2004/11/24 by Diego Rattaggi, Rattaggi, Diego
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA) #math.GR #math.RA
paper · pdf · doi:10.48550/arxiv.math/0411547
16 pages, some minor changes, this is the final version
arxiv created 2005/07/21 · arxiv updated 2009/12/01
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 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 Γp,l originally studied by Mozes [15]. 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 SO3(ℚ). As an application, we prove that the multiplicative group generated by the two quaternions 1+2i and 1+4k is not free.