Anti-tori in Square Complex Groups |
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Authors: | Email author" target="_blank">Diego?RattaggiEmail author |
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Institution: | (1) Section de Mathématiques, Université de Genève, 2-4 Rue du Lièvre, CH-1211, CP 64, Genève 4, Switzerland |
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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 X such that a and b do not have any commuting nontrivial 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 Israel J. Math. 90(1–3) (1995), 253–294]. 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
. 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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Keywords: | anti-torus square complex quaternion free subgroup commutative transitive |
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