Resumen de Free Actions of Abelian P-Groups on the N -Torus

Daciberg Lima Gonçalves, Joao Peres Vieira

• 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 × (Zpk2)h2 × ¿ × (Zpkr)hr, r = 1, k1 = k2 = ¿ = kr = 1, p prime. Suppose that the group H acts freely on Tn and the induced representation on ?1(Tn) ¿ Zn is faithful and has first Betti number b. We show that the numbers n, p, b, ki and hi, (i=1,¿,r) satisfy some relation. In particular, when H ¿ (Zp)h, the minimum value of n is F(p)+b when b = 1. Also when H ¿ Zpk1 × Zp the minimum value of n is F(pk1)+p-1+b for b = 1. Here F denotes the Euler function.