Anti-tori in square complex groups

An anti-torus is a subgroup $$ in the fundamental group of a compact non-positively curved space $X$, acting in a specific way on the universal covering space $\tilde{X}$ such that $a$ and $b$ do not have any commuting non-trivial powers. We construct and investigate anti-tori in a class of commutative transitive fundamental groups of finite square complexes, in particular for the groups $\Gamma_{p,l}$ originally studied by Mozes [15]. It turns out that anti-tori in $\Gamma_{p,l}$ directly correspond to non-commuting pairs of Hamilton quaternions. Moreover, free anti-tori in $\Gamma_{p,l}$ are related to free groups generated by two integer quaternions, and also to free subgroups of $\mathrm{SO}_3(\mathbb{Q})$. As an application, we prove that the multiplicative group generated by the two quaternions $1+2i$ and $1+4k$ is not free.Comment: 16 pages, some minor changes, this is the final versio

Abelian subgroup structure of square complex groups and arithmetic of quaternions

A square complex is a 2-complex formed by gluing squares together. This article is concerned with the fundamental group $\Gamma$ of certain square complexes of nonpositive curvature, related to quaternion algebras. The abelian subgroup structure of $\Gamma$ is studied in some detail.Comment: 13 page

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 was introduced by Burger and Mozes [M. Burger and S. Mozes. Finitely presented simple groups and products of trees. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), 747-752.], [M. Burger and S. Mozes. Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math. 92 (2001), 151-194.]. We refine their methods and construct Σ0 as an index 4 subgroup of a group presented by 10 generators and 24 short relations. For comparison, the smallest virtually simple group of [M. Burger and S. Mozes. Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math. 92 (2001), 151-194., Theorem 6.4] needs more than 18000 relations, and the smallest simple group constructed in [M. Burger and S. Mozes. Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math. 92 (2001), 151-194., §6.5] needs even more than 360000 relations in any finite presentatio

Anti-tori in Square Complex Groups

An anti-torus is a subgroup 〈a,b 〉 in the fundamental group of a compact non-positively curved space X, acting in a specific way on the universal covering space X such that a and b do not have any commuting nontrivial powers. We construct and investigate anti-tori in a class of commutative transitive fundamental groups of finite square complexes, in particular for the groups Γp,l originally studied by Mozes [Israel J. Math. 90(1-3) (1995), 253-294]. It turns out that anti-tori in Γp,l directly correspond to non commuting pairs of Hamilton quaternions. Moreover, free anti-tori in Γp,l are related to free groups generated by two integer quaternions, and also to free subgroups of $$SO_3(\mathbb{Q})$$ . As an application, we prove that the multiplicative group generated by the two quaternions 1+2i and 1+4k is not fre

Finite and infinite quotients of discrete and indiscrete groups

These notes are devoted to lattices in products of trees and related topics. They provide an introduction to the construction, by M. Burger and S. Mozes, of examples of such lattices that are simple as abstract groups. Two features of that construction are emphasized: the relevance of non-discrete locally compact groups, and the two-step strategy in the proof of simplicity, addressing separately, and with completely different methods, the existence of finite and infinite quotients. A brief history of the quest for finitely generated and finitely presented infinite simple groups is also sketched. A comparison with Margulis' proof of Kneser's simplicity conjecture is discussed, and the relevance of the Classification of the Finite Simple Groups is pointed out. A final chapter is devoted to finite and infinite quotients of hyperbolic groups and their relation to the asymptotic properties of the finite simple groups. Numerous open problems are discussed along the way.Comment: Revised according to referee's report; definition of BMW-groups updated; more examples added in Section 4; new Proposition 5.1