Realizing Baumslag-Solitar groups as functions of the $n$-adic integers - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T09:33:41Z http://mathoverflow.net/feeds/question/52146 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52146/realizing-baumslag-solitar-groups-as-functions-of-the-n-adic-integers Realizing Baumslag-Solitar groups as functions of the $n$-adic integers dan 2011-01-15T04:26:22Z 2011-01-15T14:11:32Z <p>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!</p> http://mathoverflow.net/questions/52146/realizing-baumslag-solitar-groups-as-functions-of-the-n-adic-integers/52156#52156 Answer by Mark Sapir for Realizing Baumslag-Solitar groups as functions of the $n$-adic integers Mark Sapir 2011-01-15T09:51:09Z 2011-01-15T14:11:32Z <p>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: <a href="http://www.emis.de/journals/JLT/13-2/galpl.ps.gz" rel="nofollow">http://www.emis.de/journals/JLT/13-2/galpl.ps.gz</a> .</p>