The structure of groups of multigerm equivalences

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.