Salem numbers and arithmetic hyperbolic groups

Arithmetic Hyperbolic Reflection Groups

A hyperbolic reflection group is a discrete group generated by reflections in the faces of an n-dimensional hyperbolic polyhedron. This survey article is dedicated to the study of arithmetic hyperbolic reflection groups with an emphasis on the results that were obtained in the last ten years and on the open problems.

Growth Series of Some Hyperbolic Graphs and Salem Numbers

Extending the analogous result of Cannon and Wagreich for the fundamental groups of surfaces, we show that, for the l-regular graphs Xl,m associated to regular tessellations of hyperbolic plane by m-gons, the denominators of the growth series (which are rational and were computed by Floyd and Plotnick [FP94]) are reciprocal Salem polynomials. As a consequence, the growth rates of these graphs a...

Salem numbers and growth series of some hyperbolic graphs

Extending the analogous result of Cannon andWagreich for the fundamental groups of surfaces, we show that, for the l-regular graphs Xl,m associated to regular tessellations of the hyperbolic plane by m-gons, the denominators of the growth series (which are rational and were computed by Floyd and Plotnick (Floyd and Plotnick, 1994)) are reciprocal Salem polynomials. As a consequence, the growth ...

Coxeter Groups, Salem Numbers and the Hilbert Metric

Finiteness of Arithmetic Hyperbolic Reflection Groups

We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.