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Graphs of Quasicircles and Quasiconformal Homeomorphisms

Katherine Williams Booth, Alexander Nolte, Yvon Verberne

TL;DR

This work provides a combinatorial characterization of the quasiconformal homeomorphism group Q(S) for a closed oriented surface S with genus g ≥ 2 by identifying it with the subgroup of automorphisms of the quasicircle graph QCdagger(S) that preserve a coarse order. The authors define an extended graph EQCext(S) to encode both essential and inessential quasicircles and prove that every automorphism of QCdagger(S) is realized by a homeomorphism of S, which is quasiconformal exactly when the automorphism is coarse-order preserving. A key technical achievement is establishing that QCdagger(S) is Gromov-hyperbolic, enabling geometric-group-theoretic methods to study Q(S). The construction leverages a blend of combinatorial graph analysis (torus/bigon/sharing structures, arc graphs) and analytic quasicircle theory (quasiconformality, Tukia-type extensions, Aseev’s theorem). Together, these results bridge combinatorial automorphisms with analytic deformation theory, opening avenues for applying hyperbolic-geometry techniques to the study of surface diffeomorphisms and quasiconformal maps.

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.

Graphs of Quasicircles and Quasiconformal Homeomorphisms

TL;DR

This work provides a combinatorial characterization of the quasiconformal homeomorphism group Q(S) for a closed oriented surface S with genus g ≥ 2 by identifying it with the subgroup of automorphisms of the quasicircle graph QCdagger(S) that preserve a coarse order. The authors define an extended graph EQCext(S) to encode both essential and inessential quasicircles and prove that every automorphism of QCdagger(S) is realized by a homeomorphism of S, which is quasiconformal exactly when the automorphism is coarse-order preserving. A key technical achievement is establishing that QCdagger(S) is Gromov-hyperbolic, enabling geometric-group-theoretic methods to study Q(S). The construction leverages a blend of combinatorial graph analysis (torus/bigon/sharing structures, arc graphs) and analytic quasicircle theory (quasiconformality, Tukia-type extensions, Aseev’s theorem). Together, these results bridge combinatorial automorphisms with analytic deformation theory, opening avenues for applying hyperbolic-geometry techniques to the study of surface diffeomorphisms and quasiconformal maps.

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.
Paper Structure (15 sections, 44 theorems, 7 equations, 5 figures)

This paper contains 15 sections, 44 theorems, 7 equations, 5 figures.

Table of Contents

  1. Outline
  2. Quasiconformal Homeomorphisms and Quasicircles
  3. Quasicircles in the Riemann sphere
  4. Surfaces with genus
  5. Graphs of Quasicircles
  6. Encoding Topological Data Into Graphs
  7. Torus pairs
  8. Bigon pairs
  9. Sharing pairs
  10. Connectivity of arc graphs
  11. Building the quasiconformal map
  12. From essential to inessential
  13. Building the homeomorphism
  14. Quasiconformality
  15. Hyperbolicity

Key Result

Theorem 1

Let $S$ be a closed orientable surface of genus $g \geq 2$. Any automorphism $\Phi$ of $\mathcal{QC}^\dagger(S)$ is induced by a homeomorphism $\varphi$ of $S$. The homeomorphism $\varphi$ is quasiconformal if and only if $\Phi$ is coarsely order preserving.

Figures (5)

  • Figure 1: Two nondegenerate torus pairs are formed by the above blue and red curves. The intersections between the curves in each torus pair are drawn in purple. The pair on the left is transverse, and the third quasicircle is the symmetric difference between the curves. The pair on the right is nontransverse due to the cusps in the symmetric difference.
  • Figure 2: Examples of type 1 and type 2 quasicircles for the nonseparating noncrossing annulus pair $\{c, d\}$.
  • Figure 3: A linked sharing pair.
  • Figure 4: The torus pairs $\{c, d'\}$ and $\{c', d\}$ and the quasicircle that forms a torus triple with both.
  • Figure 5: The torus pairs $\{c, c'\}$ and $\{d, d'\}$ and the quasicircle that forms a torus triple with both.

Theorems & Definitions (79)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Definition 2.1
  • Theorem 2.2: Ahlfors ahlfors1963quasiconformal
  • Theorem 2.3: Aseev aseev2014quasiconformal
  • Theorem 2.4: Tukia tukia1981extension
  • Theorem 2.5
  • Definition 2.6
  • Remark 1
  • ...and 69 more