Deformation of finite-volume hyperbolic Coxeter polyhedra, limiting growth rates and Pisot numbers

Deformation of finite-volume hyperbolic Coxeter polyhedra, limiting growth rates and Pisot numbers

A connection between real poles of the growth functions for Coxeter groups acting on hyperbolic space of dimensions three and greater and algebraic integers is investigated. In particular, a certain geometric convergence of fundamental domains for cocompact hyperbolic Coxeter groups with finite-volume limiting polyhedron provides a relation between Salem numbers and Pisot numbers. Several examp...

Eriksson's numbers game and finite Coxeter groups

The numbers game is a one-player game played on a finite simple graph with certain “amplitudes” assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. This game and its interactions with Coxeter/Weyl group theory and Lie theory have been studied by many author...

The Volume of Hyperbolic Coxeter Polytopes of Even Dimension

On the Monotonicity of the Volume of Hyperbolic Convex Polyhedra

We give a proof of the monotonicity of the volume of nonobtuse-angled compact convex polyhedra in terms of their dihedral angles. More exactly we prove the following. Let P and Q be nonobtuse-angled compact convex polyhedra of the same simple combinatorial type in hyperbolic 3-space. If each (inner) dihedral angle of Q is at least as large as the corresponding (inner) dihedral angle of P , then...

Pisot Numbers and Greedy Algorithm