AA MODEL FOR RANDOM THREE–MANIFOLDS
BRAM PETRI AND JEAN RAIMBAULT
Abstract.
We study compact three-manifolds with boundary obtained by randomly gluingtogether truncated tetrahedra along their faces. We prove that, asymptotically almost surelyas the number of tetrahedra tends to infinity, these manifolds are connected and have a singleboundary 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 andwe 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 alaw of large numbers for the volume of this metric, prove that the associated Laplacian has auniform spectral gap and show that the diameter of our manifolds is logarithmic as a functionof their volume. Finally, we determine the Benjamini–Schramm limit of our sequence of randommanifolds. Introduction
Context.
Random constructions of compact manifolds can be seen as an analogue of thewell-established theory of random graphs and serve similar purposes. First of all, they makethe notion of a “typical” manifold rigorous. Secondly, they can be used as a testing ground forconjectures of which the proof is still out of reach. Finally, there is what is often called the probabilistic method – using probability theory to prove the existence of objects with extremalproperties. In this paper we are mostly interested in the first aspect.Let us be more specific about what kind of objects we are intetested in. As is the case for graphs,there are countably many homeomorphism types of compact manifolds. Thus a random manifoldconsists not in one random variable but rather a family of random variables—say M n , n ≥ n is some measure of “complexity”, usually in relation with a particular construction that isused to define the random objects. For graphs this will often be the number of vertices. Randommodels for 3–manifolds that have been well-studied are random Heegaard splittings and randomfibered manifolds; here the complexity depends on two integers: the genus g of the handlebodyor the fiber, and the number of steps k used to generate a random mapping class [19, Section2.10]. A basic property should be that the union of the support of the M n is the whole set of themanifolds one is interested in studying. The models above satisfy this requirement if one takesboth k, g → + ∞ (though only virtually for the second one) but the asymptotic results pertainingto them (in particular hyperbolicity) are mostly studied in terms of the mapping class (that is,when k → + ∞ ). If one is interested in studying typical 3–manifolds this does not seem satisfying.A more direct measure of the complexity of a 3–manifold is the minimal number of tetrahedraneeded to triangulate it. A natural way to construct random 3–manifolds is thus to start with amodel for a random triangulation on n tetrahedra and condition it to be a manifold. However,studying such a model of a random manifold is hard because if one randomly glues the faces of n tetrahedra together in pairs, the probability that the result is a manifold tends to 0 as n → ∞ (seefor instance [19, Proposition 2.8]). So we cannot rely on the study of a generic triangulation to Date : September 28, 2020. a r X i v : . [ m a t h . G T ] S e p BRAM PETRI AND JEAN RAIMBAULT establish a.a.s. properties of the manifolds and we have to instead study probabilities conditionedon a set of conditions that is hard to manage.We will not adress this issue in this paper, but note that even counting the number of triangu-lations is a hard problem (the best known bounds we are aware of are due to Chapuy–Perarnau[14]). Instead we will consider compact manifolds with boundary associated with random trian-gulations. The only points in a 3-dimensional triangulation that might not admit neighbourhoodshomeomorphic to open sets in R are the vertices. As such, we obtain a random 3–manifold withboundary by randomly gluing together n tetrahedra that are truncated near the vertices (seeFigure 1). Moreover all compact 3–manifolds with non-empty boundary can be obtained in thisway (see for example [16, Corollary 1.3]). Figure 1.
A truncated tetrahedron. M n is built by randomly gluing n copies ofthis polyhedron together along their hexagonal faces.We are interested in the asymptotic behaviour as n → + ∞ of geometric and topological prop-erties of M n . We are particularly interested in finding properties whose probability of occurence isasymptotically 1 (for regular graphs this can for instance be connectivity or expansion, dependingon the model). For 3–manifolds the most obvious candidate for such a property is hyperbolicity.We will prove that a.a.s. our manifolds are hyperbolic (with totally geodesic boundary) withvolume proportional to the number of tetrahedra, their Heegaard genus goes to infinity, and getestimates on their Betti numbers. We also prove some finer results about their geometry: theyare expanders and we show the converge to an explicit limit in a probabilistic version of theGromov–Hausdorff topology.1.2. Results.
