Rips-Segev torsion-free groups without the unique product property

TL;DR

This paper extends graphical small cancellation theory to free products, providing new examples of Gromov hyperbolic torsion-free groups lacking the unique product property, and explores their genericity among group presentations.

Contribution

It generalizes Gromov's small cancellation theorem to free products and constructs uncountably many non-isomorphic torsion-free groups without the unique product property.

Findings

First examples of Gromov hyperbolic groups without the unique product property.

Constructed uncountably many non-isomorphic torsion-free groups.

Showed these groups' presentations are not generic among finite group presentations.

Abstract

We generalize the graphical small cancellation theory of Gromov to a graphical small cancellation theory over the free product. We extend Gromov's small cancellation theorem to the free product. We explain and generalize Rips-Segev's construction of torsion-free groups without the unique product property by viewing these groups as given by graphical small cancellation presentations over the free product. Our graphical small cancellation theorem then provides first examples of Gromov hyperbolic groups without the unique product property. We construct uncountably many non-isomorphic torsion-free groups without the unique product property. We show that the presentations of generalized Rips-Segev groups are not generic among finite presentations of groups.