Hyperbolic Graphs of Surface Groups

TL;DR

This paper establishes conditions under which the fundamental group of a reglued graph of surfaces is hyperbolic, providing new examples that differ from previously known surface-by-free groups.

Contribution

It introduces a sufficient condition for hyperbolicity of reglued graphs of surfaces and constructs novel examples not commensurate with surface-by-free groups.

Findings

Identified conditions ensuring hyperbolicity of fundamental groups

Constructed explicit examples with unique properties

Demonstrated differences from known hyperbolic groups

Abstract

We give a sufficient condition under which the fundamental group of a reglued graph of surfaces is hyperbolic. A reglued graph of surfaces is constructed by cutting a fixed graph of surfaces along the edge surfaces, then regluing by pseudo-Anosov homeomorphisms of the edge surfaces. By carefully choosing the regluing homeomorphism, we construct an example of such a reglued graph of surfaces, whose fundamental group is not abstractly commensurate to any surface-by-free group, i.e., which is different from all the examples given in Mosher's paper 'A hyperbolic-by-hyperbolic hyperbolic group'.