Sergei Yakovenko's blog: on Math and Teaching

Wednesday, December 12, 2007

“Auxiliary Lesson” שעור עזר) #8) December 13, 2007

Finitely generated subgroups of \text{Diff}(\mathbb C^1,0), I. Formal theory. 

  1. Formal normal form for a single holomorphic self-map from \text{Diff}(\mathbb C^1,0). Parabolic germs.
  2. Bochner theorem on holomorphic linearization of finite groups.
  3. Stratification of the subgroup of parabolic germs \text{Diff}_1(\mathbb C^1,0).
  4. Tits alternative for finitely generated subgroups of \text{Diff}(\mathbb C^1,0): every such subgroup is either metabelian (its commutator is commutative, e.g., trivial), or non-solvable (all iterated commutators are nontrivial).
  5. Centralizers and symmetries: formal classification of solvable subgroups.
  6. Integrable germs and their holomorphic linearizability.

Recommended reading: Section 6 (first part) from the book (printing disabled)

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