Boundary rigidity of 3D CAT(0) cube complexes

John Haslegrave, Alex D. Scott, Youri Tamitegama, Jane Tan

Abstract: The boundary rigidity problem is a classical question from Riemannian geometry: if (M,g)<math><mrow is="true"><mo is="true">(</mo><mi is="true">M</mi><mo is="true">,</mo><mi is="true">g</mi><mo is="true">)</mo></mrow></math> is a Riemannian manifold with smooth boundary, is the geometry of M<math><mi is="true">M</mi></math> determined up to isometry by the metric dg<math><msub is="true"><mrow is="true"><mi is="true">d</mi></mrow><mrow is="true"><mi is="true">g</mi></mrow></msub></math> induced on the boundary ∂M<math><mrow is="true"><mi is="true">∂</mi><mi is="true">M</mi></mrow></math>? 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<math><msup is="true"><mrow is="true"><mi mathvariant="double-struck" is="true">R</mi></mrow><mrow is="true"><mn is="true">3</mn></mrow></msup></math> 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.