Normal Subgroups of Profinite Groups of Non-negative Deficiency

TL;DR

This paper studies profinite groups with non-negative deficiency, introducing p-deficiency to analyze their structure, especially when they contain finitely generated normal subgroups of infinite index, revealing strong cohomological and structural properties.

Contribution

It introduces the concept of p-deficiency for profinite groups and establishes structural results relating normal subgroups, cohomological dimensions, and Poincaré duality groups.

Findings

Positive p-deficiency implies specific cohomological dimension relations.

Normal subgroups of infinite index have constrained p-Sylow subgroup structures.

Results apply to profinite completions of certain group extensions and arithmetic groups.

Abstract

We initiate the study of profinite groups of non-negative deficiency. The principal focus of the paper is to show that the existence of a finitely generated normal subgroup of infinite index in a profinite group $G$ of non-negative deficiency gives rather strong consequences for the structure of $G$. To make this precise we introduce the notion of $p$-deficiency ($p$ a prime) for a profinite group $G$. This concept is more useful in the study of profinite groups then the notion of deficiency. We prove that if the $p$-deficiency of $G$ is positive and $N$ is a finitely generated normal subgroup such that the $p$-Sylow subgroup of $G/N$ is infinite and $p$ divides the order of $N$ then we have $\cd_p(G)=2$, $\cd_p(N)=1$ and $\vcd_p(G/N)=1$ for the cohomological $p$-dimensions; moreover either the $p$-Sylow subgroup of $G/N$ is virtually cyclic or the $p$-Sylow subgroup of $N$ is cyclic. A profinite Poincar\'e duality group $G$ of dimension 3 at a prime $p$ ($PD^3$-group) has deficiency 0. In this case we show that for $N$ and $p$ as above either $N$ is $PD^1$ at $p$ and $G/N$ is virtually $PD^2$ at $p$ or $N$ is $PD^2$ at $p$ and $G/N$ is virtually $PD^1$ at $p$. In particular if $G$ is pro-$p$ then either $N$ is infinite cyclic and $G/N$ is virtually Demushkin or $N$ is Demushkin and $G/N$ is virtually infinite cyclic. We apply this results to deduce structural information on the profinite completions of ascending HNN-extensions of free groups. We also give some implications of our theory to the congruence kernels of certain arithmetic groups.