Absolute Continuity and Large-Scale Geometry of Polish Groups

2018/02/28 by Jake Herndon, Herndon, Jake
Mathematics · #Advanced Topology and Set Theory #Mathematics and Applications #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1802.10239

Abstract

We apply the theory of large-scale geometry of Polish groups to groups of absolutely continuous homeomorphisms. Let M be either the compact interval or circle. We prove that the Polish group AC₊(M) of orientation-preserving homeomorphisms f:M→ M such that f and f^-1 are absolutely continuous has a trivial quasi-isometry type. We also prove that the Polish group AC_{\mathbb Z}^loc(\mathbb R) of homeomorphisms f:\mathbb R→\mathbb R such that f commutes with integer translations and both f and f^-1 are locally absolutely continuous is quasi-isometric to the group of integers. To study AC₊(\mathbb S¹) and AC_{\mathbb Z}^loc(\mathbb R) we use the observation that these groups are Zappa-Szép products.

Related

• Generation and genericity of the group of absolutely continuous homeomorphisms of the interval
• Aspects of automatic continuity
• Large scale geometry of automorphism groups
• Group topologies on groups of bi-absolutely continuous homeomorphisms
• Classification and Non-classification of Homeomorphism Relations