Absolute Continuity and Large-Scale Geometry of Polish Groups

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 ${\mathrm{AC}}_{+}(M)$ 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 ${\mathrm{AC}}_{\mathbb{Z}}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathbb{R})$ 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 ${\mathrm{AC}}_{+}\left({\mathbb{S}}^1\right)$ and $\operatorname{AC}_{\mathbb Z}^$\mathrm{l}\mathrm{o}\mathrm{c}$(\mathbb R)$ we use the observation that these groups are Zappa-Sz'ep products.