Poorly connected groups

Hume, David; Mackay, John

doi:10.1090/proc/15128

Poorly connected groups
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by David Hume and John M. Mackay

Proc. Amer. Math. Soc. 148 (2020), 4653-4664

DOI: https://doi.org/10.1090/proc/15128

Published electronically: August 14, 2020

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Abstract:

We investigate groups whose Cayley graphs have poorly connected subgraphs. We prove that a finitely generated group has bounded separation in the sense of Benjamini–Schramm–Timár 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 Baumslag–Solitar subgroup is hyperbolic, and prove it for groups with at most quadratic Dehn function.

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