Small volume closed hyperbolic 4-manifolds

Long, Cormac D.

doi:10.1112/blms/bdn077

Cited by 13 publications

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“…(1) Can one find an explicit hyperbolic example (such as the Davis manifold or the manifolds described in [11]) that satisfies the properties of Proposition 2.2? Recall that the Davis manifold has b 1 = 24 and b + 2 = 36 [16], so that all moduli spaces have odd dimension.…”

Section: Discussionmentioning

confidence: 99%

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Hyperbolic four-manifolds with vanishing Seiberg-Witten invariants

Agol¹,

Lin²

2018

Preprint

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Section: Discussionmentioning

confidence: 99%

“…we would have χ(X) ≤ 4; on the other hand, in all known examples of closed orientable hyperbolic 4-manifolds χ ≥ 16 [14,11] (recall that by Gauss-Bonnet, volume and Euler characteristic are proportional).…”

Section: A Vanishing Criterion For the Seiberg-witten Invariantsmentioning

confidence: 98%

Hyperbolic four-manifolds with vanishing Seiberg-Witten invariants

Agol¹,

Lin²

2018

Preprint

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“…In 2008 Long discovered more examples [40] of torsion-free subgroups of Γ 1 of index 11520 and hence of non-orientable closed hyperbolic four-manifolds with χ = 8.…”

Section: The Davis Manifoldmentioning

confidence: 99%

Hyperbolic four-manifolds

Martelli¹

2015

Preprint

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This is a short survey on finite-volume hyperbolic four-manifolds. We first describe some general theorems, and then focus on the geometry of the concrete examples that we found in the literature.The starting point of most constructions is an explicit reflection group Γ acting on H 4 , together with its Coxeter polytope P . Hyperbolic manifolds then arise either algebraically from the determination of torsion-free subgroups of Γ, or more geometrically by assembling copies of P .We end the survey by raising a few open questions.

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“…Further examples with χ(M ) = 16 have been obtained by Long in [18] by considering a homomorphism from W 1 onto the finite simple group PSp 4 (4).…”

Section: Theorem 2 (Belolipetskymentioning

confidence: 99%

On compact hyperbolic manifolds of Euler characteristic two

Emery

2014

Algebr. Geom. Topol.

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Abstract. 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 4.1. Main result and discussion 1.1. Smallest hyperbolic manifolds. Let H n be the hyperbolic n-space. By a hyperbolic n-manifold we mean an orientable manifold M = Γ\H n , where Γ is a torsion-free discrete subgroup Γ ⊂ Isom + (H n ). The set of volumes of hyperbolic n-manifolds being well ordered, it is natural to try to determine for each dimension n the hyperbolic manifolds of smallest volume. For n = 3 this problem has recently been solved in [15], the smallest volume being achieved by a unique compact manifold, the Weeks manifold. When n is even the volume is proportional to the Euler characteristic, and this allows to formulate the problem in terms of finding the hyperbolic manifolds M with smallest |χ(M )|. In particular this observation solves the problem in the case of surfaces. For n > 3, noncompact hyperbolic n-manifolds M with |χ(M )| = 1 have been found for n = 4, 6 [14].In the present paper we consider the case of compact manifolds of even dimension. In particular, such manifolds have even Euler characteristic (see [17, Theorem 1.2]). We restrict ourselves to the case of arithmetic manifolds, where Prasad's formula [20] can be used to study volumes. We complete the proof of the following result. The result for n > 10 already follows from the work of Belolipetsky [4, 5], also based on Prasad's volume formula. More precisely, Belolipetsky determined the smallest Euler characteristic |χ(Γ)| for arithmetic orbifold quotients Γ\H n (n even). This smallest value grows fast with the dimension n, and for compact quotients we have |χ(Γ)| > 2 for n > 10. That the result of nonexistence holds for n high enough is already a consequence of Borel-Prasad's general finiteness result [9], which was the first application of Prasad's formula. The proof of Theorem 1 for n = 6, 8, 10 requires a more precise analysis of the Euler characteristic of arithmetic subgroups Γ ⊂ PO(n, 1), and in particular of the special values of Dedekind zeta functions that appear as factors of χ(Γ).For n = 4, the corresponding problem is not solved, but there is the following result [5]. Theorem 2 (Belolipetsky). If M = Γ\H 4 is a compact arithmetic manifold with χ(M ) ≤ 16, then Γ arises as a (torsion-free) subgroup of the following hyperbolic Coxeter group:An arithmetic (orientable) hyperbolic 4-manifold of Euler characteristic 16 has been first constructed by Conder and Maclachlan in [12], using the presentation of W 1 to obtain a torsion-free subgroup with the help of a computer. Further examples with χ(M ) = 16 have been obtained by Long in [18] by considering a homomorphism from W 1 onto the finite simple group PSp 4 (4).1.2. Hyperbolic homology spheres. Our original motivation for Theorem 1 was the problem of existence of hyperbolic homology spheres. A homo...

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Small volume closed hyperbolic 4-manifolds

Cited by 13 publications

References 3 publications

Hyperbolic four-manifolds with vanishing Seiberg-Witten invariants

Hyperbolic four-manifolds with vanishing Seiberg-Witten invariants

Hyperbolic four-manifolds

On compact hyperbolic manifolds of Euler characteristic two

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