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English

Anti-tori in square complex groups

ContributorsRattaggi, Diego
Published inGeometriae dedicata, vol. 114, no. 1, p. 189-207
Publication date2005
Abstract

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 $ 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.

Classification
  • arxiv : math.GR
Note16 pages, some minor changes, this is the final version
Citation (ISO format)
RATTAGGI, Diego. Anti-tori in square complex groups. In: Geometriae dedicata, 2005, vol. 114, n° 1, p. 189–207. doi: 10.1007/s10711-005-5538-9
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Article (Accepted version)
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ISSN of the journal0046-5755
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