Géométrie et topologie des variétés hyperboliques de grand volume

Raimbault, Jean

doi:10.5802/tsg.299

“…Their injectivity radius has been studied in [41] and torsion in their homology in [3]. Moreover, even if random Heegaard splittings turn out to be hyperbolic with probability 1 [34], unlike for instance random regular graphs [7] and random hyperbolic surfaces [10,37,38], they do not Benjamini-Schramm converge to their universal cover -i.e. already at a bounded scale, the geometry of these manifolds ceases to be that of H 3 .…”

Section: Some Remarks About These Resultsmentioning

confidence: 99%

A model for random three--manifolds

Petri¹,

Raimbault²

2020

Preprint

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We study compact three-manifolds with boundary obtained by randomly gluing together truncated tetrahedra along their faces. We prove that, asymptotically almost surely as the number of tetrahedra tends to infinity, these manifolds are connected and have a single boundary component. We prove a law of large numbers for the genus of this boundary component, we show that the Heegaard genus of these manifolds is linear in the number of tetrahedra and we bound their first Betti number.We also show that, asymptotically almost surely as the number of tetrahedra tends to infinity, our manifolds admit a unique hyperbolic metric with totally geodesic boundary. We prove a law of large numbers for the volume of this metric, prove that the associated Laplacian has a uniform spectral gap and show that the diameter of our manifolds is logarithmic as a function of their volume. Finally, we determine the Benjamini-Schramm limit of our sequence of random manifolds.

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“…For surfaces one can formulate similar questions using random models and they have a positive answer for discrete models (see e.g. [63,Appendix B]) or continuous ones (see [54]). In these dimensions it also makes sense to restrict to arithmetic manifolds for which Wang finiteness holds [18]; for surfaces it seems to be very likely that the answer would then be positive, see [50].…”

Section: Betti Numbersmentioning

confidence: 99%

Homotopy type and homology versus volume for arithmetic locally symmetric spaces

Frączyk¹,

Hurtado²,

Raimbault³

2022

Preprint

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We study locally symmetric spaces associated with arithmetic lattices in semisimple Lie groups. We prove the following results about their topology: the minimal number of tetrahedra needed for a triangulation is at most linear in the volume and the Betti numbers are sub-linear in the volume except possibly in middle degree. The proof of these results uses the geometry of these spaces, namely the study of their thin parts. In this regard we prove that these spaces converge in the Benjamini-Schramm sense to their universal covers and give an explicit bound for the volume of the thin part for trace fields of large degree. The main technical ingredients for our proofs are new estimates on orbital integrals, a counting result for elements of small displacement, and a refined version of the Margulis lemma for arithmetic locally symmetric spaces.1.1. Estimates on the volume of the thin part and the Gelander conjecture. Our first result is the following, which was conjectured to hold in [35]; we call an arithmetic locally-X manifold any quotient Γ\X where Γ is a torsion-free arithmetic lattice in G.

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“…Their injectivity radius has been studied in [42] and torsion in their homology in [3]. Moreover, even if random Heegaard splittings turn out to be hyperbolic with probability 1 [35], unlike for instance random regular graphs [7] and random hyperbolic surfaces [11,38,39], they do not Benjamini-Schramm converge to their universal cover -i.e. already at a bounded scale, the geometry of these manifolds ceases to be that of H 3 .…”

Section: Notes and Referencesmentioning

confidence: 99%

A model for random three-manifolds

Petri

¹

,

Raimbault

²

2022

Comment. Math. Helv.

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We study compact three-manifolds with boundary obtained by randomly gluing together truncated tetrahedra along their faces. We prove that, asymptotically almost surely as the number of tetrahedra tends to infinity, these manifolds are connected and have a single boundary component. We prove a law of large numbers for the genus of this boundary component, we show that the Heegaard genus of these manifolds is linear in the number of tetrahedra and we bound their first Betti number.We also show that, asymptotically almost surely as the number of tetrahedra tends to infinity, our manifolds admit a unique hyperbolic metric with totally geodesic boundary. We prove a law of large numbers for the volume of this metric, prove that the associated Laplacian has a uniform spectral gap and show that the diameter of our manifolds is logarithmic as a function of their volume. Finally, we determine the Benjamini-Schramm limit of our sequence of random manifolds.

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Géométrie et topologie des variétés hyperboliques de grand volume

Cited by 4 publications

References 36 publications

A model for random three--manifolds

A model for random three--manifolds

Homotopy type and homology versus volume for arithmetic locally symmetric spaces

A model for random three-manifolds

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