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)

Disclaimer applies, as usual 😦

1 Comment »

  1. The formula (6.10) on page 88 contains a typo: the right formula should be G_{p,0}'=\{b\cdot \exp t F_{p,0}\colon b\in\mathbb C^*,\ t\in\mathbb C\}.

    This comment is at the same time an invitation to submit corrections of typos and more grave errors in the textbook. The list will be maintained by AMS and serve the community of future readers.

    Comment by Sergei Yakovenko — Saturday, December 15, 2007 @ 8:32 | Reply

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