Property and Property restricted to a representation without non-zero invariant vectors
Mamoru Tanaka
Tohoku University, Sendai, Japan
![Property $(T_B)$ and Property $(F_B)$ restricted to a representation without non-zero invariant vectors cover](/_next/image?url=https%3A%2F%2Fcontent.preview.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-ggd-volume-8-issue-4.png&w=3840&q=90)
Abstract
In this paper, we give a necessary and sufficient condition for a finitely generated group to have a property like Kazhdan's Property 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 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 space on a finitely generated group, these conditions are regarded as an estimation of the spectrum of the -Laplace operator on the space and on the -Dirichlet finite space respectively.
Cite this article
Mamoru Tanaka, Property and Property restricted to a representation without non-zero invariant vectors. Groups Geom. Dyn. 8 (2014), no. 4, pp. 1141–1160
DOI 10.4171/GGD/258