Boundary rigidity of systolic and Helly complexes

2025/06/16 by Blufstein, Martín, Chalopin, Jérémie, Chepoi, Victor
#05C99 #20F65 #51F99 #52C99 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2506.13219

Abstract

In this article, we prove that finite (weakly) systolic and Helly complexes can be reconstructed from their boundary distances (computed in their 1-skeleta). Furthermore, Helly complexes and 2-dimensional systolic complexes can be reconstructed by an algorithm that runs in polynomial time with respect to the number of vertices of the complex. Both results can be viewed as a positive contribution to a general question of Haslegrave, Scott, Tamitegama, and Tan (2025). The reconstruction of a finite cell complex from the boundary distances is the discrete analogue of the boundary rigidity problem, which is a classical problem from Riemannian geometry.

Citations

• Boundary rigidity of 3D CAT(0) cube complexes
• Distance labeling schemes for K₄-free bridged graphs
• Helly meets Garside and Artin
• Dismantlability of weakly systolic complexes and applications
• Boundaries of systolic groups
• Two dimensional compact simple Riemannian manifolds are boundary distance rigid
• Boundary rigidity of finite CAT(0) cube complexes

Related

• Boundary rigidity of finite CAT(0) cube complexes
• Boundary rigidity of 3D CAT(0) cube complexes
• Helly's theorem for systolic complexes
• Minimal displacement set for weakly systolic complexes
• A note on the combinatorial structure of finite and locally finite simplicial complexes of nonpositive curvature