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Department of Mathematical Sciences


Publication details for Pavel Tumarkin

Felikson, A. & Tumarkin, P. (2014). Essential hyperbolic Coxeter polytopes. Israel Journal of Mathematics 199(1): 113-161.

Author(s) from Durham


We introduce a notion of an essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter polytopes. We determine a potentially large combinatorial class of polytopes containing, in particular, all the compact hyperbolic Coxeter polytopes of dimension at least 6 which are known to be essential, and prove that this class contains finitely many polytopes only. We also construct an effective algorithm of classifying polytopes from this class, realize it in the four-dimensional case, and formulate a conjecture on finiteness of the number of essential polytopes.