Profinite Groups

Abstract

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Free-by-finite pro-p groups and integral p-adic representations

“…Remark also that in both theorems Z p can be replaced by the field of p elements F p . Basic results on profinite groups, rings and modules used in the paper can be found in Ribes and Zalesski [10]. All groups in the paper are pro-p, all subgroups (including the commutator subgroup) are closed, all homomorphisms are continuous; r and n denote natural numbers, p a prime number, Z p the ring of p-adic integers, F p the field of p elements, C p r the cyclic group of order p r .…”

“…We need a pro-p version of the Kurosh Subgroup Theorem that can be found in Ribes and Zalesskii [10] (appendix D3 and Theorem D 3.8). Let G = i∈I G i be a free pro-p product of finitely many pro-p groups G i and B an open subgroup of G. Then G contains an abstract free product G abs = * i∈I G i as a dense subgroup of G and B abs := B ∩ G abs is a dense subgroup of B.…”

“…basis of a free abstract group F abs (Z), where j ∈ I is a fixed index (see Kurosh System in appendix D3 of [10]). By taking the pro-p completion of B abs one obtains the Kurosh type decomposition for B.…”

A virtually free pro-p need not be the fundamental group of a profinite graph of finite groups

“…Basic material on profinite groups can be found in [10,14]. The proofs of Theorems 1.1 and 1.2 rely on the profinite analogue of Bass-Serre theory which has been initiated by Gildenhuys and Ribes [3] and developed by O. V Mel'nikov and the second author of the present article [7,15,18,19].…”

“…The Frattini subgroup of G is denoted by Φ(G) and for H ≤ G we denote by |G : H| its index. By Tor(G) we mean the set of all torsion elements in G. For profinite groups we shall employ (standard) notations as found in [10] or [11].…”