Absolute Continuity and Large-Scale Geometry of Polish Groups

TL;DR

This paper explores the large-scale geometric properties of groups of absolutely continuous homeomorphisms on intervals, circles, and real lines, revealing their quasi-isometry types and structural decompositions.

Contribution

It establishes the quasi-isometry types of certain Polish groups of absolutely continuous homeomorphisms and demonstrates their structural decomposition as Zappa-Szép products.

Findings

_+(M) has trivial quasi-isometry type.

_{\u211d}^{ ext{loc}}(\u211d) is quasi-isometric to .

Groups are Zappa-Sze9 products.

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 $\operatorname{AC}_+(M)$ of orientation-preserving homeomorphisms $f:M\to M$ such that $f$ and $f^{-1}$ are absolutely continuous has a trivial quasi-isometry type. We also prove that the Polish group $\operatorname{AC}_{\mathbb Z}^\mathrm{loc}(\mathbb R)$ of homeomorphisms $f:\mathbb R\to\mathbb R$ 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 $\operatorname{AC}_+\left(\mathbb S^1\right)$ and $\operatorname{AC}_{\mathbb Z}^\mathrm{loc}(\mathbb R)$ we use the observation that these groups are Zappa-Sz\'ep products.