The 3rd MSJ-SI

The Mathematical Society of Japan, Seasonal Institute

Development of Galois-Teichmüller Theory
and Anabelian Geometry

October 30 Saturday, 11:30--12:20
Y. Hoshi (RIMS, Kyoto University)
Survey on the combinatorial anabelian geometry of hyperbolic curves
In this talk, I will give a survey on the combinatorial anabelian geometry of hyperbolic curves. First, I will review briefly the notion of a semi-graph of anabelioids of PSC-type, which is one of the main objects of interest in combinatorial anabelian geometry, and discuss Grothendieck conjecture-type results for outer isomorphisms between the fundamental groups of semi-graphs of anabelioids of PSC-type equipped with certain outer representations. Next, I will explain various consequences of these Grothendieck conjecture-type results: (1) the injectivity portion of combinatorial cuspidalization, (2) faithfulness of the outer Galois representations associated to hyperbolic curves, (3) a version of the Grothendieck conjecture for universal curves over moduli spaces of curves over algebraically closed fields. Finally, I will discuss a generalization of Yves Andre's result concerning the intersection of the outer Galois representation associated to a tripod over a number fie! ld and the group of outer automorphisms of the tempered fundamental group of the tripod.