Betti numbers of Shimura curves and arithmetic three-orbifolds

Abstract

We show that asymptotically the first Betti number b1 of a Shimura curve satisfies the Gauss-Bonnet equality 2π(b1 − 2) = vol where vol is hyperbolic volume; equivalently 2g−2 = (1+o(1)) vol where g is the arithmetic genus. We also show that the first Betti number of a congruence hyperbolic 3-orbifolds asymptotically vanishes relatively to hyperbolic volume, that is b1/ vol → 0. This generalises results from [1] and [10] and we rely on results and techniques from these works, most importantly the notion of Benjamini-Schramm convergence of locally symmetric spaces.

Homotopy type and homology versus volume for arithmetic locally symmetric spaces

“…The present work is in part a follow-up to the work of the first author [33] where convergence was proven to hold for all torsion-free lattices in PGL 2 (C) and PGL 2 (R), and subsequent work with the last author [34] dealing with the presence of torsion elements. Using the same approach as in these papers we show that Benjamini-Schramm convergence holds for sequences of lattices where the degree of their trace field is bounded.…”

“…We expect that with a better understanding of the conical singularities 6 in locally symmetric spaces, our proof of Gelander's conjecture for manifolds could be extended to orbifolds. In dimension 3 results in this direction are given in [34], and the techniques of [5] might also be relevant.…”

“…In the case where the degree of the trace field is bounded we use the same approach as in [33,34], combining the notion of Benjamini-Schramm convergence of locally symmetric spaces introduced in [2] with an estimate on the geometric side of Selberg's trace formula [7]. Following [33], we can estimate the volume of the thin part by the trace of a certain convolution operator.…”

“…These bounds are quite delicate and difficult to obtain for conjugacy classes with large centralizers. Using a reduction argument adapted from [34] (see Theorem 8.3 and Lemma 10.8 for precise statements) we can restrict W to be the set of highly regular conjugacy classes and still get the conclusion about the Benjamin-Schramm convergence. This part of the argument uses the Zariski density of invariant random subgroups proved in [2].…”