2023/03/16 by Marco Lotz, Lotz, Marco
Mathematics · #20F55 #20F65 (Primary) 51M20 #51M10 (Secondary) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematics and Applications #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2303.09300
openalex publication_date 2023/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In a discrete group generated by hyperplane reflections in the n-dimensional hyperbolic space, the reflection length of an element is the minimal number of hyperplane reflections in the group that suffices to factor the element. For a Coxeter group that arises in this way and does not split into a direct product of spherical and affine reflection groups, the reflection length is unbounded. The action of the Coxeter group induces a tessellation of the hyperbolic space. After fixing a fundamental domain, there exists a bijection between the tiles and the group elements. We describe certain points in the visual boundary of the n-dimensional hyperbolic space for which every neighbourhood contains tiles of every reflection length. To prove this, we show that two disjoint hyperplanes in the n-dimensional hyperbolic space without common boundary points have a unique common perpendicular.