Hyperbolic groups with logarithmic separation profile

TL;DR

This paper characterizes hyperbolic groups with logarithmic separation profiles, showing they split over cyclic groups and can be constructed from simpler groups, but not all such groups have this profile.

Contribution

It establishes a link between separation profiles and group splittings, and provides a counterexample illustrating limitations of the profile.

Findings

Hyperbolic groups with logarithmic separation profiles split over cyclic groups.

Such groups can be built from Fuchsian and free groups via amalgamations and HNN extensions.

Not all groups with a hierarchy have logarithmic separation profiles, demonstrated by a counterexample.

Abstract

We prove that hyperbolic groups with logarithmic separation profiles split over cyclic groups. This shows that such groups can be inductively built from Fuchsian groups and free groups by amalgamations and HNN extensions over finite or virtually cyclic groups. However, we show that not all groups admitting such a hierarchy have logarithmic separation profile by providing an example of a surface amalgam over a cyclic group with superlogarithmic separation profile.