(M)$ vertices, all of degrees bounded by $B$.