# Poorly connected groups

## TL;DR

This paper characterizes finitely generated groups with bounded separation as virtually free, establishes a gap theorem for finitely presented groups' connectivity, and explores a conjecture relating hyperbolicity to group properties.

## Contribution

It provides a characterization of groups with bounded separation, proves a new gap theorem for finitely presented groups, and formulates and verifies a connectivity conjecture for certain hyperbolic groups.

## Key findings

- Finitely generated groups with bounded separation are exactly virtually free groups.
- A gap theorem for connectivity of finitely presented groups is established.
- The connectivity conjecture is proved for groups with at most quadratic Dehn function.

## 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 $F$ with no Baumslag-Solitar subgroup is hyperbolic, and prove it for groups with at most quadratic Dehn function.