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 Γ_p,l originally studied by Mozes Israel J. Math. 90(1–3) (1995), 253–294]. It turns out that anti-tori in Γ_p,l directly correspond to non commuting pairs of Hamilton quaternions. Moreover, free anti-tori in Γ_p,l are related to free groups generated by two integer quaternions, and also to free subgroups of . As an application, we prove that the multiplicative group generated by the two quaternions 1+2i and 1+4k is not free.