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The preprint "Reflection length at infinity in hyperbolic reflection groups" by Marco Lotz has been recently uploaded to the arXiv!

Link to the paper: arXiv:2303.09300

Abstract:

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.

As of August 2021 Petra Schwer has joined the Editorial board of Innovations in Incidence Geometry - Algebraic, topological and combinatorial.

IIG  publishes carefully selected and peer-reviewed original research papers of the highest quality about all aspects of incidence geometry and its applications. These include

• finite and combinatorial geometry,
• rank-2 geometries,
• geometry of groups,
• Tits-buildings and diagram geometries,
• incidence geometric aspects of algebraic geometry,
• incidence geometric aspects of algebraic combinatorics,
• computational aspects,
• arrangements of hyperplanes,
• abstract polytopes and convex polytopes,
• tropical and F₁ geometry,
• Coxeter groups and root systems,
• topological geometry,
• applications of incidence geometry (including coding theory, cryptography, quantum information theory).