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

TL;DR

This paper proves a uniform boundedness result for the skinning map's image diameter in acylindrical reflection groups, linking it to topological complexity and extremal width, thus advancing understanding of disk pattern rigidity.

Contribution

It establishes a uniform bounded diameter for the skinning map in acylindrical reflection groups, confirming Minsky's conjecture in this setting.

Findings

Bounded diameter depends only on boundary complexity

Connection between skinning map diameter and extremal width

Results imply a form of uniform rigidity for disk patterns

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.