The structure of groups of multigerm equivalences

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.