Abstract:We study the growth of dimHj(U,Fp), where U is an open subgroup of G∈L and scriptL is a special class of pro‐p groups defined in . Furthermore for G∈L non‐abelian we prove the core property: for pro‐p subgroups N≤H≤G such that H is finitely generated and N is non‐trivial normal in G the index [G:H] is always finite.
We study when a pro-p subdirect product S ⩽ G1 × . . . × Gn is of type FPm for m ⩾ 2 for some special pro-p groups Gi. In particular we treat the case when Gi is a finitely generated non-trivial free pro-p product different from C2 ∐ C2 if p = 2 or a non-abelian pro-p group from the class $\mathcal{L}$ defined in [12].
We study when a pro-p subdirect product S ⩽ G1 × . . . × Gn is of type FPm for m ⩾ 2 for some special pro-p groups Gi. In particular we treat the case when Gi is a finitely generated non-trivial free pro-p product different from C2 ∐ C2 if p = 2 or a non-abelian pro-p group from the class $\mathcal{L}$ defined in [12].
“…The purpose of this note is to investigate the following question which was raised by D.H. Kochloukova and P.A. Zalesskii in [14], where they answered the analogous question for pro-p limit groups.…”
Motivated by their study of pro-p limit groups, D.H. Kochloukova and P.A. Zalesskii formulated in [14, Remark after Thm. 3.3] a question concerning the minimum number of generators d(N ) of a normal subgroup N of index p in a non-abelian limit group G (cf. Question*). It is shown that the analogous question for the rational rank has an affirmative answer (cf. Thm. A). From this result one may conclude that the original question of D.H. Kochloukova and P.A. Zalesskii has an affirmative answer if the abelianization G ab of G is torsion free and d(G) = d(G ab ) (cf. Cor. B), or if G has the IF-property (cf. Thm C).
“…Some other works that study discrete and profinite groups through the numbers of generators of finite index subgroups are [1,2,9,15,16,17,18,23,26,29,30,31].…”
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