Volume versus rank of lattices

Abstract

We show the rank (i.e. minimal size of a generating set) of lattices cannot grow faster than the volume.

Betti numbers of finite volume orbifolds

“…Gelander has also proved that if torsion-free, then has a presentation for which the number of generators and the total length of the relations is bounded linearly by the volume of X= (with constants depending on X ). Recently, Gelander [9] extended this result and proved that even for lattices with torsion, the minimal number of generators of is bounded from above by a constant (depending on G ) times the volume of X= . In particular, this provides a linear bound to the first Betti number of .…”

On the number of finite subgroups of a lattice

“…Otherwise, the elements of the k − 1-iterated commutator, where k is the nilpotency rank of N , are central, and are parabolic. Similar considerations are made in [13].…”

Subgroup growth of right‐angled Artin and Coxeter groups

“…We will see here that this is not the case for lattices in three-dimensional hyperbolic space. There is still some relation, though not as precise and only in one direction: a result of T. Gelander [13] states that there is a constant C such that, if Γ Ă PSL 2 pCq is a discrete subgroup of finite covolume (a lattice-the result is proven more generally for all lattices in semisimple Lie groups) then it is generated by at most C volpΓzH 3 q elements. The subgroup growth rate is at most the number of gerenators minus one so we see that it is linearly bounded by the volume.…”