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Title 
Antitori in square complex groups 

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Published in  Geometriae Dedicata. 2005, vol. 114, no. 1, p. 189207  
Abstract  An antitorus is a subgroup $<a,b>$ in the fundamental group of a compact nonpositively 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 nontrivial powers. We construct and investigate antitori 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 antitori in $Gamma_{p,l}$ directly correspond to noncommuting pairs of Hamilton quaternions. Moreover, free antitori 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://archiveouverte.unige.ch/unige:12148  
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arXiv: math/0411547 

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