We will impose one further condition: we condition on two tetrahedra sharing atmost one face and every face being incident to two distinct tetrahedra. This is strictly weakerthan asking that the complex be simplicial . The resulting random manifold will be called M n .A detailed description of the model can be found in Section 2.1.The first question now is what the topology of the resulting manifold is. We prove: Theorem 1.1 (Topology) . (a) We have lim n →∞ P [ M n is connected and has a single boundary component ] = 1 (b) The genus g ( ∂M n ) of the boundary of M n satisfies g ( ∂M n ) ∼ n as n → ∞ Even if it can be argued that this is not a very unnatural constraint, our main reason for setting this constraintis a technical one: we need it in the proof of Lemma 3.6
MODEL FOR RANDOM THREE–MANIFOLDS 3 in probability.(c) Let D M n denotes the double of M n along its boundary and g ( D M n ) its Heegard genus.Then lim n → + ∞ P [ n − θ ( n ) ≤ g ( D M n ) ≤ n + θ ( n )] = 1 , for any function θ : N → R that grows super-logarithmically .(d) There exists C such that the Betti numbers b ( M n ) and b ( M n , ∂M n ) satisfy lim n → + ∞ P [ b ( M n , ∂M n ) ≤ θ ( n )] = 1 , lim n → + ∞ P [ | b ( M n ) − n | ≤ θ ( n )] = 1 for any function θ : N → R that grows super-logarithmically. This is a combination of Corollary 2.2, 2.5 and 2.6. Moreover, in Theorem 2.4 below we provevarious combinatorial properties of the interior edges in our random complex. In item (c) we lookat the Heegaard genus of the double rather than the usual notion of Heegard genus of the manifolditself (defined in terms of decompositions with compression bodies, cf. [40, 2.2]) because the latteris bounded below by the genus of the boundary, so (c) says something that is not covered by (b).In low dimensions it turns out that typical objects are often hyperbolic and in that sense, ourmodel is no different. Note that it follows from Mostow rigidity that if M n caries a hyperbolicmetric with totally geodedic boundary, then this metric is unique up to isometry. As such, onecan also ask for the geometric properties of this metric. We prove: Theorem 1.2 (Geometry) . We have lim n → + ∞ P [ M n carries a hyperbolic metric with totally geodesic boundary ] = 1 . This metric has the following properties:(a) The hyperbolic volume vol( M n ) of M n satisfies: vol( M n ) ∼ n · v O as n → ∞ in probability. Here v O denotes the volume of the regular right angled ideal hyperbolicoctahedron.(b) There exists a constant c λ > so that the first discrete Laplacian eigenvalue λ ( M n ) of M n satisfies lim n → + ∞ P [ λ ( M n ) > c λ ] = 1 . (c) There exists a constant c d > such that the diameter diam( M n ) of M n satisfies: lim n → + ∞ P [diam( M n ) < c d log(vol( M n ))] = 1 (d) There exists a constant c s > such that the systole sys( M n ) of M n satisfies: lim n → + ∞ P [sys( M n ) > c s ] = 1 (e) For every ε > , lim n → + ∞ P (cid:20) sys( D M n ) < n / − ε (cid:21) = 1 . The same holds for the minimal length among arcs in M n that are homotopically non-trivial relative to ∂M n . Some remarks about these results : By this we mean that lim n →∞ θ ( n )log( n ) = + ∞ . The systole of a compact manifold is the smallest length of a closed geodesic; we do not take it to includelengths of arcs with endpoints on the boundary ; see next item for this.
BRAM PETRI AND JEAN RAIMBAULT • Our proof of hyperbolisation for M n does not rely on Perelman’s proof of the Geometri-sation conjecture. Instead, we use Andreev’s theorem [39] and recent work by Futer–Purcell–Schleimer [22] on Dehn fillings. Note that there is also a “Ricci-flow-free” proofof hyperbolisation of random Heegaard splittings [20]. • (b) admits a more geometric reformulation, as it follows from it together with classicalwork by Buser [13] that the Cheeger constant of M n is also (asymptotically almost surely)uniformly bounded from below. It also implies (with (a) and a theorem of Lackenby [30])a weaker version of (c) in our topological theorem. • (c) also implies that D M n has logarithmic diameter. It also follows from an easy volumeargument that the diameter of a closed hyperbolic 3-manifold M satisfiesdiam( M ) ≥
12 log(vol( M )) − C for some uniform constant C > • By arguments similar to those we use for (d) and (e) we could give probabilistic upperbounds for sys( M n ), and a lower bound for sys( D M n ). The former are a bit awkward tostate, and the latter would not be sharp. See also Question 3 below.Besides expansion, another way of looking efficiently at the global geometry of a (possiblyrandom) compact Riemannian manifold that has recently seen much interest is the so-calledBenjamini–Schramm topology (see [24] for a survey). We determine the Benjamini–Schrammlimit of the sequence ( M n ) n as a consequence of our proof of hyperbolisation. This is moretechnical than our other results, so we will not give precise statements here but just a sketch ofwhat this means. Very roughly, a sequence of finite volume random hyperbolic manifolds ( M n ) n converges in the Benjamini–Schramm sense to a limit M ∞ (a random pointed manifold) if forevery fixed R >
0, the R -neighbourhood of a uniformly random point in M n converges (in pointedGromov–Hausdorff topology) to the R -neighbourhood of a random point in M ∞ . It turns outthat the Benjamini–Schramm limit of M n can be identified with a tree of right angled octahedrapointed at a uniform random point (which makes sense since this manifold has a cofinite groupof isometries). A rigorous exposition of these notions, and a precise statement for the resultdiscussed above, are given in Section 3.6.2.1.3. Notes and references.
