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

TL;DR

This paper investigates the structure of volume-preserving subgroups within certain geometric groups and their impact on singularity classification in unimodular geometry, revealing conditions for trivial and infinite-dimensional moduli spaces.

Contribution

It characterizes the G_V-moduli space for various volume-preserving subgroup actions, identifying when these spaces are trivial or infinite-dimensional.

Findings

Moduli space vanishes for A-equivalence with volume-preserving target diffeomorphisms.

Moduli space vanishes for K-equivalence with volume-preserving source diffeomorphisms.

Existence of A-stable maps with infinite-dimensional moduli space under volume-preserving source diffeomorphisms.

Abstract

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.