morphism from a compact group to Z ? - MathOverflow

morphism from a compact group to Z ?

2011-11-15T09:01:22Z

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).

For compact Lie groups, using the exponential map, the answers is no, but in general I don't know.

Answer by Andreas Thom for morphism from a compact group to Z ?

2011-11-15T09:39:15Z

The answer is no in general, but this is a rather deep fact.

Theorem: (Nikolov, Segal) If $G$ is any compact Hausdorff topological group, then every finitely generated (abstract) quotient of $G$ is finite.

N. Nikolov and D. Segal, Generators and commutators in finite groups; abstract quotients of compact groups, arXiv, http://arxiv.org/abs/1102.3037

Answer by Alain Valette for morphism from a compact group to Z ?

2011-11-15T12:33:09Z

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.