On largeness of hyperbolic discrete groups

E. B. Vinberg

A group $\Gamma$ is called large if there exists a finite index subgroup in $\Gamma$ admitting a homomorphism onto a non-abelian free group. In this case, for any $m$ there exists a finite index subgroup in $\Gamma$ admitting a homomorphism onto a free abelian group of rank $m$. It is known that any group that can be presented by $p$ generators and $q<p-1$ defining relations, is large.

It follows from an old result of D. Kazhdan that, for any lattice $\Gamma$ in a semisimple Lie group without factors locally isomorphic to $\mathrm{SO}(n,1)$ or $\mathrm{SU}(n,1)$, the group $\Gamma/(\Gamma,\Gamma)$ is finite and, hence, $\Gamma$ is not large. For example, the group $\mathrm{SL}(3,\Bbb Z)$ is not large. On the other hand, many discrete subgroups of the group $\mathrm{Isom}\, \mathrm{L}^n$ of motions of the $n$-dimensional Lobachevsky space $\mathrm{L}^n$ are known to be large. In particular, the following results were recently obtained:

1) Any standard arithmetic lattice in $\mathrm{SO}(n,1)$ is large [1].

2) Any subgroup of a standard arithmetic lattice in $\mathrm{SO}(n,1)$ is either large or virtually abelian [2].

3) Any finitely generated reflection subgroup in $\mathrm{Isom}\, \mathrm{L}^n$ is either large or virtually abelian [1,2].

4) Any subgroup of a finitely generated reflection subgroup in $\mathrm{Isom}\, \mathrm{L}^n$ is either large or virtually abelian [3].

It seems plausible that any discrete subgroup in $\mathrm{Isom}\, \mathrm{L}^n$ is either large or virtually abelian, but proving this requires new methods.