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

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!

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

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 .

• Should that be $|m| \ne |n| \ge 2$ ? Jan 15, 2011 at 12:18
• Yes, I fixed that.
– user6976
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.
– user6976
Jan 15, 2011 at 22:09
• @Derek: I consider 'countable" to be of cardinality $\aleph_0$.
– user6976
Jan 17, 2011 at 12:25