Poorly connected groups
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 BenjaminiSchrammTimár 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 $F$ with no BaumslagSolitar subgroup is hyperbolic, and prove it for groups with at most quadratic Dehn function.
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 DOI:
 10.48550/arXiv.1904.04639
 arXiv:
 arXiv:1904.04639
 Bibcode:
 2019arXiv190404639H
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Combinatorics;
 20F65 (Primary);
 05C40;
 20E05;
 20F67 (Secondary)
 EPrint:
 14 pages. Changes to v2: Proof of the Theorem 1.2 shortened, Theorem 1.4 added completing the nogap result outlined in v1