Various models for random manifolds are known in dimensions two(eg. [10, 25, 37, 11, 33]) and three (eg. [19]) and all three types of questions mentioned abovehave been explored: the models in [10] and [37] are plausible as models of typical (hyperbolic)surfaces, the original motivation for the introduction of random Heegaard splittings by Dunfieldand Thurston was to study the (at that point still unsolved) virtual Haken conjecture and [25],[11] and [32] are applications of the probabilistic method to produce hyperbolic surfaces withoutshort pants decompositions, hyperbolic surfaces with near-minimal diameter and infinitely manyclosed hyperbolic homology three-spheres with a fixed Casson invariant and Heegaard genusrespectively.The most studied models for random three-manifolds are those of random Heegaard splittingsand random mapping tori. Both of these are hyperbolic with probability 1 [34, 35]. Moreover,like our manifolds (Theorem 1.2(b)) they satisfy a law of large numbers for volume [42], with aconstant depending on the underlying random walk. Their spectral gap behaves differently: it isinversely quadratic in volume [27]. Their injectivity radius has been studied in [41] and torsionin their homology in [3]. Moreover, even if random Heegaard splittings turn out to be hyperbolicwith probability 1 [34], unlike for instance random regular graphs [7] and random hyperbolicsurfaces [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 . MODEL FOR RANDOM THREE–MANIFOLDS 5
Whether the model studied in this paper is plausible as a model for a “typical” three-manifoldwith boundary, we leave to the reader. We note that the question of studying this model hasbeen evoked before (see for instance [18, Question 6.2]) but we aren’t aware of any prior otherresults on it.Finally, it is possible to derive models for random closed manifolds from our random manifoldswith boundary. The simplest would be obtained by just doubling it; however this is supportedonly on manifolds admitting an involution with codimension-1 fixed locus. There are variousways to break this symmetry: the fact that the boundary is triangulated allows us to identifyit to a fixed (depending only on the genus) model surface “up to a finite ambiguity”. So wecan talk about random mapping classes of the boundary almost as usual, and these allow usto perform various more complicated constructions such as gluing back a copy using a randommapping class. We can also glue the appropriate handlebody (or indeed any other manifold withconnected boundary of the correct genus). By Geometrisation all these models are hyperbolic;however we do not know how their volume behaves, whether they are expanders (say for a choiceof the mapping class with “few steps”) or not, or whether they admit a Benjamini–Schrammlimit. The investigation of such questions would certainly require a different set of tools thanwhat we use here.1.4.
Proof ideas.
The order in which we prove our results is very different from the order inwhich we presented them above. The two big steps consist of understanding the combinatorialproperties of the complex we build and then using those to understand the geometry and topology.The first observation is that all results above are of the form P n [ P n ] → n → ∞ for somesequence of properties P n of M n . It follows from classical results in graph theory [7, 43] that forsuch statements it is sufficient to prove the analogous statement for the random manifolds N n obtained by randomly gluing the building blocks together (without setting the condition on faceswe set for M n ).In what follows, we will try to avoid repeating the phrase “asymptotically almost surely” andwill often just say that M n has this or that property when we mean it has the given propertyasymptotically almost surely.The proofs now start with the combinatorics (in Section 2). Using the observation above, theidea is to study the properties of N n and then turn these into properties of M n . First of all,we prove, using elementary but tedious combinatorics, that the number of boundary componentsof N n (and hence M n ) is 1. The next step, which is responsible for the largest part of thecombinatorial arguments, is to study the combinatorics of interior edges in N n . We ask twoquestions: how many edges are there? And, given some number k ∈ N , how many edges arethere that are incident to k truncated tetrahedra? To answer these questions, we will use peeling techniques. These are techniques coming from the world of random planar maps (see for instance[17]). The basic idea is to explore the random cell complex N n using a specific algorithm –adapted to the problem at hand – to determine in which order cells are explored. These lead tobounds on the expected number of interior edges (the total and the number that is incident toa fixed number of 3-cells) that we think might be