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arXiv · 2608.19101

Growth gaps and generating sets

Aleksander Skenderi·Gal Yehuda
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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 $Λ$ in a higher rank semisimple Lie group $G$ with Kazhdan's property (T), the group $Λ\times Λ$ 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.

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Aleksander Skenderi, Gal Yehuda. 2026-08-19. Growth gaps and generating sets. https://arxiv.org/abs/2608.19101

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