Counting arithmetic lattices and surfaces

Abstract

We give estimates on the number AL H .x/ of conjugacy classes of arithmetic lattices of covolume at most x in a simple Lie group H . In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most x. Our main result is for the classical case H D PSL.2; ‫/ޒ‬ where we show thatThe proofs use several different techniques: geometric (bounding the number of generators of as a function of its covolume), number theoretic (bounding the number of maximal such ) and sharp estimates on the character values of the symmetric groups (to bound the subgroup growth of ).

Volume versus rank of lattices

“…Corollary 1.4 was proved in [4] for rank one groups (see [4], Theorem 1.1). Using Theorem 1.1 instead of the rank one case given in [4], Corollary 1.4 is proved by the argument given in [4] also for higher rank groups.…”

“…When considering the class of finite index subgroups G in some given finitely generated group D, it is easy to show that dðGÞ is at most À dðDÞ À 1 Á ½D : G þ 1 and in particular bounded linearly by the index. We prove the analog statement when D is replaced by a connected semisimple Lie group G. for every irreducible lattice G < G. Theorem 1.1 for rank one groups was proved in [4], Section 2. For torsion free lattices Theorem 1.1 was proved in [10], where also bounds and the number of short relations were given.…”

“…Using Theorem 1.1 instead of the rank one case given in [4], Corollary 1.4 is proved by the argument given in [4] also for higher rank groups. It was also shown in [4] that for SOðn; 1Þ this estimate is sharp in the sense that AL SOðn; 1Þ ðvÞ f v av for some constant a > 0 and v su‰-ciently large, and for G ¼ SLð2; RÞ the following precise asymptotic formula was given:…”

Embedding arithmetic hyperbolic manifolds

“…The manifolds that we build in order to prove Theorem 1.3 are non-arithmetic. Indeed, there is an upper bound of the form v b(log v) (and v b in the compact case) for the growth rate of commensurabilty classes of arithmetic hyperbolic manifolds of any dimension n ≥ 2 [4,6]. In other words, "most" hyperbolic manifolds are non-arithmetic.…”

Families of mutually isospectral Riemannian orbifolds

“…In fact, our result is somewhat stronger in the sense that we prove that within a fixed commensurability class, the set of all maximal arithmetic subgroups which have covolume less than V is bounded by a function polynomial in V . Although the existence of such a bound is implicit in the proof of Theorem 1.6 of Belolipetsky, Gelander, Lubotzky and Shalev [1], no explicit bound is currently known. To that end, we prove the following theorem.…”