interesting in their own right (see Theorem2.4). This in turn yields the Euler characteristic and hence the genus of the boundary of N n (andhence M n ). Note that all these combinatorial resuls can also be interpreted in terms of the cellcomplex obtained from gluing tetrahedra according to the same pattern – a pseudo-manifold.After this, we deal with the geometric questions in Section 3. Some of our topological resultsalso follow from these. Our first goal is hyperbolisation. The main idea behind our proof ofthis is to see our manifold as a Dehn filling of a non-compact hyperbolic three-manifold (similarideas were used, with very different objectives, in [16]). This non-compact manifold is obtained by BRAM PETRI AND JEAN RAIMBAULT gluing hyperbolic right-angled octahedra, using four alternating faces out of eight per octahedron,along the same pattern as M n . We then first apply Andreev’s theorem to fill cusps with “few”octahedra around them. The number such cusps is controlled by our combinatorial bounds. Thiscreates another non-compact hyperbolic manifold, but without “small” cusps. We then fill theremaining cusps and rely on results by Futer–Purcell–Schleimer [22] to guarantee the result ishyperbolic. These same results also give us information about the way the geometry changesbetween the non-compact and the compact manifold.Once we have proved that M n is hyperbolic, we use results on random regular graphs togetherwith a version of the Brooks–Burger transfer principle to show that λ ( M n ) can be uniformlybounded from below. Together with the law of larger number for volumes, which follows es-sentially directly from the geometric control we have over our hyperbolisations, and results byLackenby we obtain the fact that the Heegaard genus of D M n grows linearly in n . We prove thelogarithmic bound on the diameter of M n by combining the fact that random 4-regular graphshave logarithmic diameter with the geometric control we have over the change of geometry duringDehn filling.Finally, we prove Benjamini–Schramm convergence, again using the geometric comparissonbetween the non-compact hyperbolic manifold and M n .1.5. Questions.
We finish this introduction with some questions.
Question 1 (Poisson-Dirichlet distribution for edges) . Let us write L = ( L , L , . . . )for the random vector that contains the lengths of all the interior edges in M n . Here the length ofan edge is the number of 3-cells incident to it and is counted with multiplicity – i.e. if an interioredges is incident to a 3-cell in multiple places then the 3-cell is counted multiple times.If we order this vector so that L ≥ L ≥ . . . and normalise it by dividing by the total length(6 n ), does the resulting partition of the interval [0 ,
1] converge in distribution to a Poisson-Dirichlet distributed variable?The analogous result is known to hold for surfaces obtained by randomly gluing polygonstogether [23, 15, 12].
Question 2 (Explicit measures of expansion) . Determine the optimal spectral gaps and Cheegerconstants that hold a.a.s. for M n . For instance, do we have ∀ ε > n → + ∞ P ( λ ( M n ) > − ε ) = 1?An analogue of this is conjectured for random hyperbolic surfaces [44, Problem 10.3], [33, Con-jecture 1.1] and holds for random regular graphs [21].Finally we can also ask for sharp estimates for the systoles of M n and D M n . Question 3.
Give an explicit sequence ( s n ) such that sys( D M n ) ∼ s n in probability. Compute(if it exists) lim E (sys( M n )).1.5.1. Finer behaviour of homology and L -invariants of the limit. In Theorem 1.1(d) we getgood bounds for the typical Betti numbers of the M n . However, in particular in view of the factthat the random Heegaard splittings of [19] typically have vanishing first Betti numbers, we askthe following question. Question 4.
Does b ( M n , ∂M n ) = 0 hold a.a.s.? MODEL FOR RANDOM THREE–MANIFOLDS 7
A positive answer is suggested by computer experiments conducted by Nathan Dunfield usingReigina. His results also suggest the following conjecture about the behaviour of the full integralhomology, denoting by H ( M n ) tors the torsion subgroup of H ( M n ): Question 5. Is H ( M n ) tors trivial a.a.s.? Or a weaker variant : do we have lim n → + ∞ log | H ( M n ) tors | n =0 in probability?Note that for random Heegaard splittings the opposite behaviour occurs : H ( M n ) tors is it aslarge as possible, i.e. of exponential size in the number of thetrahedra (see [29, Section 2.2]).In view of the convergence discussed in Section 3.6.2 the last question could be related to the L -invariants of the infinite cover O ∞ → O (see [28] for an introduction to this topic). Ourresult on Betti numbers implies, via generalisations of the L¨uck Approximation Theorem (see[28, 5.4.3]) that the L -Betti numbers of O ∞ → O relative to the boundary vanish. We can askabout other L -invariants: Question 6.
