A Finitely Presented Torsion-free Simple Group

We construct a finitely presented torsion-free simple group Σ0, 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 Σ0 as an index 4 subgroup of a group Σ < Aut(T12)×Aut(T8) 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. 0. Introduction Burger-Mozes have constructed in [2, 4] the first examples of groups which are simultaneously finitely presented, torsion-free and simple. Moreover, they are CAT(0), bi-automatic, and have finite cohomological dimension. These groups can be realized in various ways: as fundamental groups of finite square complexes, as cocompact lattices in a product of automorphism groups of regular trees Aut(T2m)×Aut(T2n) for sufficiently large m,n ∈ N, or as amalgams of finitely generated free groups. The groups of Burger-Mozes have positively answered several open questions: for example Neumann’s question ([9]) on the existence of simple amalgams of finitely generated free groups, or a question of G. Mess (see [7, Problem 5.11 (C)]) on the existence of finite aspherical complexes with simple fundamental group. The construction is based on a “normal subgroup theorem” ([4, Theorem 4.1]) which shows for a certain class of irreducible lattices acting on a product of trees, that any non-trivial normal subgroup has finite index. This statement and its remarkable proof are adapted from the famous analogous theorem of Margulis ([8, Theorem IV.4.9]) in the context of irreducible lattices in higher rank semisimple Lie groups. Another important ingredient in the construction of BurgerMozes is a sufficient criterion ([4, Proposition 2.1]) for the non-residual finiteness of groups acting on a product of trees. Even the bare existence of such non-residually finite groups is remarkable, since for example finitely generated linear groups, or cocompact lattices in Aut(Tk) are always residually finite. The non-residually finite groups of Burger-Mozes coming from their criterion always do have non-trivial normal subgroups of infinite index, but appropriate embeddings into groups satisfying the normal subgroup theorem immediately lead to virtually simple groups. Unfortunately, these groups and their simple subgroups have very large finite presentations. We therefore modify the constructions by taking a small non-residually finite group of Wise ([11, Section II.5]), embed it into a group Σ < Aut(T12)×Aut(T8) satisfying the normal subgroup theorem, and detect a simple subgroup Σ0 < Σ of index 4. Several GAP-programs ([5]) have enabled us to find very quickly the groups Σ and Date: April 20, 2004. 1