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Anti-tori in square complex groups

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Published in Geometriae Dedicata. 2005, vol. 114, no. 1, p. 189-207
Abstract 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 $ ilde{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.
Note 16 pages, some minor changes, this is the final version
Stable URL https://archive-ouverte.unige.ch/unige:12148
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Deposited on : 2010-10-19

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