MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.