How to "synthesize" sub-fixed point sets into the global one just by looking inside groups?

Thursday, January 12, 2017
Ora Inizio: 
Ora Fine: 
Masato Mimura
Tohoku university e EPF Lausanne

Consider a group G action on a metric space X by isometries, and subgroups M_1,..., M_l. One interesting and important problem is to study conditions under which non-emptyness of each M_i-fixed points ensures that of global (G-)fixed points. One extreme case is where G is a simple Lie group, X is an origin-excluded Hilbert space, and the action is given by a (strongly continuous) unitary representation. Then, the Howe--Moore property, based on the Mautner phenomenon,  implies that for each(!) non-compact closed subgroup M, the existence of M-fixed points suffices that of G-fixed points. This is too nice to be expected for the case of discrete group actions. Another example is a Helly-type theorem, which imposes some dimension condition on X.


In this talk, we will present a new criterion for such problems on actions of finitely generated groups, that is stated only in terms of (intrinsic) group structures. One such a criterion was previously given by Yehuda Shalom in 1999 (Publ. IHES.) in terms of "Bounded Generation". We remove that hypothesis. Applications of our theorem are to Kazhdan's property (T), and more...