# On the isometrisability of group actions on p-spaces

## TL;DR

This paper investigates the $p$-isometrisability property of discrete groups, establishing that groups with non-abelian free subgroups are not $p$-isometrisable for any $p$ in (1, ∞), and explores related open questions.

## Contribution

It introduces the concept of $p$-isometrisability, proves non-$p$-isometrisability for groups with free subgroups, and discusses connections with the Littlewood exponent.

## Key findings

- Groups with non-abelian free subgroups are not $p$-isometrisable for any $p$ in (1, ∞).
- $p$-isometrisability generalizes unitarisability when $p=2$.
- Open questions relate $p$-isometrisability to the Littlewood exponent.

## Abstract

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 ${\rm Lit}(\Gamma)$.