On the isometrisability of group actions on p-spaces

In this note we consider a $p$-isometrisability property of discrete groups. If $p=2$ this property is equivalent to unitarisability. We prove that any group containing a non-abelian free subgroup is not $p$-isometrisable for any $p\in (1, \infty)$. We also discuss some open questions and possible relations of $p$-isometrisability with the recently introduced Littlewood exponent ${\rm Lit}(Γ)$.

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