Topics surrounding the combinatorial anabelian geometry of hyperbolic curves III: Tripods and tempered fundamental groups

Let Σ be a subset of the set of prime numbers that is either equal to the entire set of prime numbers or of cardinality one. In the present paper, we continue our study of the pro-Σ fundamental groups of hyperbolic curves and their associated configuration spaces over algebraically closed fields in which the primes of Σ are invertible. The focus of the present paper is on applications of the theory developed in previous papers to the theory of tempered fundamental groups, in the style of André. These applications are motivated by the goal of surmounting two fundamental technical difficulties that appear in previous work of André—namely, (a) the fact that the characterization of the local Galois groups in the global Galois image associated to a hyperbolic curve that is given in earlier work of André is proved only for a quite limited class of hyperbolic curves—that is, a class that is far from generic; (b) the proof given in earlier work of André of a certain key injectivity result, which is of central importance in establishing the theory of a p-adic local analogue of the well-known global theory of the Grothendieck–Teichmüller group, contains a fundamental gap. In the present paper, we surmount these technical difficulties by introducing the notion of an M-admissible or metric-admissible outer automorphism of the profinite geometric fundamental group of a p-adic hyperbolic curve. Roughly speaking, M-admissible outer automorphisms are outer automorphisms that are compatible with the data constituted by the indices at the various nodes of the special fiber of the p-adic curve under consideration. By combining this notion with combinatorial anabelian results and techniques developed in earlier papers by the authors, together with the theory of cyclotomic synchronization (also developed in earlier papers by the authors), we obtain a generalization of André’s characterization of the local Galois groups in the global Galois image associated to a hyperbolic curve to the case of arbitrary hyperbolic curves [cf. (a)]. Moreover, by applying the theory of local contractibility of p-adic analytic spaces developed by Berkovich, we show that the techniques developed in the present and earlier papers by the authors allow one to relate the groups of M-admissible outer automorphisms treated in the present paper to the groups of outer automorphisms of tempered fundamental groups of higher-dimensional configuration spaces (associated to the given p-adic hyperbolic curve). These considerations allow one to repair the gap in André’s proof—albeit at the expense of working with M-admissible outer automorphisms—and hence, to realize the goal of obtaining a local analogue of the Grothendieck–Teichmüller group [cf. (b)].