Combinatorial anabelian geometry is a modern branch of anabelian geometry which deals with those aspects of anabelian geometry which manifest themselves over algebraically closed fields of characteristic zero. The origin of combinatorial anabelian geometry is in S. Mochizuki’s pioneering papers from 2007, in which he reinterpreted and generalised some key components of his earlier famous proof of the Grothendieck conjecture. S. Mochizuki discovered that one can separate arguments which work over algebraically closed fields from arithmetic arguments, and study the former by using combinatorial methods. This led to a very nontrivial development of the theory of combinatorial anabelian geometry by S. Mochizuki and Y. Hoshi and other mathematicians. In this talk, after introducing the theory of combinatorial anabelian geometry I will discuss applications of combinatorial anabelian geometry to the study of the absolute Galois group of number fields and of p-adic local fields and to the study of the Grothendieck-Teichmueller group. In particular, I will talk about the recent construction of a splitting of the natural inclusion of the absolute Galois group of p-adic numbers to the (largest) p-adic Grothendieck–Teichmueller group and a splitting of the natural embedding of the absolute Galois group of rationals into the commensurator of the absolute Galois group of the maximal abelian extension of rationals in the Grothendieck–Teichmueller group.
- Algebra Seminar