Rosen fractions and Veech groups, an overly brief introduction

Schmidt, Thomas A.

Actes des rencontres du CIRM, Tome 1 (2009) no. 1, pp. 61-67.

We give a very brief, but gentle, sketch of an introduction both to the Rosen continued fractions and to a geometric setting to which they are related, given in terms of Veech groups. We have kept the informal approach of the talk at the Numerations conference, aimed at an audience assumed to have heard of neither of the topics of the title.

The Rosen continued fractions are a family of continued fraction algorithms, each gives expansions of real numbers in terms of elements of a corresponding algebraic number field. A Veech group is comprised of the Jacobians of locally affine self-maps on a “flat” surface to itself. The Rosen fractions are directly related to a certain family of (projective) matrix groups; these groups are directly related to W.  Veech’s original examples of surfaces with “optimal” dynamics.

Classification : 37-01, 37-02, 11-01, 11-02, 11J70, 37E35, 37F30

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