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 graphs of polynomials, we exhibit a non-trivial example of a degree 6 polynomial failing to have this property. This example settles a problem raised in a recent paper of the second author regarding constant weighted sums of polynomials in the complex plane. We also show that degree 6 is exceptional in another regard, as it is the lowest degree for which the monodromy group of a polynomial is not determined by the combinatorics of the post-critical set. These results give new applications of iterated monodromy groups to complex dynamics.     

Realizing alternating groups as monodromy groups of genus one covers

We prove that if , a generic Riemann surface of genus 1 admits a meromorphic function (i.e., an analytic branched cover of ) of degree such that every branch point has multiplicity and the monodromy group is the alternating group . To prove this theorem, we construct a Hurwitz space and show that it maps (generically) onto the genus one moduli space.

     

Skew-symmetric vanishing lattices and their monodromy groups

Mathematische Annalen -     

Exceptional polynomials with 2-transitive affine monodromy groups

In this paper all exceptional polynomials having a doubly transitive affine arithmetic monodromy group are classified. We show that these polynomials belong to well-known families of exceptional polynomials. We construct such polynomials explicitly.     

Arithmetic groups and Salem numbers

We show that the existence of a sequence of elements from cocompact torsion-free arithmetic subgroups ofSL(2,R) converging to the identity is equivalent to the density of Salem numbers in [1,∞).     

Certain unipotent representations of finite Chevalley groups and Picard-Lefschetz monodromy

The purpose of this note is to give a strange relation between the dimension of certain unipotent representations of finite Chevalley groups of type G₂, F₄, and E₈ on the one hand, and the minimal polynomials of the Picard-Lefschetz monodromy on the other hand.     

Arithmetic embeddings of algebraic groups

Embeddings of algebraic groups are considered. The action of an algebraic group on these embeddings preserving their arithmertic properties is defined. A formula for the number of orbits is presented. Bibliography 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 146–149. Translated by M. N. Kubenskii.     

Arithmetic groups with isomorphic finite quotients

Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its profinite completion. We show that for a wide class of S-arithmetic groups, this map is finite to one, while the fibers are of unbounded size.