Polishability of some groups of interval and circle diffeomorphisms

2017/09/13 by Cohen, Michael P.
#22F99 #26A45 #26A46 #37E05 #37E10 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1709.04523

Abstract

Let M=I or M=\mathbbS¹ and let k≥ 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 C¹ diffeomorphisms whose derivative has bounded variation, admits no Polish group topology whatsoever.

Related

• \mathop\rm PL₊(I) is not a Polish group
• Diffeomorphism groups of critical regularity
• Absolute Continuity and Large-Scale Geometry of Polish Groups
• Polish topologies on groups of non-singular transformations
• Sharper changes in topologies