Polishability of some groups of interval and circle diffeomorphisms

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.