The galaxy of Coxeter groups
arXiv:2211.17038 · doi:10.1016/j.jalgebra.2023.12.006
Abstract
In this paper we introduce the galaxy of Coxeter groups -- an infinite dimensional, locally finite, ranked simplicial complex which captures isomorphisms between Coxeter systems. In doing so, we would like to suggest a new framework to study the isomorphism problem for Coxeter groups. We prove some structural results about this space, provide a full characterization in small ranks and propose many questions. In addition we survey known tools, results and conjectures. Along the way we show profinite rigidity of triangle Coxeter groups -- a result which is possibly of independent interest.
arXiv comment30 pages, 6 figures. v2: Incorporated referee's suggestions; Corrected a mistake in the proof of Theorem 4.25 (formerly 4.24), improved other proofs and text; v3: Corrected a few typos, final version. (Journal of Algebra (2023), in press.)
References in corpus (6)
- Twist-rigid Coxeter groups
- Bipolar Coxeter groups
- On the direct indecomposability of infinite irreducible Coxeter groups and the Isomorphism Problem of Coxeter groups
- Angle-deformations in Coxeter groups
- Matching theorems for systems of a finitely generated Coxeter group
- Cohomology of Fuchsian Groups and Non-Euclidean Crystallographic Groups