[Paper Review] The profinite completion of a group localised at a subgroup

Colin D. Reid|arXiv (Cornell University)|Aug 17, 2012

Geometric and Algebraic Topology4 references3 citations

TL;DR

This paper introduces the profinite completion of a group localized at a commensurated subgroup, constructing a totally disconnected, locally compact (t.d.l.c.) group $ψ_G^K$ that embeds the original group $G$ densely and realizes the profinite completion of the subgroup $K$ as an open compact subgroup. The key contribution is a universal property: any homomorphism from $G$ to a t.d.l.c. group with profinite image of $K$ factors uniquely through this completion.

ABSTRACT

Let $G$ be a group and let $K$ be a commensurated subgroup of $G$. Then there is a totally disconnected, locally compact (t.d.l.c.) group $\hat{G}_K$ that contains the profinite completion of $K$ as an open compact subgroup and also contains $G$ (modulo the finite residual of $K$) as a dense subgroup. Moreover, given an arbitrary group $G$, then every t.d.l.c. group containing an image of $G$ as a dense subgroup can be realised as a quotient of $\hat{G}_K$ for some commensurated subgroup $K$.

Motivation & Objective

• To generalize the profinite completion construction to groups with commensurated subgroups, extending its utility beyond residually finite groups.
• To define a canonical t.d.l.c. group $ψ_G^K$ that captures the local profinite structure of $G$ at a commensurated subgroup $K$.
• To establish a universal property ensuring that every homomorphism from $G$ to a t.d.l.c. group with profinite image of $K$ factors uniquely through $ψ_G^K$.
• To clarify the relationship between the localised completion and standard profinite completions, especially for $K$ and subgroups of $\mathrm{Comm}_G(K)$.

Proposed method

• Define the localised profinite completion $\hat{G}_K$ as the completion of $G$ with respect to the filter of subgroups of $K$ of finite index, using equivalence classes of filters on $G$.
• Equip $\hat{G}_K$ with a group topology generated by basic open sets $B(gN) = \{[f] \in \hat{G}_K \mid gN \in f\}$ for $g \in G$, $N \in \mathcal{F}(K)$, making it a t.d.l.c. group.
• Construct a canonical homomorphism $\theta: G \to \hat{G}_K$ whose image is dense and such that $\theta|_K$ maps $K$ continuously onto an open profinite subgroup $\overline{\theta(K)}$.
• Prove the universal property: for any continuous homomorphism $\phi: G \to R$ with $\overline{\phi(K)}$ profinite, there exists a unique continuous $\psi: \hat{G}_K \to R$ such that $\phi = \psi \circ \theta$.
• Show that $\theta|_K$ induces the standard profinite completion of $K$, so $\hat{K}_K = \hat{K}$, and that $\theta$ is injective if and only if $K$ is open in $G$.
• Establish that $\hat{G}_K$ is independent of the choice of commensurate subgroup: if $K$ and $L$ are commensurate, then $\hat{G}_K = \hat{G}_L$.

Experimental results

Research questions

• RQ1 How can the profinite completion construction be generalized to groups that are not residually finite but contain a commensurated subgroup with rich finite quotients?
• RQ2 What universal property characterizes the t.d.l.c. group that arises as the completion of $G$ localized at a commensurated subgroup $K$?
• RQ3 To what extent does the localised completion $\hat{G}_K$ encode the structure of $G$ and its homomorphisms into t.d.l.c. groups?
• RQ4 How does the localised completion relate to the standard profinite completion of $K$ and to the commensurator $\mathrm{Comm}_G(K)$?
• RQ5 Under what conditions is the homomorphism $\theta: G \to \hat{G}_K$ continuous or an isomorphism?

Key findings

• The localised profinite completion $\hat{G}_K$ is a t.d.l.c. group that contains $G$ as a dense subgroup and the profinite completion of $K$ as an open compact subgroup.
• The homomorphism $\theta: G \to \hat{G}_K$ is universal: any continuous homomorphism $\phi: G \to R$ to a t.d.l.c. group $R$ with $\overline{\phi(K)}$ profinite factors uniquely through $\theta$ via a continuous $\psi: \hat{G}_K \to R$.
• The restriction $\theta|_K$ induces the standard profinite completion of $K$, so $\hat{K}_K = \hat{K}$, and $\overline{\theta(K)}$ is isomorphic to $\hat{K}$ as a topological group.
• The kernel of $\theta$ is the intersection of all closed subgroups of $K$ of finite index, and $\theta^{-1}(\overline{\theta(N)}) = N$ for every such $N$.
• If $K$ and $L$ are commensurate subgroups of $G$, then $\hat{G}_K = \hat{G}_L$, meaning the completion depends only on the commensurability class of $K$.
• When $G$ is discrete and $K$ is commensurated, the completion $\hat{G}_K$ accounts for all homomorphisms from $G$ to t.d.l.c. groups via the universal property, and $\theta$ is continuous if and only if $K$ is open in $G$.