Hyperbolic groups with logarithmic separation profile
Nir Lazarovich, Corentin Le Coz
TL;DR
This work characterizes hyperbolic groups with logarithmic-scale separation profiles, proving that such groups necessarily split over finite or virtually cyclic subgroups or are Fuchsian, and hence can be built from Fuchsian and free groups via amalgamations and HNN extensions over finite or virtually cyclic edge groups. The proof hinges on converting a bound on $\mathrm{sep}_G(n)$ into bounded spheres separation, then into a finite topological separator of the boundary, and finally applying Bowditch's boundary theory to deduce a splitting. The paper also demonstrates the sharpness of the converse by constructing a surface amalgam whose universal cover has superlogarithmic separation, illustrating that conformal dimension one does not force logarithmic separation. Additional discussion connects separation profiles to Poincaré inequalities in the boundary and raises questions about sharpness and behavior for non-filling gluings. Overall, the results illuminate the interplay between coarse geometric invariants, boundary topology, and hierarchical decompositions in hyperbolic groups.
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.