Groups, Geometry, and Dynamics


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Volume 8, Issue 4, 2014, pp. 1141–1160
DOI: 10.4171/GGD/258

Published online: 2014-12-31

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

Mamoru Tanaka[1]

(1) Tohoku University, Sendai, Japan

In this paper, we give a necessary and sufficient condition for a finitely generated group to have 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 a finitely generated group to have a property like Property $(FH)$ restricted to the set of the affine isometric actions whose linear part is a given 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.

Keywords: Finitely generated groups, isometric action, strictly convex Banach spaces

Tanaka Mamoru: Property $(T_B)$ and Property $(F_B)$ restricted to a representation without non-zero invariant vectors. Groups Geom. Dyn. 8 (2014), 1141-1160. doi: 10.4171/GGD/258