Free actions of abelian p-groups on the n-Torus

• Autor(a) principal: Gonçalves, Daciberg Lima
• Data de Publicação: 2005
• Outros Autores: Vieira, João Peres [UNESP]
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Institucional da UNESP
• Texto Completo: http://hdl.handle.net/11449/224509
• topic: Bieberbach groups; Free actions; Integral representation; p-groups

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.