One-ended subgroups of graphs of free groups with cyclic edge groups

TL;DR

This paper investigates the structure of one-ended hyperbolic groups, showing they are either surface groups or contain specific subgroups, with implications for limit groups and a characterization of boundary surfaces in free groups.

Contribution

It provides a new dichotomy for one-ended hyperbolic groups as graphs of free groups with cyclic edge groups, and characterizes boundary surfaces among free groups with peripheral structures.

Findings

Hyperbolic groups are either surface groups or contain special subgroups.

The results extend to limit groups.

Characterization of boundary surfaces in free groups.

Abstract

Consider a one-ended word-hyperbolic group. If it is the fundamental group of a graph of free groups with cyclic edge groups then either it is the fundamental group of a surface or it contains a finitely generated one-ended subgroup of infinite index. As a corollary, the same holds for limit groups. We also obtain a characterisation of surfaces with boundary among free groups equipped with peripheral structures.