The structure of groups of multigerm equivalences

TL;DR

This paper extends known results about the structure of equivalence groups from monogerms to multigerms, demonstrating the existence and decomposition of maximal compact subgroups for finitely determined multigerms.

Contribution

It generalizes the structure theory of equivalence groups to multigerms, including the decomposition of maximal compact subgroups and their relation to monogerm components.

Findings

Maximal compact subgroup $MC(\A_f)$ decomposes as a product for multigerms.

The structure of $MC(\A_f)$ is similar to that of monogerms.

Maximal compact subgroups are small and computationally accessible.

Abstract

We study the structure of classical groups of equivalences for smooth multigerms $f \colon (N,S) \to (P,y)$, and extend several known results for monogerm equivalences to the case of mulitgerms. In particular, we study the group $\A$ of source- and target diffeomorphism germs, and its stabilizer $\A_f$. For monogerms $f$ it is well-known that if $f$ is finitely $\A$-determined, then $\A_f$ has a maximal compact subgroup $MC(\A_f)$, unique up to conjugacy, and $\A_f/MC(\A_f)$ is contractible. We prove the same result for finitely $\A$-determined multigerms $f$. Moreover, we show that for a ministable multigerm $f$, the maximal compact subgroup $MC(\A_f)$ decomposes as a product of maximal compact subgroups $MC(\A_{g_i})$ for suitable representatives $g_i$ of the monogerm components of $f$. We study a product decomposition of $MC(\A_f)$ in terms of $MC(\mathscr{R}_f)$ and a group of target diffeomorphisms, and conjecture a decomposition theorem. Finally, we show that for a large class of maps, maximal compact subgroups are small and easy to compute.