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Let $\mathbb{Z}_n$ denote the ring of the $n$-adic integers. I recently read a paper which used the fact that the Baumslag-Solitar groups BS($\pm$1,n) and BS(n,$\pm$1) can be realized as functions $\mathbb{Z}_n \rightarrow \mathbb{Z}_n$. Can BS(m,n) (for m and n arbitrary) be realized as a group of functions $\mathbb{Z}_r \rightarrow \mathbb{Z}_r$ for some $r$? Thanks!

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What sort of functions did you have in mind? Group automorphisms? Ring automorphisms? Continuous? –  Jim Belk Jan 15 '11 at 4:51
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up vote 6 down vote accepted

If you mean action by automorphisms, then the answer is "no" since the Baumslag-Solitar groups $BS(m,n)$, $|m|\ne |n|\ge 2$ are not residually finite. The groups $BS(m,n)$ do act nicely on the products of a tree and the Hyperbolic space: http://www.emis.de/journals/JLT/13-2/galpl.ps.gz .

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Should that be $|m| \ne |n| \ge 2$ ? –  Derek Holt Jan 15 '11 at 12:18
Yes, I fixed that. –  Mark Sapir Jan 15 '11 at 14:11
Can these groups act just as functions (not automorphisms)? –  dave Jan 15 '11 at 20:49
Yes. Every countable group acts faithfully by permutations on every countable (and more than countable) set. –  Mark Sapir Jan 15 '11 at 22:09
@Derek: I consider 'countable" to be of cardinality $\aleph_0$. –  Mark Sapir Jan 17 '11 at 12:25
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