Disk patterns, quasi-duality and the uniform bounded diameter conjecture

TL;DR

This work establishes a uniform bound on the diameter of the skinning map in the deformation space of acylindrical reflection groups, with the bound depending only on the topological complexity of the boundary components. It achieves this by translating skinning-map data into combinatorial extremal-width control on Coxeter graphs via disk patterns, and proving a uniform quasi-duality between discrete and conformal widths that remains stable under degeneration. The approach provides a uniform rigidity statement for disk patterns and connects combinatorial graph width theory with Teichmüller theory, yielding new insights into the structure of the deformation spaces of reflection groups. The results advance our understanding of when and how disk patterns rigidly determine hyperbolic structures and open pathways to effective hyperbolization-type conclusions through quantitative width estimates.

Abstract

We show that the diameter of the image of the skinning map on the deformation space of an acylindrical reflection group is bounded by a constant depending only on the topological complexity of the components of its boundary, answering a conjecture of Minsky in the reflection group setting. This result can be interpreted as a uniform rigidity theorem for disk patterns. Our method also establishes a connection between the diameter of the skinning image and certain discrete extremal width on the Coxeter graph of the reflection group.

Figures (10)

• Figure 1.1: A reflection group, the corresponding disk pattern, and the skinning image.
• Figure 2.1: Coxeter graph for Proposition \ref{['prop:prism']}. A vertex labelled $b$ is put at infinity.
• Figure 2.2: Two disk patterns with the same graph but different weights. The one on the right contains an elliptic connection.
• Figure 2.3: Two disk patterns with the same graph but different weights. The one on the right contains a right-angled 2-connection.
• Figure 3.1: Some examples to illustrate Theorem \ref{['thm:combunifbound']}.

Theorems & Definitions (65)

• Conjecture 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Koebe-Andreev-Thurston
• proof
• Proposition 2.4: Koebe-Andreev-Thurston for triangular prism
• Lemma 2.5
• proof
• Proposition 2.6