Random methods in 3-manifold theory

Lubotzky, Alexander; Maher, Joseph F.; Wu, Conan

doi:10.48550/arxiv.1405.6410

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“…We say that a property P holds for a random 3-manifold of Heegaard genus g and maximal homology rank if the following holds: the proportion of 3-manifolds with P which are defined by a gluing with an element of the n-th step of the walk tends to one as n → ∞, independently of µ. To motivate this model, note that any 3-manifold M with Heegaard genus g and first Betti number b 1 (M ) = g is obtained as N ϕ for some ϕ ∈ I g (compare Section 2) and by [DT06] (see also [LMW14]), the Heegaard genus of a 3-manifold obtained from a random gluing in this sense is g. Furthermore, by a theorem by Maher [Ma10], a random manifold with Heegaard genus g and maximal homology rank is indeed hyperbolic.…”

Section: Introductionmentioning

confidence: 99%

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Exponential Torsion Growth for Random 3-Manifolds

Baik

Bauer

Gekhtman

et al. 2017

International Mathematics Research Notices

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Abstract. We show that a random 3-manifold with positive first Betti number admits a tower of cyclic covers with exponential torsion growth. IntroductionGiven a manifold M and a tower of coverings of M , i.e. a sequenceof finite covers, one can ask about the growth of topological invariants for the manifolds in the sequence. In the case that M is a hyperbolic 3-manifold of finite volume, the study of such questions led to interesting conjectures which relate the growth of invariants of the sequence to invariants of hyperbolic 3-space.More concretely, in the case of a tower of congruence covers of a closed arithmetic hyperbolic 3-manifold, conjecturally the growth rate of the torsion H 1 (M i , Z) tor in the first homology group coincides with the ℓ 2 -torsion of H 3 , which equals As it became apparent in recent years, the existence of towers of covers with exponential torsion homology growth should be abundant for 3-manifolds. The recent work [BGS16] explains that however, such towers do not exist for manifolds of higher dimension. The goal of this paper is to study the existence of towers of cyclic covers with exponential torsion growth for random 3-manifolds in a sense that we make precise next.Any closed 3-manifold M admits a Heegaard decomposition. This means that M can be obtained by gluing two handlebodies of some genus g ≥ 0 with a diffeomorphism of their boundaries. The smallest genus of a handlebody which gives rise to M in this way is called the Heegaard genus of M .For a fixed base identification, the manifold M only depends on the element in the mapping class group Mod(S g ) of the boundary surface S g of the handlebody defined by the gluing diffeomorphism. We denote by N ϕ the closed 3-manifold defined by the gluing map ϕ ∈ Mod(S g ). Thus topological properties of closed 3-manifolds N ϕ are directly related to properties of the mapping class ϕ.This viewpoint was used by Dunfield and Thurston [DT06] to define the notion of a random 3-manifold using a random walk on the mapping class group. Embarking from [DT06], the purpose of this work is to study cyclic covers 1 of random hyperbolic 3-manifolds with positive first Betti number.Let I g be the Torelli subgroup of Mod(S g ), i.e. the subgroup formed by all those mapping classes which act trivially on H 1 (S g ; Z). For g ≥ 3 this is a finitely generated group. We use the following model for random 3-manifolds with large Betti number, which is inspired by (but slightly different from) the Dunfield-Thurston model. Take any probability measure µ on I g whose support equals a finite set which generates I g as a semigroup. Such a µ defines a random walk on I g . We say that a property P holds for a random 3-manifold of Heegaard genus g and maximal homology rank if the following holds: the proportion of 3-manifolds with P which are defined by a gluing with an element of the n-th step of the walk tends to one as n → ∞, independently of µ. To motivate this model, note that any 3-manifold M with Heegaard genus g and first Betti number b 1 (M ) = g is obtain...

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Section: Introductionmentioning

confidence: 99%

“…A precise version of this result is Theorem 6.1 in Section 6. We do not discuss the rate of convergence although we believe that it can be derived from careful analysis of Benoist and Quint's work on random walks on reductive groups [BQ14,BQ16] (see also Section 7.6 of [Ko08] and [LMW14]).…”

Section: Introductionmentioning

confidence: 99%