Monodromy in local groups

LOCAL MONODROMY OF p-DIVISIBLE GROUPS

A p-divisible group over a field K admits a slope decomposition; associated to each slope λ is an integer m and a representation Gal(K) → GLm(Dλ), where Dλ is the Qp-division algebra with Brauer invariant [λ]. We call m the multiplicity of λ in the p-divisible group. Let G0 be a p-divisible group over a field k. Suppose that λ is not a slope of G0, but that there exists a deformation of G in wh...

Iterated Monodromy Groups

We associate a group IMG(f) to every covering f of a topological space M by its open subset. It is the quotient of the fundamental group π1(M) by the intersection of the kernels of its monodromy action for the iterates fn. Every iterated monodromy group comes together with a naturally defined action on a rooted tree. We present an effective method to compute this action and show how the dynamic...

Monodromy Groups and Poincaré Series

Independence of ` of Monodromy Groups

Let X be a smooth curve over a finite field of characteristic p. Let E be a number field, and consider an E-compatible system L = {Lλ} of lisse sheaves on X . This means that for every place λ of E not lying over p, we are given a lisse Eλ-sheaf Lλ on X , and these lisse sheaves are E-compatible with one another in the sense that, for every closed point x of X , the polynomial det(1 − T Frobx,L...

Blocks of monodromy groups in complex dynamics

Motivated by a problem in complex dynamics, we examine the block structure of the natural action of iterated monodromy groups on the tree of preimages of a generic point. We show that in many cases, including when the polynomial has prime power degree, there are no large blocks other than those arising naturally from the tree structure. However, using a method of construction based on real grap...