Abstract
AbstractLet $\mathcal{C}$ 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.
Publisher
Cambridge University Press (CUP)
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