Semi-graphs of Anabelioids

Abstract

In this paper, we discuss various “general nonsense” aspects of the geometry of semi-graphs of profinite groups [cf. [Mzk3], Appendix], by applying the language of anabelioids introduced in [Mzk4]. After proving certain basic properties concerning various commensurators associated to a semi-graph of anabelioids, we show that the geometry of a semi-graph of anabelioids may be recovered from the category-theoretic structure of certain naturally associated categories — e.g., “temperoids” [in essence, the analogue of a Galois category for the “tempered fundamental groups” of [André]] and “categories of localizations”. Finally, we apply these techniques to obtain certain results in the absolute anabelian geometry [cf. [Mzk3], [Mzk8]] of tempered fundamental groups associated to hyperbolic curves over p-adic local fields.