Amenability of finite energy path and loop groups

TL;DR

This paper proves that groups of finite energy Sobolev paths and loops in a compact Lie group are amenable, extending previous results from continuous paths to Sobolev class $H^1$ paths and loops.

Contribution

It establishes the amenability of Sobolev class $H^1$ path and loop groups in a compact Lie group, a significant generalization of earlier continuous path results.

Findings

Finite energy path and loop groups are amenable.

Amenability extends from continuous to Sobolev $H^1$ paths.

Supports invariant measures for group actions.

Abstract

It is shown that the groups of finite energy (that is, Sobolev class $H^1$) paths and loops with values in a compact Lie group are amenable in the sense of Pierre de la Harpe, that is, every continuous action of such a group on a compact space admits an invariant regular Borel probability measure. To our knowledge, the strongest previously known result concerned the amenability of groups of continuous paths and loops (Malliavin and Malliavin 1992).