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

TL;DR

This paper proves that for discrete groups and 1<p<q< infty, Property (F_{l_q}) implies Property (F_{l_p}), extending known relations between properties for Banach space representations.

Contribution

It establishes a new implication between properties (F_{l_q}) and (F_{l_p}) for discrete groups when 1<p<q< finite, p≠2.

Findings

Property (F_{l_q}) implies Property (F_{l_p}) for 1<p<q< finite, p≠2.

Connections between properties (T_{l_p}) and (F_{l_q}) are clarified.

Extension of known properties relations in Banach space group representations.

Abstract

It is known that for $\sigma$-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 $(\tau)$. 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 $(F_{l_p})$.