Let $\mathbb{Z}_n$ denote the ring of the $n$adic integers. I recently read a paper which used the fact that the BaumslagSolitar 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!

If you mean action by automorphisms, then the answer is "no" since the BaumslagSolitar 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/132/galpl.ps.gz . 

