Surface subgroups from homology

TL;DR

This paper demonstrates that certain hyperbolic groups contain surface subgroups when their second homology is nontrivial, and describes the geometric structure of their Gromov-Thurston norm unit ball.

Contribution

It proves the existence of surface subgroups in specific hyperbolic groups based on homological properties and characterizes the Gromov-Thurston norm unit ball as a finite-sided rational polyhedron.

Findings

Hyperbolic groups with nonzero second homology contain surface subgroups.

The Gromov-Thurston norm unit ball is a finite-sided rational polyhedron.

Provides a homological criterion for surface subgroup existence.

Abstract

Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H_2(G;Q) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov-Thurston norm on H_2(G;R) is a finite-sided rational polyhedron.