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

• DOI: 10.48550/arxiv.2408.10344
• Publication Date: 2024-01-01
• Authors: Luo, Yusheng; Zhang, Yongquan
• Keywords: Mathematics - Geometric Topology; Mathematics - Complex Variables; FOS: Mathematics; Geometric Topology (math.GT); Dynamical Systems (math.DS); Mathematics - Dynamical Systems; Complex Variables (math.CV)

50 pages, 11 figures<br/>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.<br/>