Graphs of Quasicircles and Quasiconformal Homeomorphisms

TL;DR

This paper characterizes quasiconformal homeomorphisms of higher-genus surfaces using a combinatorial graph of essential quasicircles, linking geometric function theory with graph automorphisms and coarse geometry.

Contribution

It provides a novel combinatorial characterization of quasiconformal homeomorphisms on surfaces of genus at least two, connecting them to graph automorphisms respecting a coarse order.

Findings

Quasiconformal homeomorphisms correspond exactly to automorphisms of a graph of essential quasicircles.

The graph of essential quasicircles respects a canonical coarse ordering.

Discussion of the coarse geometric properties of this graph.

Abstract

We give a combinatorial characterization of the group of quasiconformal homeomorphisms of a closed, oriented surface $S$ of genus at least $2$. In particular, we prove they are exactly the automorphisms of a graph of essential quasicircles on $S$ that respect a canonical coarse ordering induced by quality constants. We also discuss the coarse geometry of this graph.