Rips construction without unique product

TL;DR

This paper constructs specific Gromov hyperbolic groups lacking the unique product property, with some having Kazhdan's Property (T), expanding the landscape of examples in geometric group theory.

Contribution

It introduces a novel method to produce torsion-free Gromov hyperbolic groups without the unique product property, including Tarski monster groups, using short exact sequences.

Findings

Constructed Gromov hyperbolic groups without the unique product property.

Produced Tarski monster groups lacking the unique product property.

Demonstrated diversity of such groups by varying the quotient Q.

Abstract

Given a finitely presented group $Q,$ we produce a short exact sequence $1\to N \hookrightarrow G \twoheadrightarrow Q \to 1$ such that $G$ is a torsion-free Gromov hyperbolic group without the unique product property and $N$ is without the unique product property and has Kazhdan's Property (T). Varying $Q,$ we show a wide diversity of concrete examples of Gromov hyperbolic groups without the unique product property. As an application, we obtain Tarski monster groups without the unique product property.