Hyperbolic surface subgroups of one-ended doubles of free groups

TL;DR

This paper proves that certain one-ended doubles of free groups contain hyperbolic surface subgroups under specific conditions, advancing understanding of Gromov's question about hyperbolic groups.

Contribution

It establishes new conditions under which one-ended doubles of free groups contain hyperbolic surface subgroups, using combinatorial and polyhedral methods.

Findings

Hyperbolic surface subgroups exist in certain one-ended doubles of free groups.

Conditions include free group rank two or uniform generator usage in amalgamating words.

Proof involves Whitehead graphs and combinatorial induction.

Abstract

Gromov asked whether every one-ended word-hyperbolic group contains a hyperbolic surface group. We prove that every one-ended double of a free group has a hyperbolic surface subgroup if (1) the free group has rank two, or (2) every generator is used the same number of times in the amalgamating words. To prove this, we formulate a stronger statement on Whitehead graphs and prove its specialization by combinatorial induction for (1) and the characterization of perfect matching polytopes by Edmonds for (2).