Commensurating actions and invariant means
I will address the following two types of group actions: commensurating actions and actions with an invariant mean.
A group has Property FW if all its commensuration actions are trivial, and has Property FM if all its actions with an invariant mean have a finite orbit.
Property FW can also be defined in terms of actions on median graphs, CAT(0) cubings, or spaces with walls.
Both properties, FW and FM, interpolate between Kazhdan's Property (T) and Serre's Property FA; these two properties turn out to be related in the case of irreducible lattices in semi-simple Lie groups, where commensurating actions can be studied using Hilbertian methods.
Yves de Cornulier
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