  The moduli space of multiply-connected Calabi-Yau threefolds is shown to
contain codimension-one loci on which the corresponding variety develops a
certain type of hyperquotient singularity. These have local descriptions as
discrete quotients of the conifold, and are referred to here as hyperconifolds.
In many (or possibly all) cases such a singularity can be resolved to yield a
distinct compact Calabi-Yau manifold. These considerations therefore provide a
method for embedding an interesting class of singularities in compact
Calabi-Yau varieties, and for constructing new Calabi-Yau manifolds. It is
unclear whether the transitions described can be realised in string theory.
