Surfaces and p-adic fields I: Dehn twists

Abstract

Following the philosophy of arithmetic topology, we describe a point of view which helps look at surfaces and $p$-adic fields in a "uniform way", and show that results on mapping class groups can be extended to this point of view, and thus be applied to $G_{K}$, the absolute Galois groups of the $p$-adic field $K$. By moving both groups to the world of pro-$p$ groups (for $G_{K}$ we take its maximal pro-$p$ quotient, and for $\pi_{1}(S)$ we take its pro-$p$ completion), we see they both are pro-$p$ Poincare duality groups of dimension $2$, also known as Demuskin groups. Such groups have a very nice classification in terms of generators and relations. By using the language of graphs of groups and examining discrete groups with Demuskin type relations, we show that all splittings of a Demuskin group come from a discrete splitting, which in turn helps us show that Dehn twists make sense in such a context. This gives us a family of infinite order Outer automorphisms of $G_{K}(p)$ the maximal pro-$p$ quotient of the Galois group, which are "arithmetic Dehn twists". On the other hand, when specializing this to Demuskin groups coming from surface groups, one gets back the usual definition of Dehn twists on surfaces. As a finally corollary, we show that there in an infinite family of non-isomorphic discrete groups, having isomorphic pro-$l$ completions for all primes $l$ (which are free pro-$l$ for $l \neq p$ and Demuskin for $l=p$).