The rates of growth in a hyperbolic group

Koji Fujiwara, Zlil Sela

Research output: Contribution to journalArticlepeer-review

Abstract

We study the countable set of rates of growth of a hyperbolic group with respect to all its finite generating sets. We prove that the set is well-ordered, and that every real number can be the rate of growth of at most finitely many generating sets up to automorphism of the group. We prove that the ordinal of the set of rates of growth is at least ω0ω0 , and in case the group is a limit group (e.g. free and surface groups) it is ω0ω0 . We further study the rates of growth of all the finitely generated subgroups of a hyperbolic group with respect to all their finite generating sets. This set is proved to be well-ordered as well, and every real number can be the rate of growth of at most finitely many isomorphism classes of finite generating sets of subgroups of a given hyperbolic group. Finally, we strengthen our results to include rates of growth of all the finite generating sets of all the subsemigroups of a hyperbolic group.

Original languageAmerican English
Pages (from-to)1427-1470
Number of pages44
JournalInventiones Mathematicae
Volume233
Issue number3
DOIs
StatePublished - Sep 2023

All Science Journal Classification (ASJC) codes

  • General Mathematics

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