Actions of finite groups and smooth functions on surfaces - Feshchenko, B. (Methods of Functional Analysis and Topology, 2016)
Let f:M→ R be a Morse function on a smooth closed surface, V be a connected component of some critical level of f, and E_V be its atom. Let also S(f) be a stabilizer of the function f under the right action of the group of diffeomorphisms Diff(M) on the space of smooth functions on M, and S_V(f) = {hS(f) | h(V) = V}. The group S_V(f) acts on the set π_0E_V of connected components of the boundary of E_V. Therefore we have a homomorphism φ:S(f)→ Aut(π_0E_V). Let also G = φ(S(f)) be the image of S(f) in Aut(π_0E_V). Suppose that the inclusion E_VMV induces a bijection π_0 E_V→π_0(MV). Let H be a subgroup of G. We present a sufficient condition for existence of a section s:H→ S_V(f) of the homomorphism φ, so, the action of H on E_V lifts to the H-action on M by f-preserving diffeomorphisms of M. This result holds for a larger class of smooth functions f:M→ R having the following property: for each critical point z of f the germ of f at z is smoothly equivalent to a homogeneous polynomial R^2→ R without multiple linear factors.