Small eigenvalues of random 3-manifolds

Hamenstädt, Ursula; Viaggi, Gabriele

doi:10.1090/tran/8564

Cited by 4 publications

(1 citation statement)

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“…Moreover, like our manifolds (Theorem 1.2 (b)) they satisfy a law of large numbers for volume [43], with a constant depending on the underlying random walk. Their spectral gap behaves differently: it is inversely quadratic in volume [28]. Their injectivity radius has been studied in [42] and torsion in their homology in [3].…”

Section: Notes and Referencesmentioning

confidence: 99%

A model for random three-manifolds

Petri

Raimbault

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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