Torsion homology of arithmetic lattices and K2 of imaginary fields

We study upper bounds for the torsion in homology of nonuniform arithmetic lattices. Together with recent results of Calegari-Venkatesh, this can be used to obtain upper bounds on K2 of the ring of integers of totally imaginary fields.Comment: Version 2 is a major update. Result for S-integers added. This allows to show case d=2 (imaginary quadratic fields) and d=4 in Theorem 1.3, previously excluded. 12 pages (previously 6

Even unimodular Lorentzian lattices and hyperbolic volume

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

On volumes of arithmetic quotients of PO(n,1), n odd

We determine the minimal volume of arithmetic hyperbolic orientable n-dimensional orbifolds (compact and non-compact) for every odd dimension n>3. Combined with the previously known results it solves the minimal volume problem for arithmetic hyperbolic n-orbifolds in all dimensions.Comment: 34 pages, final revision, to appear in Proc. LM

Hermitian lattices and bounds in K-theory of algebraic integers

Elaborating on a method of Soul\'e, and using better estimates for the geometry of hermitian lattices, we improve the upper bounds for the torsion part of the K-theory of the rings of integers of number fields.Comment: 11 pages. Minor chang

Arbitrarily large families of spaces of the same volume

In any connected non-compact semi-simple Lie group without factors locally isomorphic to $${SL_2(\mathbb {R})}$$ , there can be only finitely many lattices (up to isomorphism) of a given covolume. We show that there exist arbitrarily large families of pairwise non-isomorphic arithmetic lattices of the same covolume. We construct these lattices with the help of Bruhat-Tits theory, using Prasad's volume formula to control their covolume

Salem numbers and arithmetic hyperbolic groups

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.Comment: The exposition in version 3 is more compact; this shortens the paper: 26 pages now instead of 37. A discussion on Lehmer's problem has been added in Section 1.2. Final version, to appear is Trans. AM