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Jun 25, 2013 at 13:37 history edited YCor CC BY-SA 3.0
added reference for Nikolov-Segal's example
Jun 5, 2012 at 19:36 comment added YCor @Benjamin: some compact groups admit a "horocyclic" action, namely all elements (and thus all finitely generated subgroups) have fixed points, but there's no global fixed point. The criterion for the existence of such an action is that the group has countable cofinality, i.e. is union of a sequence of proper subgroups; some profinite groups satisfy this property as an abstract group (e.g. those infinite abelian, or infinite linear over a field) and some others don't.
Jun 5, 2012 at 18:40 comment added Benjamin Steinberg This is a nice answer. Compact groups have fixed points on trees.
Jun 5, 2012 at 8:34 history answered YCor CC BY-SA 3.0