Existence and uniqueness of dual circle configurations for arbitrary cellular decompositions of the plane (the infinite Koebe–Andreev–Thurston conjecture of Rehwinkel–Whitehead–Yang–Yang)

Rehwinkel, Whitehead, Yang, and Yang (2023, published 2024) introduced a class of fractal circle packings built from a "base" circle configuration B and a "dual" configuration B̂ whose tangency graphs are dual cellular decompositions of the plane, generalizing the polyhedral packings of Kontorovich and Nakamura. The construction's validity in full generality rests on their Conjecture 3.3, an infinite analogue of the Koebe–Andreev–Thurston theorem: every pair of simple, 3-connected dual plane graphs realizing a cellular decomposition of the plane and its dual should arise as the tangency graphs of such dual circle configurations, with the pair unique up to similarity. This is known only for triangulations (via discrete uniformization) and for the square grid. Conjecture 3.3 remains open.

Background

A circle packing is a collection of oriented circles in the Riemann sphere $\hat{\mathbb{C}}$ with pairwise disjoint interiors, some pairs tangent. Its tangency graph (or nerve) has one vertex per circle and an edge for each tangency; it is automatically a planar graph. A generalized circle is either a round circle or a straight line (a circle through $\infty$). The classical Koebe–Andreev–Thurston circle packing theorem (Koebe 1936; Andreev 1970; Thurston 1979) states, in the dual-pair formulation of Brightwell and Scheinerman, that any pair of finite, simple (no loops or parallel edges), 3-connected plane graphs $G, \hat G$ corresponding to a cellular decomposition of the sphere and its dual can be realized simultaneously as the tangency graphs of two collections of circles $B, \hat B$, with intersections between the two collections orthogonal and occurring at incident vertex–face pairs, and the realization unique up to Möbius transformation (a conformal automorphism of $\hat{\mathbb{C}}$). A cellular decomposition of a surface is a decomposition into faces by a planar graph whose faces are all topological disks; 3-connected means the graph stays connected after deleting any two vertices. A triangulation is the special case where every face is a triangle.

For infinite graphs the picture is partial. Beardon and Stephenson's uniformization theorem for circle packings, Schramm's existence and rigidity theorems for infinite packings, and Stephenson's Discrete Uniformization Theorem imply that every infinite, simple, 3-connected plane graph $G$ representing a triangulation of the plane arises as the tangency graph of a circle configuration unique up to similarity of the plane (a map $z \mapsto az+b$ or $z \mapsto a\bar z+b$ with $a \neq 0$, i.e. a conformal or anti-conformal automorphism of $\mathbb{C}$ fixing $\infty$). For non-triangulating combinatorics much less is known; the main other settled cases are Schramm's theorem on circle patterns with the combinatorics of the square grid, and the triangular lattice, whose uniqueness was proved by Rodin and Sullivan (in an appendix to their circle-packing convergence paper) as a crucial step in their proof.

Rehwinkel, Whitehead, Yang, and Yang used these infinite versions to build a new class of packings. They start from a base configuration $B$ and a dual configuration $\hat B$, two collections of oriented generalized circles satisfying: (1) within each collection the circles are pairwise disjoint or tangent, with disjoint interiors; (2) the tangency graphs $G_B$ and $G_{\hat B}$ are each nontrivial, connected, and dual to each other; (3) a circle of $B$ and a circle of $\hat B$ either intersect orthogonally and correspond to an incident vertex–face pair of the two graphs, or have disjoint interiors; (4) $B \cup \hat B$ has at most one accumulation point in $\hat{\mathbb{C}}$. The packing $P$ is then the orbit of $B$ under the group generated by reflections in the circles of $\hat B$ (the Apollonian group of the configuration), a fractal packing that generalizes the polyhedral packings of Kontorovich and Nakamura, which arise from the finite theorem applied to a polyhedron and its dual. The authors' symmetry structure theorems and their arithmetic examples (integrality and superintegrality of curvatures — the curvature of a circle is the reciprocal of its radius, and superintegrality is a stronger arithmetic property arising in the Kontorovich–Nakamura classification of crystallographic packings) presuppose that the underlying base–dual configuration exists and is essentially unique. The finite Koebe–Andreev–Thurston theorem and the triangulation theorem guarantee this for polyhedral packings and for triangulation-based examples, but not for an arbitrary cellular decomposition of the plane. The authors therefore proposed, as their central open problem (Conjecture 3.3), a common infinite extension of the two theorems, noting that when $G$ is not a triangulation the configurations $B$ and $\hat B$ must be constructed together and both are needed for uniqueness.

Problem Statement

Conjecture 3.3 of Rehwinkel–Whitehead–Yang–Yang (the infinite Koebe–Andreev–Thurston theorem for dual pairs). Let $G$ and $\hat G$ be a pair of simple (no loops or parallel edges), 3-connected plane graphs, corresponding to the 1-skeleton of a cellular decomposition of the plane $\mathbb{C}$ and the 1-skeleton of its dual cellular decomposition. Do there exist two collections of oriented generalized circles $B$ and $\hat B$ (the base and dual configurations; a generalized circle is a round circle or a straight line, and orienting it means choosing one of the two sides into which it divides $\hat{\mathbb{C}}$ as its interior) such that: (1) the circles within each of $B$ and $\hat B$ are pairwise disjoint or tangent, with disjoint interiors; (2) the tangency graph of $B$ (the graph with one vertex per circle and one edge for each tangency) is isomorphic to $G$ and the tangency graph of $\hat B$ is isomorphic to $\hat G$, where in each case the isomorphism is of graphs and of the associated cellular decompositions; (3) if a circle of $B$ and a circle of $\hat B$ intersect, they do so orthogonally and correspond to an incident vertex–face pair of the two decompositions, and otherwise their interiors are disjoint; and (4) $B \cup \hat B$ has at most one accumulation point in the Riemann sphere $\hat{\mathbb{C}}$? Moreover, are the configurations $B$ and $\hat B$ unique up to similarity of the plane, i.e. up to $z \mapsto az+b$ or $z \mapsto a\bar z+b$ with $a \neq 0$? Condition (3) is one-directional: it forces every intersection of a circle of $B$ and a circle of $\hat B$ to be orthogonal at an incident vertex–face pair of the two decompositions, but does not require every incident vertex–face pair to intersect. A complete proof that such configurations exist and are unique for every such pair $(G, \hat G)$, or an explicit admissible pair for which no such configurations exist or for which two non-similar realization pairs exist, resolves the problem. The problem should not be reduced to the known triangulation or square-grid cases; it concerns arbitrary cellular decompositions of the plane.

References

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