Séminaire Lotharingien de Combinatoire, 89B.26 (2023), 12 pp.
Amanda Burcroff
Compact Hyperbolic Coxeter d-Polytopes with d+4 Facets and Related Dimension Bounds
Abstract.
We complete the classification of compact hyperbolic Coxeter d-polytopes with d+4 facets for d=4 and 5. By previous work of Felikson and Tumarkin, the only remaining dimension where new polytopes may arise is d=6. We derive a new method for generating the combinatorial types of these polytopes via the classification of point set order types. In dimensions 4 and 5, there are 348 and 51 polytopes, respectively, yielding many new examples for further study.
We furthermore provide new upper bounds on the dimension d of compact hyperbolic Coxeter polytopes with d+k facets for k ≤ 10. It was shown by Vinberg in 1985 that there are no compact hyperbolic Coxeter polytopes in dimensions higher than 29, and no better dimension bounds have previously been published for k ≥ 5. As a consequence of our bounds, we prove that a compact hyperbolic Coxeter 29-polytope has at least 40 facets.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
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