Tripod-degrees

Abstract

Let p, l be distinct prime numbers. A tripod-degree over p at l is defined to be an l-adic unit obtained by forming the image, by the l-adic cyclotomic character, of some continuous automorphism of the geometrically pro-l fundamental group of a split tripod over a finite field of characteristic p. The notion of a tripod-degree plays an important role in the study of the geometrically pro-l anabelian geometry of hyperbolic curves over finite fields, e.g., in the theory of cuspidalizations of the geometrically pro-l fundamental groups of hyperbolic curves over finite fields. In the present paper, we study the tripod-degrees. In particular, we prove that, under a certain condition, the group of tripod-degrees over p at l coincides with the closed subgroup of the group of l-adic units topologically generated by p. As an application of this result, we also conclude that, under a certain condition, the natural homomorphism from the group of automorphisms of the split tripod to the group of outer continuous automorphisms of the geometrically pro-l fundamental group of the split tripod that lie over the identity automorphism of the absolute Galois group of the basefield is surjective.