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Let $T$ be a regular tree, and suppose that $G \leq \mathrm{Aut}(T)$ has finite quotient graph, $T / G$. Is it true (in general) that $G$ will have trivial centralizer in the full automorphism group? If not, are there more specialized situations when this is true?

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    Nope, that's$a$perfectly good counter-example! A result along these lines is used in a paper I'm reading, but it's not backed up, so perhaps it's just that there need to be more restraints in the statement.2012-11-06

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