Volume rigidity and algebraic shifting

We study the generic volume-rigidity of $(d-1)$-dimensional simplicial complexes in $R^{d-1}$, 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 not characterized by a corresponding hypergraph sparsity property.