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!

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

1 Answer 1


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 .

  • $\begingroup$ Should that be $|m| \ne |n| \ge 2$ ? $\endgroup$
    – Derek Holt
    Jan 15, 2011 at 12:18
  • $\begingroup$ Yes, I fixed that. $\endgroup$
    – user6976
    Jan 15, 2011 at 14:11
  • $\begingroup$ Can these groups act just as functions (not automorphisms)? $\endgroup$
    – dave
    Jan 15, 2011 at 20:49
  • $\begingroup$ Yes. Every countable group acts faithfully by permutations on every countable (and more than countable) set. $\endgroup$
    – user6976
    Jan 15, 2011 at 22:09
  • 1
    $\begingroup$ @Derek: I consider 'countable" to be of cardinality $\aleph_0$. $\endgroup$
    – user6976
    Jan 17, 2011 at 12:25

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