Finiteness Properties of Formal Lie Group Actions

Abstract

Abstract. Following ideas of Arnold and Seigal-Yakovenko, we prove that the space of matrix coefficients of a formal Lie group action belongs to a Noetherian ring. Using this result we extend the uniform intersection multiplicity estimates of these authors from the abelian case to general Lie groups. We also demonstrate a simple new proof for a jet-determination result of Baouendi. et al.In the second part of the paper we use similar ideas to prove a result on embedding formal diffeomorphisms in one-parameter groups extending a result of Takens. In particular this implies that the results of Arnold and Seigal-Yakovenko are formal consequence of our result for Lie groups.

Recurrent orbits of subgroups of local complex analytic diffeomorphisms

“…Let us define the derived groups of G. Definition 2. 1 We define the derived group G (1) (or [G, G]) of G as the group generated by the commutators [ f, g] := f g f −1 g 1 of elements of G. Analogously we define…”

“…Definition 4. 1 We define the group of formal diffeomorphisms Diff (C n , 0) as the projective limit lim ← −k∈N D k .…”

“…Let us prove that if it holds for some k ≥ j + 1 so it does for k + 1. Since [L j , L k ] ⊂ L k we obtain [L j , (L k ) (1) ] ⊂ (L k ) (1) and then [L j , (L k ) (2) ] ⊂ (L k ) (2) by the Jacobi identity. We haveL k+1 CK n = (L k ) (a) CK n for some a ∈ {1, 2} by Remark 4.6.…”

“…We denote by Diff (C n , 0) the group of local complex analytic diffeomorphisms defined in a neighborhood of the origin of C n . The algebraic nature of subgroups of Diff (C n , 0) is studied from different points of view in the literature, for instance in the context of groups of real analytic diffeomorphisms in compact manifolds [7], the existence of faithful analytic actions of mapping class groups of surfaces on surfaces [6], the study of integrability properties of one-dimensional foliations [5,13], local intersection dynamics [1,17], the study of the derived length [10,15]. This paper generalizes two results introduced by Rebelo and Reis [14] (Theorems 1 and 2) that relate algebraic properties of a subgroup G of Diff (C 2 , 0) (being non-virtually solvable)…”

“…Conditions (1) and (2) in Theorem 1 can be weakened by checking them out on all subgroups of G. For instance Theorem 1 holds if we replace Condition (1) with the existence of a non-solvable unipotent subgroup of G.…”