Boundary rigidity of finite CAT(0) cube complexes

TL;DR

This paper proves that finite CAT(0) cube complexes can be uniquely reconstructed from boundary distances, advancing the understanding of boundary rigidity in discrete geometric structures.

Contribution

It establishes the boundary rigidity property for finite CAT(0) cube complexes, confirming a recent conjecture and connecting discrete geometry with classical boundary rigidity problems.

Findings

Finite CAT(0) cube complexes are reconstructible from boundary distances.

The proof uses median graphs and corner peelings techniques.

Confirms conjecture by Haslegrave et al. (2023).

Abstract

In this note, we prove that finite CAT(0) cube complexes can be reconstructed from their boundary distances (computed in their 1-skeleta). This result was conjectured by Haslegrave, Scott, Tamitegama, and Tan (2023). The reconstruction of a finite cell complex from the boundary distances is the discrete version of the boundary rigidity problem, which is a classical problem from Riemannian geometry. In the proofs, we use the bijection between CAT(0) cube complexes and median graphs and the corner peelings of median graphs.