The Geometry of Hyperbolic Curvoids

— The main purposes of the present paper are to introduce the notion of a hyperbolic curvoid and to study the geometry of hyperbolic curvoids. A hyperbolic curvoid is defined to be a certain profinite group and may be considered to be “group-theoretic abstraction” of the notion of a hyperbolic curve from the viewpoint of anabelian geometry. One typical example of a hyperbolic curvoid is a profinite group isomorphic to the étale fundamental group of a hyperbolic curve either over a number field or over a mixed-characteristic nonarchimedean local field. The first part of the present paper centers around establishments of a construction of the “geometric subgroup” of hyperbolic curvoids and a construction of the “collection of cuspidal inertia subgroups” of hyperbolic curvoids. Moreover, we also consider respective analogues for hyperbolic curvoids of the theory of partial compactifications of hyperbolic curves and the theory of quotient orbicurves of hyperbolic curves by actions of finite groups.