morphism from a compact group to Z ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T06:19:23Z http://mathoverflow.net/feeds/question/80966 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80966/morphism-from-a-compact-group-to-z morphism from a compact group to Z ? Florent MARTIN 2011-11-15T09:01:22Z 2011-11-16T17:11:01Z <p>I wonder if it there exists a topological compact group $G$ (by compact, I mean Hausdorff and quasi-compact) and a non-zero group morphism $\phi : G \to \mathbb{Z}$ (without assuming any topological condition on this morphism).</p> <p>For compact Lie groups, using the exponential map, the answers is no, but in general I don't know. </p> http://mathoverflow.net/questions/80966/morphism-from-a-compact-group-to-z/80968#80968 Answer by Andreas Thom for morphism from a compact group to Z ? Andreas Thom 2011-11-15T09:39:15Z 2011-11-16T17:11:01Z <p>The answer is no in general, but this is a rather deep fact.</p> <blockquote> <p><strong>Theorem:</strong> (Nikolov, Segal) If $G$ is any compact Hausdorff topological group, then every finitely generated (abstract) quotient of $G$ is finite.</p> </blockquote> <p>N. Nikolov and D. Segal, <em>Generators and commutators in finite groups; abstract quotients of compact groups</em>, arXiv, <a href="http://arxiv.org/abs/1102.3037" rel="nofollow">http://arxiv.org/abs/1102.3037</a></p> http://mathoverflow.net/questions/80966/morphism-from-a-compact-group-to-z/80978#80978 Answer by Alain Valette for morphism from a compact group to Z ? Alain Valette 2011-11-15T12:33:09Z 2011-11-15T12:45:28Z <p>Andreas shot first, but I still encourage everybody to have a look at the lemma on p.263 of R. Alperin,Locally compact groups acting on trees and property $T$. Monatsh. Math. 93 (1982), no. 4, 261–265: any homomorphism from a locally compact group to $\mathbb{Z}$, is continuous. This answers Florent's question.</p>