On largeness of hyperbolic discrete groups

E. B. Vinberg

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

It follows from an old result of D. Kazhdan that, for any lattice in a semisimple Lie group without factors locally isomorphic to or , the group is finite and, hence, is not large. For example, the group is not large. On the other hand, many discrete subgroups of the group of motions of the -dimensional Lobachevsky space are known to be large. In particular, the following results were recently obtained:

1) Any standard arithmetic lattice in is large [1].

2) Any subgroup of a standard arithmetic lattice in is either large or virtually abelian [2].

3) Any finitely generated reflection subgroup in is either large or virtually abelian [1,2].

4) Any subgroup of a finitely generated reflection subgroup in is either large or virtually abelian [3].

It seems plausible that any discrete subgroup in is either large or virtually abelian, but proving this requires new methods.