Word fibres in finite p-groups and pro-p groups
Ainhoa Iniguez-Goizueta · Oxford University Research Archive (ORA) (University of Oxford) · 2016
Given a group word w in k variables, a group GM and g ∈ G, we consider the set Sw(g) of k-tuples (g1,..., gk) ∈ G(k) such that w(g1,..., gk) = g and when G is finite, the size of Sw(g), Nw(g). N. Amit conjectured that for any finite nilpotent group G and any word in k variables, Nw(1) ≥ |G|k-1. In this thesis we first prove Amit’s conjecture for finite groups of nilpotency class 2. This was independently proved by Levy in [1]. More generally, we study the class functions Nw for this class of groups and show that the inequality can be improved to Nw(1) ≥ |G|k/|Gw (Gw is the set of w-values in G) if G has odd order. This last result is explained by the fact that the functions Nw are characters of G in this case. For groups of even order, all that can be said is that Nw is a generalized character, something that is false in general for groups of nilpotency class greater than 2. We characterize group theoretically when Nxn is a character if G is a 2-group of nilpotency class 2. We also address the (much harder) problem of studying if Nw(g) ≥ |G|k-1 for g ∈ Gw, proving that this is the case for the free p-groups of nilpotency class 2 and exponent p. Finally, we look at the analogous problem for finitely generated pro-p groups. Let G be a finitely generated pro-p group and {Gn} some filtration. We define the dimension of a closed subset H ⊆ G as Dim{Gn}(H) = lim infn→∞ logp | HG(k)n/ G(k)n|logp | (G / Gn)(k) | · In this setting, a rather natural way to define the metric is by using the filtration Gn = /\xpn : x ∈ G\/. For this filtration, we ask whether for any word w in k variables, Dim{Gn} Sw(1) ≥ k - 1/k. We show that for free pro-p groups, using the filtration given by its dimension subgroups, this is not true in general.