On tilings defined by discrete reflection groups

TL;DR

This paper generalizes tiling results from Weyl groups to cocompact hyperbolic reflection groups and provides clearer proofs of existing theorems.

Contribution

It extends previous tiling theorems to hyperbolic reflection groups and simplifies the proofs of Waldspurger and Meinrenken's results.

Findings

Tilings by sets $(1-w)C^ ext{circ}$ form a partition in hyperbolic reflection groups.

Provides simplified proofs of existing tiling theorems.

Generalizes tiling results from affine Weyl groups to hyperbolic groups.

Abstract

The recent articles of Waldspurger and Meinrenken contained the results of tilings formed by the sets of type $(1-w)C^\circ$, $w\in W$, where $W$ is a linear or affine Weyl group, and $C^\circ$ is an open kernel of a fundamental chamber $C$ of the group $W$. In this article we generalize these results to cocompact hyperbolic reflection groups. We also give more clear and simple proofs of the Waldspurger and Meinrenken theorems.