A finitely presented torsion-free simple group

Abstract

We construct a finitely presented torsion-free simple group $\Sigma_0$, acting cocompactly on a product of two regular trees. An infinite family of such groups has been introduced by Burger-Mozes ([2,4]). We refine their methods and get $\Sigma_0$ as an index 4 subgroup of a group $\Sigma < \mathrm{Aut}(\mathcal{T}_{12}) \times \mathrm{Aut}(\mathcal{T}_{8})$ presented by 10 generators and 24 short relations. For comparison, the smallest virtually simple group of [4, Theorem 6.4] needs more than 18000 relations, and the smallest simple group constructed in [4, Section 6.5] needs even more than 360000 relations in any finite presentation.