Property $(FL_p)$ implies property $(FL_q)$ for $1<q<p<\infty$

It is known that for $σ$-compact groups Kazhdan's Property $(T)$ is equivalent to Serre's Property $(FH)$. Generalized versions of those properties, called properties $(T_{B})$ and $(F_{B})$, can be defined in terms of the isometric representations of a group on an arbitrary Banach space $B$. Property $(F_{B})$ implies $(T_{B})$. It is known that a group with Property $(T_{l_p})$ shares some properties with Kazhdan's groups, for example compact generation and compact abelianization. Moreover in the case of discrete groups, Property $(T_{l_p})$ implies Lubotzky's Property $(τ)$. In this paper we prove that in the case of discrete groups and $1<p<q<\infty$ and $p\not=2$, Property $(F_{l_q})$ implies Property $(F_{l_p})$.

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