Sala Seminari (Dip. Matematica).
Consider a group G action on a metric space X by isometries, andsubgroups M_1,…, M_l. One interesting and important problem isto studyconditions underwhichnon-emptyness of each M_i-fixed points ensures thatof global (G-)fixed points. One extreme case is where G is a simple Lie group, Xisan origin-excludedHilbert 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, theexistence of M-fixed points suffices that of G-fixed points. This is too nice tobe expectedfor the case of discrete group actions. Another example is a Helly-typetheorem, whichimposes some dimension condition on X. In this talk,we will present a new criterion for such problems onactions of finitelygenerated 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. Applicationsofour theoremareto Kazhdan’s property (T), and more…