Abelian subgroup structure of square complex groups and arithmetic of quaternions

Abstract

A square complex is a 2-complex formed by gluing squares together. This article is concerned with the fundamental group Γ of certain square complexes of nonpositive curvature, related to quaternion algebras. The abelian subgroup structure of Γ is studied in some detail. Before outlining the results, it is necessary to describe the construction of Γ. In [Moz, Section 3], there is constructed a lattice subgroup Γ = Γp,l of G = PGL2(Qp)×PGL2(Ql), where p, l ≡ 1 (mod 4) are two distinct primes. This restriction was made because −1 has a square root in Qp if and only if p ≡ 1 (mod 4), but the construction of Γ is generalized in [Rat, Chapter 3] to all pairs (p, l) of distinct odd primes. The affine building ∆ of G is a product of two homogeneous trees of degrees (p + 1) and (l + 1) respectively. The group Γ is a finitely presented torsion free group which acts freely and transitively on the vertices of ∆, with a finite square complex as quotient ∆/Γ. Here is how Γ is constructed. Let