The present monograph further develops the study, via the techniques of combinatorial anabelian geometry, of the profinite fundamental groups of
configuration spaces
associated to
hyperbolic curves
over algebraically closed fields of characteristic zero.
The starting point of the theory of the present monograph is a
combinatorial anabelian
result which allows one to reduce issues concerning the anabelian geometry of configuration spaces to issues concerning the anabelian geometry of hyperbolic curves, as well as to give
purely group-theoretic characterizations
of the
cuspidal inertia subgroups
of one-dimensional subquotients of the profinite fundamental group of a configuration space.
We then turn to the study of
tripod synchronization
, i.e., of the phenomenon that an outer automorphism of the profinite fundamental group of a log configuration space associated to a stable log curve inducesthe
same
outer automorphism on certain subquotients of such a fundamental group determined by
tripods
[i.e., copies of the projective line minus three points]. The theory of tripod synchronization shows that such outer automorphisms exhibit somewhat different behavior from the behavior that occurs in the case of discrete fundamental groups and, moreover, may be applied to obtain various strong results concerning
profinite Dehn multi-twists
.
In the final portion of the monograph, we develop a theory of
localizability
, on the dual graph of a stable log curve, for the condition that an outer automorphism of the profinite fundamental group of the stable log curve
lift
to an outer automorphism of the profinite fundamental group of a corresponding log configuration space. This localizability is combined with the theory of tripod synchronization to construct a purely combinatorial analogue of the natural outer surjection from the étale fundamental group of the moduli stack of hyperbolic curves over the field of rational numbers to the absolute Galois group of the field of rational numbers.