On Compact Hyperbolic Coxeter Polytopes with Few Facets

This thesis is concerned with classifying and bounding the dimension of compact hyperbolic Coxeter polytopes with few facets. We derive a new method for generating the combinatorial type of these polytopes via the classification of point set order types. We use this to complete the classification of d-polytopes with d+4 facets for d=4 and 5. In dimensions 4 and 5, there are 341 and 50 polytopes, respectively, yielding many new examples for further study. By previous work of Felikson and Tumarkin, the only remaining dimension where new polytopes may arise is d=6. We furthermore show that any polytope of dimension 6 must have a missing face of size 3 or 4. The second portion of this thesis provides new upper bounds on the dimension of compact hyperbolic Coxeter d-polytopes with d+k facets for k <= 10. It was shown by Vinberg in 1985 that for any k, we have d <= 29, and no better bounds have previously been published for k >= 5. In the process of proving the present bounds, we additionally show that there are no compact hyperbolic Coxeter 3-free polytopes of dimension higher than 13. As a consequence of our bounds, we prove that a compact hyperbolic Coxeter 29-polytope has at least 40 facets.