Property $(T_B)$ and Property $(F_B)$ restricted to a representation without non-zero invariant vectors

TL;DR

This paper characterizes when finitely generated groups exhibit properties similar to Kazhdan's Property (T) and Property (FH) on strictly convex Banach spaces without invariant vectors, linking these to spectral estimates of the p-Laplace operator.

Contribution

It provides necessary and sufficient conditions for these properties on Banach spaces, extending classical properties to new geometric and spectral contexts.

Findings

Characterization of Property (T_B) and (F_B) on strictly convex Banach spaces.

Connection between these properties and the spectrum of the p-Laplace operator.

Extension of classical group properties to Banach space representations.

Abstract

In this paper, we give a necessary and sufficient condition for which a finitely generated group has a property like Kazhdan's Property $(T)$ restricted to one isometric representation on a strictly convex Banach space without non-zero invariant vectors. Similarly, we give a necessary and sufficient condition for which a finitely generated group has a property like Property $(FH)$ restricted to the set of the affine isometric actions whose linear part are one isometric representation on a strictly convex Banach space without non-zero invariant vectors. If the Banach space is the $\ell^p$ space ($1<p<\infty$) on a finitely generated group, these conditions are regarded as an estimation of the spectrum of the $p$-Laplace operator on the $\ell^p$ space and on the $p$-Dirichlet finite space respectively.