Poorly connected groups
Group Theory
2020-11-09 v2 Combinatorics
Abstract
We investigate groups whose Cayley graphs have poor\-ly connected subgraphs. We prove that a finitely generated group has bounded separation in the sense of Benjamini--Schramm--Tim\'ar if and only if it is virtually free. We then prove a gap theorem for connectivity of finitely presented groups, and prove that there is no comparable theorem for all finitely generated groups. Finally, we formulate a connectivity version of the conjecture that every group of type with no Baumslag-Solitar subgroup is hyperbolic, and prove it for groups with at most quadratic Dehn function.
Keywords
Cite
@article{arxiv.1904.04639,
title = {Poorly connected groups},
author = {David Hume and John M. Mackay},
journal= {arXiv preprint arXiv:1904.04639},
year = {2020}
}
Comments
14 pages. Changes to v2: Proof of the Theorem 1.2 shortened, Theorem 1.4 added completing the no-gap result outlined in v1