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If $G$ is a group which has a semigroup action on a regular rooted tree via prefix-preserving, continuous transformations (I give the tree the path metric), what kinds of algebraic restrictions can we put on $G$? I have seen a proof, for example, that any such group has decidable word problem, but do these groups need to be residually finite, for example? Thanks!

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Dan, can you define precisely what you mean by prefix preserving? Do you mean if f is the map and w=uv is a word, then f(uv)=f(u)x for some word x? –  Benjamin Steinberg Aug 29 '11 at 23:48
I would also like to know what a semigroup action of a group is. The identity does not act as an identity? –  Mark Sapir Aug 29 '11 at 23:52
@ Mark: I just mean that the identity does not need to pointwise fix the tree. The identity just needs to be an idempotent e such that eg=ge=g for all g. –  dan Aug 29 '11 at 23:58
@Ben: sorry for the lack of clarification. Yeah, I mean what you say above. –  dan Aug 29 '11 at 23:59
Then I would suggest first to look how an idempotent (1-element semigroup) can act on a tree preserving prefixes. If you know how the identity element acts you should be able to find out everything else. –  Mark Sapir Aug 30 '11 at 0:05

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