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Abstract. We introduce an algorithm which transforms every fourdimensional cubulation into an orientable cusped finite-volume hyperbolic four-manifold. Combinatorially distinct cubulations give rise to topologically distinct manifolds.Using this algorithm we construct the first examples of finite-volume hyperbolic four-manifolds with one cusp. More generally, we show that the number of k-cusped hyperbolic four-manifolds with volume V grows like C V ln V for any fixed k. As a corollary, we deduce that the 3-torus bounds geometrically a hyperbolic manifold.

The present paper regards the volume function of a doubly truncated hyperbolic tetrahedron. Starting from the previous results of J. Murakami, U. Yano and A. Ushijima, we have developed a unified approach to expressing the volume in different geometric cases by dilogarithm functions and to treat properly the many analytic strata of the latter. Finally, several numeric examples are given.

RésuméNous proposons une nouvelle approcheà construire des variétés hyperboliques M en dimension quatre, par moyens d'un polyèdre de Coxeter P ⊂ H 4 munit avec un coloriage de ses faces. Aussi, nous utilisons notre méthode pour obtenir des sous-surfaces totalement géodésiques plongées dans M, et pour décrire le résultat de mutations par rapportà ses surfaces. Comme application, nous construisons une variété complète hyperbolique non-compacte X avec un bout cuspide non-torique, et une variété complète hyperbolique non-compacte Y avec un bout cuspide torique. Elles sont des nouveaux exemples de variétés hyperboliques complètes non-compactes en dimension quatre avec un seul bout cuspide et volume relativement petit. AbstractWe develop a way of seeing a complete orientable hyperbolic 4-manifold M as an orbifold cover of a Coxeter polytope P ⊂ H 4 that has a facet colouring. We also develop a way of finding a totally geodesic sub-manifold N in M, and describing the result of mutations along N . As an application of our method, we construct an example of a complete orientable hyperbolic 4-manifold X with a single non-toric cusp and a complete orientable hyperbolic 4-manifold Y with a single toric cusp. Both X and Y have twice the minimal volume among all complete orientable hyperbolic 4-manifolds.

On the optimality of the ideal right-angled 24-cell ALEXANDER KOLPAKOVWe prove that among four-dimensional ideal right-angled hyperbolic polytopes the 24-cell is of minimal volume and of minimal facet number. As a corollary, a dimension bound for ideal right-angled hyperbolic polytopes is obtained.