Polishability of some groups of interval and circle diffeomorphisms

TL;DR

This paper introduces new Polish groups of diffeomorphisms with absolutely continuous derivatives, contrasting with the non-Polish nature of groups with bounded variation derivatives.

Contribution

It establishes a new class of Polish groups of diffeomorphisms with absolutely continuous derivatives and shows the non-existence of Polish topology for bounded variation derivative groups.

Findings

Groups with absolutely continuous derivatives admit natural Polish topologies.

Groups with derivatives of bounded variation do not admit any Polish group topology.

New classes of Polish groups extend the understanding of diffeomorphism groups.

Abstract

Let $M=I$ or $M=\mathbb{S}^1$ and let $k\geq 1$. We exhibit a new infinite class of Polish groups by showing that each group $\mathop{\rm Diff}_+^{k+AC}(M)$, consisting of those $C^k$ diffeomorphisms whose $k$-th derivative is absolutely continuous, admits a natural Polish group topology which refines the subspace topology inherited from $\mathop{\rm Diff}_+^k(M)$. By contrast, the group $\mathop{\rm Diff}_+^{1+BV}(M)$, consisting of $C^1$ diffeomorphisms whose derivative has bounded variation, admits no Polish group topology whatsoever.