Resumen de The p-period of an infinite group
For G a group of finite virtual cohomological dimension and a prime p, the p-period of G is defined to be the least positive integer d such that Farrell cohomology groups Hi(G; M) and Hi+d(G; M) have naturally isomorphic ZG modules M.
We generalize a result of Swan on the p-period of a finite p-periodic group to a p-periodic infinite group, i.e., we prove that the p-period of a p-periodic group G of finite vcd is 2LCM(|N(áxñ) / C(áxñ)|) if the G has a finite quotient whose a p-Sylow subgroup is elementary abelian or cyclic, and the kernel is torsion free, where N(-) and C(-) denote normalizer and centralizer, áxñ ranges over all conjugacy classes of Z/p subgroups. We apply this result to the computation of the p-period of a p-periodic mapping class group. Also, we give an example to illustrate this formula is false without our assumption.
