Property $(T_{L^{\mathrm{\Phi }}})$ and property $(F_{L^{\mathrm{\Phi }}})$ for Orlicz spaces $L^{\mathrm{\Phi }}$

An Orlicz space $L^{\mathrm{\Phi }}(\mathrm{\Omega })$ is a Banach function space defined by using a Young function $\mathrm{\Phi }$, which generalizes the $L^p$ spaces. We show that, for a reflexive Orlicz space $L^{\mathrm{\Phi }}([0,1])$, a locally compact second countable group has Kazhdan’s property $(T)$ if and only if it has property $(T_{L^{\mathrm{\Phi }}([0,1])})$, which is a generalization of Kazhdan’s property $(T)$ for linear isometric representations on $L^{\mathrm{\Phi }}([0,1])$. We also prove that, for a Banach space $B$ whose modulus of convexity is sufficiently large, if a locally compact second countable group has Kazhdan’s property $(T)$, then it has property $(F_B)$, which is a fixed point property for affine isometric actions on $B$. Moreover, we see that, for an Orlicz sequence space ${\mathrm{\ell }}^{\mathrm{\Phi }\mathrm{\Psi }}$ such that the Young function $\mathrm{\Psi }$ sufficiently rapidly increases near $0$, hyperbolic groups (with Kazhdan’s property $(T)$) don’t have property $(F_{{\mathrm{\ell }}^{\mathrm{\Phi }\mathrm{\Psi }}})$. These results are generalizations of the results for $L^p$-spaces.