Polishability of some groups of interval and circle diffeomorphisms

Abstract

Let $M=I$ or $M={\mathbb{S}}^1$ and let $k\ge 1$. We exhibit a new infinite class of Polish groups by showing that each group ${\mathrm{Diff}}_{+}^{k+AC}(M)$, 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 ${\mathrm{Diff}}_{+}^k(M)$. By contrast, the group ${\mathrm{Diff}}_{+}^{1+BV}(M)$, consisting of $C^1$ diffeomorphisms whose derivative has bounded variation, admits no Polish group topology whatsoever.