  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.
