# characterization of residual finiteness for finitely generated groups

Where can I find a proof of the fact that a finitely generated group is residually finite if and only if it acts faithfully on a locally finite rooted tree?

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One way is actually quite easy - that every group acting faithfully on a locally finite rooted tree is residually finite - and can be found in a paper of Mark Sapir.

Indeed, every automorphism $f$ of the tree $T$ must fix the root and so it fixes the levels of the tree. If $f \neq 1$ on level number $n$, we consider the homomorphism from $\operatorname{Aut}(T)$ to the (finite) group of automorphisms of the finite tree consisting of the first $n$ levels of $T$. The homomorphisms restricting automorphisms of $T$ to vertices of levels at most $n$. The automorphism $f$ survives this homomorphism. Thus $\operatorname{Aut}(T)$ and all its subgroups are residually finite.

He then implies that the converse is a result of Kaluzhnin, and gives a reference as,

"R. I. Grigorchuk, V. V. Nekrashevich, V. I. Sushchanskii, Automata, dynamical systems, and groups. Tr. Mat. Inst. Steklova 231 (2000), Din. Sist., Avtom. i Beskon. Gruppy, 134–214"

However, I have more than just skimmed this paper (it is 76 pages long, and I couldn't find a searchable version) but I couldn't find the proof of this result.

(As a side note, the paper of Sapir is actually very interesting, and well worth a read. Although I found his proper paper of the result easier to follow...)

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