Volume preserving subgroups of A and K and singularities in unimodular geometry

For a germ of a smooth map f and a subgroup G_V of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form V in the source or in the target we study the G_V-moduli space of f that parameterizes the G_V-orbits inside the G-orbit of f. We find, for example, that this moduli space vanishes for A-equivalence with volume-preserving target diffeomorphisms and A-stable maps f and for K-equivalence with volume-preserving source diffeomorphisms and K-simple maps f. On the other hand, there are A-stable maps f with infinite-dimensional moduli space for A-equivalence with volume-preserving source diffeomorphisms.