Hyperbolic diagram groups are free

TL;DR

This paper proves that hyperbolic diagram groups are free by showing they cannot contain a subgroup isomorphic to , providing a clear criterion for the freeness of diagram groups.

Contribution

It establishes a precise condition for when diagram groups are free, specifically linking hyperbolicity to the absence of  subgroups.

Findings

Hyperbolic diagram groups are necessarily free.

Diagram groups are free iff they contain no  subgroup.

Answer to Guba and Sapir's question on hyperbolic diagram groups.

Abstract

In this paper, we study the so-called diagram groups. Our main result is that diagram groups are free if and only if they do not contain any subgroup isomorphic to $\mathbb{Z}^2$. As an immediate corollary, we get that hyperbolic diagram groups are necessarily free, answering a question of Guba and Sapir.