  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.
