Vincent Emery

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.

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 Prasads volume to compute the volumes of the smallest hyperbolic arithmetic orbifolds.

The three smallest compact arithmetic hyperbolic 5–orbifolds

We determine the three hyperbolic 5-orbifolds of smallest volume among compact arithmetic orbifolds, and we identify their fundamental groups with hyperbolic Coxeter groups. This gives two different ways to compute the volume of these orbifolds.

On compact hyperbolic manifolds of Euler characteristic two

We prove that for n>4 there is no compact arithmetic hyperbolic n-manifold whose Euler characteristic has absolute value equal to 2. In particular, this shows the nonexistence of arithmetically defined hyperbolic rational homology n-sphere with n even different than 4.

Arbitrarily large families of spaces of the same volume

In any connected non-compact semi-simple Lie group without factors locally isomorphic to

Covolumes of nonuniform lattices in PU(n, 1)

{SL_2(\mathbb {R})}

Hermitian lattices and bounds in K-theory of algebraic integers

, 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 covolumes.

Quaternionic hyperbolic lattices of minimal covolume

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