Surfaces and p-adic fields I: Dehn twists

Abstract

Under the philosophy of arithmetic topology, one would like to have an analogy between surfaces S, and p-adic fields K.In the following 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 GK , the absolute Galois groups of the p-adic field K.By moving both groups to the world of pro-p groups (for GK we take its maximal pro-p quotient, and for π1(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.Next, in order to move geometric ideas to a purely group theoretic setting, we use the language of graphs of groups and Bass-Serre trees.By restating results for surfaces purely in such a manner, we can (after proving a few technical results) get similar results for p-adic fields.We start by examining discrete groups with Demuskin type relations, for which we show a few pro-p rigidity type results.Using this 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 GK (p), the maximal pro-p quotient of GK , 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.We also get a curve complex for p-adic fields, namely the complex whose edges are graph of groups with GK (p) as their fundamental group, and 2 vertices have an edge between them if the graphs corresponding to the vertices are compatible. By the relations between discrete and pro-p Demuskin groups, we get a nice way to understand said curve complex.As a finally corollary, we show that there in an infinite family of nonisomorphic discrete groups, having isomorphic pro-l completions for all primes l (which are free pro-l for l = p and Demuskin for l = p). his DPhil and before it. In particular he would like to thank his viva examiners, Kobi Kremnitzer and Tomer Schlank, for helpful comments and corrections. He would also like to thank Jonathan Fruchter, Ido Grayevsky and Alex Lubotzky, for helpful discussions on materials related to the current paper, and to Dani Wise for suggesting the use of Whitehead's algorithm in the proof of the final corollary.The author would also like to thank Uri Bader and Tsachik Gelander, and their Midrasha on groups, where he learned most of his geometric group theory background.