## On the images of profinite groups

### Jeudi 1er octobre 2009 14:00-15:00 - Nikolov Nikolai - Imperial College London

Résumé : Let G be a profinite group. What happens if we ignore the topology of G and consider it as an abstract group ? It is easy to construct ’pathological’ non-continuous homomorphisms of G even in the class of pro-p groups and so initially it looks that this question is unlikely to lead to any interesting theory.
However if we put the sensible restriction that G is topologically finitely generated then it turns out that the group structure of G is controlled by ts topology.
$G^q$ is an open subgroup of G for any integer q and hence its open subgroups coincide with those of finite index. It turns out that we have control over other images of G : For example when G is finitely generated prosoluble then the only perfect image of G is the trivial one and the only dense normal subgroups of $G$ contain $[G,G]$ . In this talk I will survey these and related results and show how they relate to width of words and conjugacy classes in families of finite groups.
Most of this relies on joint work with Dan Segal in Oxford.

Lieu : bât. 425 - 121-123

On the images of profinite groups  Version PDF
septembre 2020 :
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