# Polygonal words in free groups

## TL;DR

This paper introduces the concept of polygonal words in free groups and demonstrates that their polygonality ensures the associated square complex contains a virtually embedded closed surface, advancing understanding of hyperbolic groups.

## Contribution

The paper defines polygonal words in free groups and proves their polygonality guarantees the existence of a virtually embedded closed surface in related complexes.

## Key findings

- Polygonality of certain words implies the presence of a virtually embedded surface.
- Many classes of words, including specific height 1 words, are shown to be polygonal.
- Polygonality is established for words without an isolated generator in rank 2.

## Abstract

A longstanding question of Gromov asks whether every one-ended word-hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. An infinite family of word-hyperbolic groups can be obtained by taking doubles of free groups amalgamated along words that are not proper powers. We define a set of polygonal words in a free group of finite rank, and prove that polygonality of the amalgamating word guarantees that the associated square complex virtually contains a $\pi_1$-injective closed surface. We provide many concrete examples of classes of polygonal words. For instance, in the case when the rank is 2, we establish polygonality of words without an isolated generator, and also of almost all simple height 1 words, including Baumslag--Solitar relator $a^p (a^q)^b$ for $pq\ne0$.