Abstract: An Orlicz space $L^{\Phi}(\Omega)$ is a Banach function space defined by using a Young function $\Phi$, which generalizes the $L^p$ spaces.
We show, for an Orlicz space $L^{\Phi}([0,1])$ which is not isomorphic to $L^\infty([0,1])$, if a locally compact second countable group has property $(T_{L^{\Phi}([0,1])})$, which is a generalization of Kazhdan's property $(T)$ for linear isometric representations on $L^{\Phi}([0,1])$, then it has Kazhdan's property $(T)$.
We also show, for a separable complex Orlicz space $L^{\Phi}(\Omega)$ with gauge norm, $\Omega =\mathbb{R},[0,1], \mathbb{N}$, if a locally compact second countable group has Kazhdan's property $(T)$, then it has property $(T_{L^{\Phi}(\Omega)})$.
We prove, for a finitely generated group $\Gamma $ and a Banach space $B$ whose modulus of convexity is sufficiently large, if $\Gamma $ 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 a hyperbolic group $\Gamma $ (which may have Kazhdan's property $(T)$) and an Orlicz sequence space $\ell^{\Phi\Psi}$ with gauge norm such that the Young function $\Psi$ sufficiently rapidly increases near $0$, $\Gamma $ doesn't have property $(F_{\ell^{\Phi\Psi}})$.
These results are generalizations of the results for $L^p$-spaces.