RESEARCH GROUP

GEOMETRY

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.

X-MATH (eXperimental MATHematics) is an online asynchronous symposium organized by the Heidelberg Experimental Geometry Lab (HEGL) showcasing projects from Heidelberg and Magdeburg. Marco Lotz is presenting a project there. Th website is https://x-math.mathi.uni-heidelberg.de/

Our working group is organizing a Mini-Workshop on "Reflection Groups", taking place 8th & 9th of February 2023 in Magdeburg,
as a satellite event of the upcoming "Young Geometric Group Theory XI" conference (organized by us, Münster, and California). More info on the homepage of the workshop:

https://www.geometry.ovgu.de/MiniReflection.html

This 2-day workshop will include overview talks, a problem session, and invited talks by
 -Michelle Chu (University of Minnesota – Twin Cities, USA),
 -Annette Karrer (McGill University, Canada),
 -Mireille Soergel (Université de Bourgogne, France / ETH Zürich, Switzerland).