Anti-tori in square complex groups

Details Anti-tori in square complex groups Anti-tori in square complex groups

2004-11-24

arxiv.org/abs/math/0411547v2

Mathematics

Group Theory

16 pages, some minor changes, this is the final version

Scientific paper

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.

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