On the isometrisability of group representations on p-spaces

In this note we consider a p-isometrisability property of discrete groups. If p=2 this property is equivalent to the well-studied notion of unitarisability. We prove that amenable groups are p-isometrisable for all p∈(1,∞). Conversely, we show that every group containing a non-abelian free subgroup is not p-isometrisable for any p∈(1,∞). We also discuss some open questions and possible relations of p-isometrisability with the recently introduced Littlewood exponent Lit(Γ).