We determine the minimal volume of arithmetic hyperbolic orientable ndimensional orbifolds (compact and non-compact) for every odd dimension n ≥ 5. Combined with the previously known results it solves the minimal volume problem for arithmetic hyperbolic n-orbifolds in all dimensions.
We compute the hyperbolic covolume of the automorphism group of each even
unimodular Lorentzian lattice. The result is obtained as a consequence of a
previous work with Belolipetsky, which uses Prasad's volume to compute the
volumes of the smallest hyperbolic arithmetic orbifolds.Comment: minor modifications. To appear in J. Reine Angew. Mat
In this paper we prove that there is a direct relationship between Salem numbers and translation lengths of hyperbolic elements of arithmetic hyperbolic groups that are determined by a quadratic form over a totally real number field. As an application we determine a sharp lower bound for the length of a closed geodesic in a noncompact arithmetic hyperbolic n-orbifold for each dimension n. We also discuss a "short geodesic conjecture", and prove its equivalence with "Lehmer's conjecture" for Salem numbers.
We determine the three hyperbolic 5-orbifolds of smallest volume among compact arithmetic orbifolds, and we identify their fundamental groups with hyperbolic Coxeter groups.
We prove that the covolume of any quasi-arithmetic hyperbolic lattice (a notion that generalizes the definition of arithmetic subgroups) is a rational multiple of the covolume of an arithmetic subgroup. As a corollary, we obtain a good description for the shape of the volumes of most of the known hyperbolic n-manifolds with n > 3.
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