Geometricity and Polygonality in Free Groups

TL;DR

This paper explores the relationship between virtual geometricity and polygonality in free groups, showing that virtual geometricity implies polygonality after certain subgroup adjustments, with counterexamples demonstrating the converse does not hold.

Contribution

It establishes that virtual geometricity implies polygonality up to finite-index subgroup and automorphism, clarifying the connection between these properties in free groups.

Findings

Virtual geometricity implies polygonality after subgroup and automorphism adjustments.

Counterexamples show the converse implication does not hold.

Provides a deeper understanding of surface group embeddings in free group doubles.

Abstract

Gordon and Wilton recently proved that the double D of a free group F amalgamated along a cyclic subgroup C of F contains a surface group if a generator w of C satisfies a certain 3-manifold theoretic condition, called virtually geometricity. Wilton and the author defined the polygonality of w which also guarantees the existence of a surface group in D. In this paper, virtual geometricity is shown to imply polygonality up to descending to a finite-index subgroup F' and applying an automorphism on F'. That the converse does not hold will follow from an example formerly considered by Manning.