Counting commensurability classes of hyperbolic manifolds

Abstract

Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about v v such manifolds of volume at most v, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, almost all hyperbolic manifolds are non-arithmetic in an appropriate sense. Moreover, by restricting attention to non-compact manifolds, our result implies the same growth type for the number of quasiisometry classes of lattices in SO(n, 1). Our method involves a geometric graph-of-spaces construction that relies on arithmetic properties of certain quadratic forms.

Embedding arithmetic hyperbolic manifolds

“…Indeed, let C n (v) denote the number of commensurability classes of hyperbolic n-manifolds admitting a representative of volume ≤ v, and B n (v) be the number of such classes represented by a geometric boundary of volume ≤ v. Of course B n (v) ≤ C n (v) ≤ µ n (v). As shown by Gelander and Levit [14], for all n ≥ 2 we have C n (v) ≥ v cv , for v large enough. By following their arguments and applying Theorem 1.1, we prove the following:…”

“…Proof of Theorem 1.3 (counting geometric boundaries). In this section we follow the idea by Gelander and Levit from [14].…”

“…By Theorem 2.2, for every x we can embed S geodesically into some orientable arithmetic M x = H n /Γ x of simplest type with Γ x ⊂ O(f x , k). We now apply [14,Proposition 4.3] in order to build orientable manifolds M x such that:…”

On volumes of quasi-arithmetic hyperbolic lattices

“…(2) Quasi-arithmetic lattices; Theorem 1.3 shows that up to commensurability their covolumes are the same as in (1). (3) Lattices that come from the interbreeding constructions (see Gromov and Piateski-Shapiro [14], and generalizations [22,12]); note that those are not quasi-arithmetic [24, Theorem 1.6]. From their construction and Corollary 1.5 we obtain that their volumes are rational linear combinations of volumes from (1), in any case (the result being already clear when the construction only involves nonseparating hypersurfaces).…”

On the Growth of L2-Invariants of Locally Symmetric Spaces, II: Exotic Invariant Random Subgroups in Rank One

“…The analysis of this section has been an inspiration for some later works regarding counting manifolds, which have already been published [48,28].…”