Anabelian geometry
For curves over algebraically closed fields of characteristic zero, these results connect geometric features with the structure of the étale fundamental group. A proper curve of genus at least two, or a punctured curve satisfying the hyperbolicity inequality, has a slim fundamental group: every open subgroup has trivial centralizer in the whole group. This controls centralizing symmetries throughout its finite covering theory.
A genus-zero curve with two punctures gives the boundary case, where the fundamental group is the profinite completion of the integers and admits a topological generator. The present results establish these geometric properties; arithmetic reconstruction of curves and the corresponding positive-characteristic results remain outside their scope.
Selected theorems and definitions
- theorem
Slimness for proper hyperbolic curves
Every open subgroup of the geometric fundamental group of a proper curve of genus at least two has trivial centralizer.
- theorem
Slimness for punctured hyperbolic curves
The same centralizer rigidity holds when the genus and number of punctures satisfy the hyperbolicity inequality.
- theorem
A genus-zero curve with two punctures
The actual étale fundamental group is the full profinite completion of the integers, with a chosen topological generator supplied by the conclusion.
References used in these guides
- Alexander Grothendieck et al.. Revêtements étales et groupe fondamental (SGA 1). Springer, Lecture Notes in Mathematics 224, 1971. Link to the updated edition with remarks by Michel Raynaud.
- Shinichi Mochizuki. The local pro-p anabelian geometry of curves. Inventiones mathematicae, 138, 319–423, 1999. Historical context for the arithmetic reconstruction problem; that reconstruction theorem is not claimed as formalized here.