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Detecting ends of residually finite groups in profinite completions

Published online by Cambridge University Press:  22 August 2013

OWEN COTTON–BARRATT
OWEN COTTON–BARRATT*
Affiliation:
Mathematical Sciences, University of Southampton, Highfield, Southampton, SO17 1BJ. e-mail: o.cotton-barratt@soton.ac.uk
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Abstract

Let $\mathcal{C}$ be a variety of finite groups. We use profinite Bass--Serre theory to show that if u : H ↪ G is a map of finitely generated residually $\mathcal{C}$ groups such that the induced map û : Ĥ → Ĝ is a surjection of the pro-$\mathcal{C}$ 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 $\mathcal{C}$ the class of finite p-groups.

We produce a monomorphism of groups u : H ↪ G such that: either G is hyperbolic but not residually finite; or û : Ĥ → Ĝ 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 3 , November 2013 , pp. 379 - 389
Copyright
Copyright © Cambridge Philosophical Society 2013 

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