Conjugacy problem in groups with quadratic Dehn function

TL;DR

This paper constructs a finitely presented group with a quadratic Dehn function that has an undecidable conjugacy problem, solving a long-standing problem posed by E. Rips in 1992.

Contribution

It provides the first example of such a group, demonstrating the independence of Dehn function growth and conjugacy problem decidability.

Findings

Finitely presented group with quadratic Dehn function

Undecidable conjugacy problem in the constructed group

Solves E. Rips' problem from 1992

Abstract

We construct a finitely presented group with quadratic Dehn function and undecidable conjugacy problem. This solves E. Rips' problem formulated in 1992. v2: misprints corrected. v3: lemmas 4.7, 4.10 corrected, more misprints fixed.