2023/09/08 by Haslegrave, John, Scott, Alex, Tamitegama, Youri +1
#57M15 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2309.04385
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 dg induced on the boundary ∂ 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 ℝ3 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.