Volume rigidity and algebraic shifting

TL;DR

This paper investigates the volume rigidity of simplicial complexes, linking it to exterior shifting, and demonstrates that volume rigidity cannot be characterized solely by hypergraph sparsity in higher dimensions.

Contribution

It establishes a connection between volume rigidity and exterior shifting, and shows that hypergraph sparsity does not characterize volume rigidity in dimensions greater than one.

Findings

Volume rigidity can be identified via exterior shifting.

Triangulations of certain 2D surfaces are volume rigid.

Volume rigidity is not characterized by hypergraph sparsity in higher dimensions.

Abstract

We study the generic volume rigidity of $(d-1)$-dimensional simplicial complexes in $\mathbb R^{d-1}$, and show that the volume rigidity of a complex can be identified in terms of its exterior shifting. In addition, we establish the volume rigidity of triangulations of several $2$-dimensional surfaces and prove that, in all dimensions $>1$, volume rigidity is {\em not} characterized by a corresponding hypergraph sparsity property.