Superintrinsic synthesis in fixed point properties (Masato Mimura, Tohoku & EPFL)

04.10.2016 10:30

For a class X of metric spaces, we say a finitely generated group G has the fixed point property (F_X), relative to X, if all isometric G-actions on every member in X have global fixed points. Fix a class X of "non-positively curved spaces" (for instance, in the sense of Busemann) stable under certain operations. We obtain new criteria to "synthesize" the "partial" (F_X) (more precisely, with respect to subgroups) into the "whole" (F_X). A basic example of such X is the class of all Hilbert spaces, and then (F_X) is equivalent to the celebrated property (T) of Kazhdan.
Our "synthesis" is intrinsic, in the sense of that our criteria do not depend on the choices of X. The point here is that, nevertheless, we exclude all of "Bounded Generation" axioms, which were the clue in previous works by Y. Shalom (Publ. IHES, 1999 and ICM 2006). As applications, we present a simpler proof of (T) for elementary groups over noncommutative rings (Ershov--Jaikin, Invent. Math., 2010). Moreover, our approach enables us to extend that to one in general L_p space settings for all finite p>1.

Lieu

Room 623, Séminaire "Groupes et Géométrie"

entrée libre