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Abstract: Based on extensive numerical experiments, Don Zagier conjectured that for any finitely generated Fuchsian group of the first kind
there is a partition of the boundary of the hyperbolic plane (circle at infinity) and a Bowen-Series-like boundary map acting in a piecewise manner by generators of the group such that its two-dimensional natural extension has an attractor with finite rectangular structure which every point enters in finite time. The finite rectangular structure property along with other properties (conjecturally equivalent to it) form, in Zagier’s terminology, a reduction theory for the group. He conjectured that the rectangular structure persists even when the partition points used in defining the boundary map are perturbed in a continuous manner.
I will talk about several results, joint with Adam Abrams and Ilie Ugarcovici, in various combinations, where the finite rectangular structure property was proved.
For the modular group, for all (a,b)-continued fractions algorithms, where a and b are real numbers satisfying b-a ≤ 1, ab ≤ -1, for surface groups, for an open set of partitions, and recently, for all finitely generated Fuchsian groups with at least one cusp, a large class of Fuchsian group which contains all subgroups of the modular group, congruence or not. If time permits, I will talk about applications of the reduction theory to symbolic coding of geodesics.