16 research outputs found

### Some hyperbolic 4-manifolds with low volume and number of cusps

We construct here two new examples of non-orientable, non-compact, hyperbolic
4-manifolds. The first has minimal volume $v_m = 4{\pi}^2/3$ and two cusps.
This example has the lowest number of cusps among known minimal volume
hyperbolic 4-manifolds. The second has volume $2\cdot v_m$ and one cusp. It has
lowest volume among known one-cusped hyperbolic 4-manifolds.Comment: 12 pages, 11 figure

### The complement of the figure-eight knot geometrically bounds

We show that some hyperbolic 3-manifolds which are tessellated by copies of
the regular ideal hyperbolic tetrahedron embed geodesically in a complete,
finite volume, hyperbolic 4-manifold. This allows us to prove that the
complement of the figure-eight knot geometrically bounds a complete, finite
volume hyperbolic 4-manifold. This the first example of geometrically bounding
hyperbolic knot complement and, amongst known examples of geometrically
bounding manifolds, the one with the smallest volume.Comment: 9 pages, 4 figures, typos corrected, improved exposition of
tetrahedral manifolds. Added Proposition 3.3, which gives necessary and
sufficient conditions for M_T to be a manifold, and Remark 4.4, which shows
that the figure-eight knot bounds a 4-manifold of minimal volume. Updated
bibliograph

### New hyperbolic 4-manifolds of low volume

We prove that there are at least 2 commensurability classes of minimal-volume
hyperbolic 4-manifolds. Moreover, by applying a well-known technique due to
Gromov and Piatetski-Shapiro, we build the smallest known non-arithmetic
hyperbolic 4-manifold.Comment: 21 pages, 6 figures. Added the Coxeter diagrams of the
commensurability classes of the manifolds. New and better proof of Lemma 2.2.
Modified statements and proofs of the main theorems: now there are two
commensurabilty classes of minimal volume manifolds. Typos correcte

### Hyperbolic four-manifolds, colourings and mutations

We develop a way of seeing a complete orientable hyperbolic $4$-manifold
$\mathcal{M}$ as an orbifold cover of a Coxeter polytope $\mathcal{P} \subset
\mathbb{H}^4$ that has a facet colouring. We also develop a way of finding
totally geodesic sub-manifolds $\mathcal{N}$ in $\mathcal{M}$, and describing
the result of mutations along $\mathcal{N}$. As an application of our method,
we construct an example of a complete orientable hyperbolic $4$-manifold
$\mathcal{X}$ with a single non-toric cusp and a complete orientable hyperbolic
$4$-manifold $\mathcal{Y}$ with a single toric cusp. Both $\mathcal{X}$ and
$\mathcal{Y}$ have twice the minimal volume among all complete orientable
hyperbolic $4$-manifolds.Comment: 24 pages, 11 figures; to appear in Proceedings of the London
Mathematical Societ

### Embedding arithmetic hyperbolic manifolds

We prove that any arithmetic hyperbolic $n$-manifold of simplest type can
either be geodesically embedded into an arithmetic hyperbolic $(n+1)$-manifold
or its universal $\mathrm{mod}~2$ Abelian cover can.Comment: 20 pages; revised version, typos corrected; Mathematical Research
Letters vol. 25, no.

### Convex plumbings in closed hyperbolic 4-manifolds

We show that every plumbing of disc bundles over surfaces whose genera
satisfy a simple inequality may be embedded as a convex submanifold in some
closed hyperbolic four-manifold. In particular its interior has a geometrically
finite hyperbolic structure that covers a closed hyperbolic four-manifold.Comment: 18 pages, 11 figure

### Arithmetic trialitarian hyperbolic lattices are not LERF

A group is LERF (locally extended residually finite) if all its finitely
generated subgroups are separable. We prove that the trialitarian arithmetic
lattices in $\mathbf{PSO}_{7,1}(\mathbb{R})$ are not LERF. This result,
together with previous work by the third author, implies that all arithmetic
lattices in $\mathbf{PO}_{n,1}(\mathbb{R})$, $n>3$, are not LERF.Comment: 8 pages, 1 figur

### Symmetries of Hyperbolic 4-Manifolds

SAGE worksheet enclosed / une feuille de calcule SAGE annexéeInternational audienceIn this paper, for each finite group G, we construct the first explicit examples of non-compact complete finite-volume arithmetic hyperbolic 4-manifolds M such that Isom M ∼ = G, or Isom + M ∼ = G. In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic 4-space, on one hand, and the combinatorics of simplicial complexes, on the other. This allows us to obtain a universal upper bound on the minimal volume of a hyperbolic 4-manifold realising a given finite group G as its isometry group in terms of the order of the group. We also obtain asymptotic bounds for the growth rate, with respect to volume, of the number of hyperbolic 4-manifolds having a finite group G as their isometry group.Pour chaque groupe G fini, nous construisons des premiers exemples explicites de 4-variétés non-compactes complètes arithmétiques hyperboliques M , ` a volume fini, telles que Isom M ∼ = G, ou Isom + M ∼ = G. Pour y parvenir, nous utilisons essentiellement la géométrie de polyèdres de Coxeter dans l'espace hyperbolique en dimension quatre, et aussi la combinatoire de complexes simpliciaux. C ¸ a nous permet d'obtenir une borne supérieure universelle pour le volume minimal d'une 4-variété hyperbolique ayant le groupe G comme son groupe d’isométries, par rapport de l'ordre du groupe. Nous obtenons aussi des bornes asymptotiques pour le taux de croissance, par rapport du volume, du nombre de 4-variétés hyperboliques ayant G comme le groupe d’isométries