Two-sided complexity bounds for Lobell manifolds

Two-sided, upper and lower, complexity bounds are obtained for a countable family of closed hyperbolic manifolds known as Löbell manifolds. The upper bound is based on the construction of fundamental polyhedra in the Lobachevsky space. The polyhedra are parametrized by an integer parameter n ≥ 5. The lower bound is based on the computation of their volumes and is asymtotic in nature. The upper and lower bounds are linear with respect to n. The Löbell manifolds is obtained by gluing together eight copies of bounded rectangular polyhedron. The vertices are chosen so that the marked vertices are mapped to each other under the identification of the faces.