Hyperbolic reflection groups are discrete Coxeter groups, they provide an important sources of finite volume hyperbolic orbifolds. Despite a systemic study of them from 1960’s by Vinberg, we are still very from the classification of them. In the joint work with Fangting Zhang, we give the classification of all the compact 4-dimensional hyperbolic Coxeter polytopes with eight facets。 The proof uses geometrical, combinatorial, and computational techniques.

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