Volume rigidity and algebraic shifting

Abstract: We study the generic volume rigidity of $(d-1)$-dimensional simplicial complexes in $\mathbb R<sup>{d-1}$, 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 $&gt;1$, volume rigidity is {\em not} characterized by a corresponding hypergraph sparsity property.