Topics Surrounding the Combinatorial Anabelian Geometry of Hyperbolic Curves: Tripods and Combinatorial Cuspidalization (2). Questo articolo non è disponibile.Topics Surrounding the Combinatorial Anabelian Geometry of Hyperbolic Curves: Tripods and Combinatorial Cuspidalization (2)

<div>The present monograph further develops the study, via the techniques of combinatorial anabelian geometry, of the profinite fundamental groups of <i>configuration spaces</i> associated to <i>hyperbolic curves</i> over algebraically closed fields of characteristic zero.</div><div><br></div><div>The starting point of the theory of the present monograph is a <i>combinatorial anabelian</i> 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 <i>purely group-theoretic characterizations</i> of the <i>cuspidal inertia subgroups</i> of one-dimensional subquotients of the profinite fundamental group of a configuration space.</div><div><br></div><div>We then turn to the study of <i>tripod synchronization</i>, 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 induces the <i>same</i> outer automorphism on certain subquotients of such a fundamental group determined by <i>tripods</i> [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 <i>profinite Dehn multi-twists</i>.</div><br><div>In the final portion of the monograph, we develop a theory of <i>localizability</i>, 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 <i>lift</i> 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.</div><div><br></div>

The present monograph further develops the study, via the techniques of combinatorial anabelian geometry, of the profinite fundamental groups of&nbsp;<i>configuration spaces</i>&nbsp;associated to&nbsp;<i>hyperbolic curves</i>&nbsp;over algebraically closed fields of characteristic zero.<div><br></div><div>The starting point of the theory of the present monograph is a&nbsp;<i>combinatorial anabelian</i>&nbsp;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&nbsp;<i>purely group-theoretic characterizations</i>&nbsp;of the&nbsp;<i>cuspidal inertia subgroups</i>&nbsp;of one-dimensional subquotients of the profinite fundamental group of a configuration space.</div><div><br></div><div>We then turn to the study of&nbsp;<i>tripod synchronization</i>, 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 induces the&nbsp;<i>same</i>&nbsp;outer automorphism on certain subquotients of such a fundamental group determined by&nbsp;<i>tripods</i>&nbsp;[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&nbsp;<i>profinite Dehn multi-twists</i>.</div><div><br></div><div>In the final portion of the monograph, we develop a theory of&nbsp;<i>localizability</i>, 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&nbsp;<i>lift</i>&nbsp;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.</div><div><br></div>