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!



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


