Graphs of Quasicircles and Quasiconformal Homeomorphisms
Katherine Williams Booth, Alexander Nolte, and Yvon Verberne

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 of genus at least . In particular, we prove they are exactly the automorphisms of a graph of essential quasicircles on that respect a canonical coarse ordering induced by quality constants. We also discuss the coarse geometry of this graph.
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Taxonomy
TopicsGeometric and Algebraic Topology · Quasicrystal Structures and Properties · Mathematical Dynamics and Fractals
