The virtual cohomological dimension of profinite groups
The chapter begins with a detailed description of tensor products of complexes of modules and the tensor product induction for a complex. It contains a theorem of Serre that asserts that torsion-free virtually free pro- p groups are free pro- p, as well as an extension due to Scheiderer of this result when the group contains torsion. A second-countable pro- p group with a free pro- p subgroup of index p is described as a free pro- p product of a free pro- p group and a continuously indexed family of groups of the form Hτ× Tτ, where Hτ is free pro- p and Tτ has order p. The chapter also includes an example of a subgroup of a free product of pro- p groups which does not admit a description along the lines of the classical Kurosh subgroup theorem. The last part of this chapter deals with the subgroup of fixed points FixF(ψ) of an automorphism ψ of a free pro- p group F: if the order of ψ is a finite power of p, the rank of that subgroup is finite, and if the order of ψ is prime to p, its rank is infinite.
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Ribes, L. (2017). The virtual cohomological dimension of profinite groups. In Ergebnisse der Mathematik und ihrer Grenzgebiete. doi:10.1007/978-3-319-61199-0_10