Detecting ends of residually finite groups in profinite completions

TL;DR

This paper investigates how the number of ends of finitely generated residually finite groups relates to their profinite completions, revealing conditions under which properties are preserved or altered, and constructing examples of pathological groups.

Contribution

It establishes new results connecting the ends of residually finite groups with their profinite completions and constructs novel examples of groups with unusual properties.

Findings

If $G$ has more than one end, $H$ has the same number of ends.

For one-ended $G$, $H$ can have more ends, especially with finite $p$-groups.

Constructed groups show pathological behaviors like non-residual finiteness and property (T) despite profinite isomorphism.

Abstract

Let $\C$ be a variety of finite groups. We use profinite Bass--Serre theory to show that if $u:H\hookrightarrow G$ is a map of finitely generated residually $\C$ groups such that the induced map $\hat{u}:\hat{H}\rightarrow\hat{G}$ is a surjection of the pro-$\C$ completions, and $G$ has more than one end, then $H$ has the same number of ends as $G$. However if $G$ has one end the number of ends of $H$ may be larger; we observe cases where this occurs for $\C$ the class of finite $p$-groups. We produce a monomorphism of groups $u:H\hookrightarrow G$ such that: either $G$ is hyperbolic but not residually finite; or $\hat{u}:\hat{H}\rightarrow\hat{G}$ is an isomorphism of profinite completions but $H$ has property (T) (and hence (FA)), but $G$ has neither. Either possibility would give new examples of pathological finitely generated groups.