16 research outputs found

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

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    We construct here two new examples of non-orientable, non-compact, hyperbolic 4-manifolds. The first has minimal volume vm=4ŌÄ2/3v_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‚čÖvm2\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

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    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

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    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

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    We develop a way of seeing a complete orientable hyperbolic 44-manifold M\mathcal{M} as an orbifold cover of a Coxeter polytope P‚äāH4\mathcal{P} \subset \mathbb{H}^4 that has a facet colouring. We also develop a way of finding totally geodesic sub-manifolds N\mathcal{N} in M\mathcal{M}, and describing the result of mutations along N\mathcal{N}. As an application of our method, we construct an example of a complete orientable hyperbolic 44-manifold X\mathcal{X} with a single non-toric cusp and a complete orientable hyperbolic 44-manifold Y\mathcal{Y} with a single toric cusp. Both X\mathcal{X} and Y\mathcal{Y} have twice the minimal volume among all complete orientable hyperbolic 44-manifolds.Comment: 24 pages, 11 figures; to appear in Proceedings of the London Mathematical Societ

    Hyperbolic 4-manifolds and the 24-cells

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    Embedding arithmetic hyperbolic manifolds

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    We prove that any arithmetic hyperbolic nn-manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic (n+1)(n+1)-manifold or its universal mod 2\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

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    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

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    A group is LERF (locally extended residually finite) if all its finitely generated subgroups are separable. We prove that the trialitarian arithmetic lattices in PSO7,1(R)\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 POn,1(R)\mathbf{PO}_{n,1}(\mathbb{R}), n>3n>3, are not LERF.Comment: 8 pages, 1 figur

    Symmetries of Hyperbolic 4-Manifolds

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    SAGE worksheet enclosed / une feuille de calcule SAGE annex√©eInternational 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 ‚ąľ = G, or Isom + M ‚ąľ = 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-vari√©t√©s non-compactes compl√®tes arithm√©tiques hyperboliques M , ` a volume fini, telles que Isom M ‚ąľ = G, ou Isom + M ‚ąľ = G. Pour y parvenir, nous utilisons essentiellement la g√©om√©trie de poly√®dres de Coxeter dans l'espace hyperbolique en dimension quatre, et aussi la combinatoire de complexes simpliciaux. C ¬ł a nous permet d'obtenir une borne sup√©rieure universelle pour le volume minimal d'une 4-vari√©t√© hyperbolique ayant le groupe G comme son groupe d‚Äôisom√©tries, 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-vari√©t√©s hyperboliques ayant G comme le groupe d‚Äôisom√©tries
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