Strong Banach property (T)

Benben Liao  (Shanghai Center for Mathematical Sciences)

10:00 am to 11:00 am, September 29nd, 2015   Science Building A1510


We say that a locally compact topological group $G$ has strong Banach property (T), if the trivial representation of $G$ is "isolated" among representations $(\pi,E)$ with small exponential growth (for example polynomial growth $\|\pi(g)\|_{\mathcal L(E)}\leq P(\ell(g)),\forall g\in G$ where $\ell:G \to \mathbb{R}_{\geq 0}$ is a length function and $P:\mathbb{R} \to \mathbb{R}$ is a polynomial) where $E$ is a Banach space of type $>1$ (for example uniformly convex Banach space).

We will discuss the existence of such groups: higher rank simple algebraic groups over non-Archimedean local fields and their co-compact lattices. For applications, a group with strong Banach property (T) must admit a fixed point for any affine isometric action on a Banach space of type $>1.$ Moreover, any family of Schreier graphs of a group with strong Banach property (T) do not admit coarse embedding in any Banach space of type $>1.$ I will also mention an on-going project with Maria Paula Gomez Aparicio and Mikael de la Salle on strengthening of geometric property (T) for coarse space.

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