Topics Surrounding the Combinatorial Anabelian Geometry of Hyperbolic Curves Ii: Tripods and Combinatorial Cuspidalization

Let Σ be a subset of the set of prime numbers which 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 starting point of the theory of the present paper is a combinatorial anabelian result which, unlike results obtained in previous papers, allows one to eliminate the hypothesis that cuspidal inertia subgroups are preserved by the isomorphism in question. This result allows us to [partially] generalize combinatorial cuspidalization results obtained in previous papers to the case of outer automorphisms of pro-Σ fundamental groups of configuration spaces that do not necessarily preserve the cuspidal inertia subgroups of the various one-dimensional subquotients of such a fundamental group. Such partial combinatorial cuspidalization results allow one in effect to reduce issues concerning the anabelian geometry of configuration spaces to issues concerning the anabelian geometry of hyperbolic curves. These results also allow us, in the case of configuration spaces of sufficiently large dimension, to give purely group-theoretic characterizations of the cuspidal inertia subgroups of the various one-dimensional subquotients of the pro-Σ fundamental group of a configuration space. We then turn to the study of tripod synchronization, i.e., roughly speaking, the phenomenon that an outer automorphism of the pro-Σ fundamental group of a log configuration space associated to a log stable curve typically induces the same outer automorphism on the various subquotients of such a fundamental group determined by tripods [i.e., copies of the projective line minus three points]. Our study of tripod synchronization allows us to show that outer automorphisms of pro-Σ fundamental groups of configuration spaces exhibit somewhat different behavior from the behavior that may be observed in the case of discrete fundamental groups, as a consequence of the classical DehnNielsen-Baer theorem. Other applications of the theory of tripod synchronization include a result concerning commuting profinite Dehn multi-twists that, a priori, arise from distinct semi-graph of 2010 Mathematics Subject Classification. Primary 14H30; Secondary 14H10.