  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.
