Polishability of some groups of interval and circle diffeomorphisms
Volume 248 / 2020
Fundamenta Mathematicae 248 (2020), 91-109
MSC: 22A05, 26A45, 26A46.
DOI: 10.4064/fm605-1-2019
Published online: 7 June 2019
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 $\mathop {\rm Diff}\nolimits _+^{k+{\rm AC}}(M)$, 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 $\mathop {\rm Diff}\nolimits _+^k(M)$. By contrast, the group $\mathop {\rm Diff}\nolimits _+^{1+{\rm BV}}(M)$, consisting of $C^1$ diffeomorphisms whose derivative has bounded variation, admits no Polish group topology whatsoever.
