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Research news: arXiv:2303.09300

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.

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Petra Schwer joins Editorial board of IIG

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).

 

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New preprint by Petra Schwer

The following is new on the ArXiv:


Chimney retractions in affine buildings encode orbits in affine flag varieties

Elizabeth Milićević, Petra Schwer, Anne Thomas

This paper determines the relationship between the geometry of retractions and the combinatorics of folded galleries for arbitrary affine buildings, and so provides a unified framework to study orbits in affine flag varieties. We introduce the notion of labeled folded galleries for any affine building X and use these to describe the preimages of chimney retractions. When X is the building for a group with an affine Tits system, such as the Bruhat-Tits building for a group over a local field, we can then relate labeled folded galleries and shadows to double coset intersections in affine flag varieties. This result generalizes the authors' previous joint work with Naqvi on groups over function fields.
Comments: 31 pages, 4 figures best viewed in color
Subjects: Group Theory (math.GR); Algebraic Geometry (math.AG); Combinatorics (math.CO); Representation Theory (math.RT)
MSC classes: 20E42 (Primary), 05E45, 14M15, 20G25, 51E24 (Secondary)
Cite as:

arXiv:2207.12923 [math.GR]

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Groups and topological groups

On Saturday 9 July, Olga Varghese will talk about "Automatic continuity" in the FU Berlin.

See you there!

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Letzte Änderung: 21.05.2021 - Ansprechpartner: Webmaster