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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!

flag	asked Jan 15 2011 at 4:26 dan 469●2●3

What sort of functions did you have in mind? Group automorphisms? Ring automorphisms? Continuous? – Jim Belk Jan 15 2011 at 4:51

Mark Sapir · Accepted Answer · 2011-01-15 14:11:32Z UTC

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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 .

edited Jan 15 2011 at 14:11

answered Jan 15 2011 at 9:51

Mark Sapir
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Should that be $|m| \ne |n| \ge 2$ ? – Derek Holt Jan 15 2011 at 12:18

Yes, I fixed that. – Mark Sapir Jan 15 2011 at 14:11

Can these groups act just as functions (not automorphisms)? – dave Jan 15 2011 at 20:49

Yes. Every countable group acts faithfully by permutations on every countable (and more than countable) set. – Mark Sapir Jan 15 2011 at 22:09

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@Derek: I consider 'countable" to be of cardinality $\aleph_0$. – Mark Sapir Jan 17 2011 at 12:25

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Realizing Baumslag-Solitar groups as functions of the $n$-adic integers

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