Rips construction without unique product

Goulnara Arzhantseva and Markus Steenbock

Vol. 322 (2023), No. 1, 1–9

DOI: 10.2140/pjm.2023.322.1

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 hyperbolic group without the unique product property and $N$ is without the unique product property and has Kazhdan’s Property (T). Varying $Q$ yields a wide diversity of concrete examples of hyperbolic groups without the unique product property. We also note, as an application of Ol’shanskiĭ’s construction of torsion-free Tarski monsters, the existence of torsion-free Tarski monster groups without the unique product property.

Keywords

small cancellation theory, unique product property, Kaplansky's zero divisor conjecture, Tarski monsters

Mathematical Subject Classification

Primary: 20F06, 20F67

Milestones

Received: 21 July 2020

Revised: 23 November 2022

Accepted: 23 November 2022

Published: 3 May 2023