Boundary rigidity of 3D CAT(0) cube complexes

TL;DR

This paper proves that the combinatorial structure of certain finite 3D CAT(0) cube complexes can be uniquely reconstructed from boundary distance data, extending previous 2D results in discrete geometry.

Contribution

It generalizes boundary rigidity results from 2D to 3D CAT(0) cube complexes with embeddings in D, establishing conditions for unique reconstruction from boundary distances.

Findings

Reconstruction of 3D CAT(0) cube complexes from boundary data is possible.

Extension of 2D boundary rigidity results to 3D complexes.

Boundary distances determine the combinatorial type under specified conditions.

Abstract

The boundary rigidity problem is a classical question from Riemannian geometry: if $(M, g)$ is a Riemannian manifold with smooth boundary, is the geometry of $M$ determined up to isometry by the metric $d_g$ induced on the boundary $\partial M$? In this paper, we consider a discrete version of this problem: can we determine the combinatorial type of a finite cube complex from its boundary distances? As in the continuous case, reconstruction is not possible in general, but one expects a positive answer under suitable contractibility and non-positive curvature conditions. Indeed, in two dimensions Haslegrave gave a positive answer to this question when the complex is a finite quadrangulation of the disc with no internal vertices of degree less than $4$. We prove a $3$-dimensional generalisation of this result: the combinatorial type of a finite CAT(0) cube complex with an embedding in $\mathbb{R}^3$ can be reconstructed from its boundary distances. Additionally, we prove a direct strengthening of Haslegrave's result: the combinatorial type of any finite 2-dimensional CAT(0) cube complex can be reconstructed from its boundary distances.