Free actions of abelian p-groups on the n-Torus
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Abstract
In this work we make some contributions to the theory of actions of abelian p-groups on the n-Torus Tn. Set H ≅ ℤpk1 h1 × ℤpk2h2 × ⋯ × ℤpkrhr, r ≥ 1, k1 ≥ k2 ≥ ⋯ ≥ kr ≥ 1, p prime. Suppose that the group H acts freely on Tn and the induced representation on π 1(Tn) ≅ ℤn is faithful and has first Betti number b. We show that the numbers n, p, b, ki and h i (i = 1, ⋯ , r) satisfy some relation. In particular, when H ≅ ℤph, the minimum value of n is φ(p) + b when b ≥ 1. Also when H ≅ ℤpk1, × ℤp the minimum value of n is φ(pk1)+ p - 1 + 6 for 6 ≥ 1. Here φ denotes the Euler function. © 2005 University of Houston.
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Keywords
Bieberbach groups, Free actions, Integral representation, p-groups
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English
Citation
Houston Journal of Mathematics, v. 31, n. 1, p. 87-102, 2005.
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