of vector fields, and to ‘resolve’ this Lie algebra via radial blowup of the underlying space.

Two papers closely related to our work are, first, Mazzeo-Vertman [@MV2012], in which the authors study analytic torsion on incomplete edge spaces, which are the $k = 1$ case of incomplete cusp edges. (They analyze the behavior of elliptic operators which have the same type of degeneracy in their principal symbols near the singular set as the Laplacian, so a weider class than considered here.) Their analysis involves a heat kernel construction using blowup analysis, which is slightly simpler in their context as the resolved double space has one less blow up (and thus the triple space is much simpler). One substantial difference between the incomplete edge and incomplete cusp edge cases is that in the incomplete edge case the space of self-adjoint extensions is substantially more subtle. In particular, a Witt space (this is a topological condition and has nothing to do with the value of $k$) that is incomplete edge may have infinitely many self-adjoint extensions if the family of induced operators on the fibers have small non-zero eigenvalues [@ALMPII]; on incomplete cusp edge spaces the small non-zero fiber eigenvalues do not 