Growth gaps and generating sets

Abstract

We show that the existence of a growth gap for infinite-index subgroups of a given finitely genrated group can depend on the finite generating set. More precisely, for any irreducible lattice $\Lambda$ in a higher rank semisimple Lie group $G$ with Kazhdan's property (T), the group $\Lambda \times \Lambda$ admits one finite symmetric generating set with a growth gap and another without a growth gap. We also prove that the growth gap can be made arbitrarily small. In contrast, for a non-elementary hyperbolic group the existence of a growth gap is independent of the finite generating set.