What are the Novikov–Shubin invariants of O ∞ → O ? Is its L -torsion equal to 0?In view of the approximation conjecture for torsion (which is wide open at present, see [28,6.5] for a survey, but much simpler to deal with in 3-dimensions when the torsion vanishes, see[31]), the vanishing of L -torsion would likely imply an affirmative answer to the weaker form ofQuestion 5. We note that our expansion results implies (via the proof of L¨uck approximation)that the zeroth Novikov-Shubin invariant is ∞ + . Acknowledgements.
We worked on and off on this project for several years and as such itbenefited from various grants. BP thanks the Max Planck Institute for Mathematics in Bonnand the ERC grant “Moduli”. JR thanks the ANR for support through the projet ANR-16-CE40-0022-01 - AGIRA and the Hausdorff institute through the junior trimester “Topologie”which was held there in 2016.We also thank Fran¸cois Costantino, Juan Souto and Gabriele Viaggi for useful discussions. BPthanks Thomas Budzinski and Nicolas Curien for teaching him how to peel a surface. We areindebted to Nathan Dunfield for various comments on a preliminary version, suggesting a moreelementary and efficient approach to Heegaard genus and sharing the results of his simulation ofthe model with us. 2.
Combinatorics
In this section we formally describe the combinatorial model we use. Moreover, we determinethe combinatorial structure of the random cell complex underlying our manifolds and derive somebasic topological properties of our manifolds from it.2.1.
The topological model.
In what follows, T n will denote a ∆-complex obtained by ran-domly gluing the faces of n tetrahedra together in pairs. The gluing that is used is picked atrandom among the three orientation reversing simplicial maps between the faces.More formally, this goes as follows:(1) We start with n labeled tetrahedra. Here labeled means that every the vertex of thesetetrahedra carries a unique label in { , . . . , n } . If a face of a tetrahedron has vertices A very mild generalization of a simplicial complex in which two k -faces are allowed to share more than one( k − k − k face on multiple sides. BRAM PETRI AND JEAN RAIMBAULT v , v , v ∈ { , . . . , n } , that moreover in this particular cyclic order induce an outwardorientation on that particular face, then we will denote the face by the cycle ( v v v ).(2) The faces are partitioned into 2 n pairs, uniformly at random. We will denote the resultingpartion by ω n = ( ω ( i ) n ) ni =1 .(3) Per pair of faces ω ( i ) n = { ( v v v ) , ( w w w ) } in this partition, one of three cyclic-order-reversing pairings between the vertices is chosen uniformly at random. The resulting2 n -tuple of pairings will be denoted σ n = ( σ ( i ) n ) ni =1 .(4) We identify each pair of faces ω ( i ) n = { ( v v v ) , ( w w w ) } in the partition ω n usingthe unique orientation reversing simplicial map that sends v j to σ ( i ) n ( v j ) for j = 1 , , T n .Let us write (Ω n , P n ) for the corresponding probability space. So Ω n is the finite set of allpossibilities for ω n and σ n and P n is the uniform probability measure on it. Note that | Ω n | = (4 n )!! 3 n , where for an even number k ∈ N , k !! = ( k − · ( k − · · · · G n to T n - the 4-valent graph whose vertices correspond to the tetrahedra of T n who share an edge per face that they have in common - is a random 4-regular graph. Themodel this induces is exactly the configuration model , one of the most studied models of randomregular graphs (see eg. [7, 43]).Note that besides the number of tetrahedra ( n ), the number of 2-faces is also deterministic inthis model (2 n ). The numbers of vertices and edges are random variables.2.2. The results.
Let N n denote the manifold with boundary obtained by truncating T n at thevertices. Figure 1 in the introduction shows what the basic building block of N n looks like.In most of this section we will think in terms of T n . However, since we are eventually interestedin N n , we will describe some of the results in terms of N n . Theorem 2.1 (Topology) . (a) We have lim n →∞ P [ N n has a single boundary component ] = 1 (b) We have E [ χ ( ∂N n )] = log( n ) − n + O (1) as n → ∞ . In particular, if we write g ( ∂N n ) for the genus of the single boundary compo-nent of M n we have that g ( ∂N n ) ∼ n as n → ∞ in probability. Part (a) follows from Proposition 2.3 and part (b) from Theorem 2.4. In the latter, we alsoprove bounds on the expected number of edges incident to a given number of tetrahedra in T n .We will need these in the geometric part of the paper.We will write M n for the random manifold we obtain if we condition on G n being simple, i.e.not having loops or multiple edges. Bollob´as [6] proved thatlim n →∞ P [ G n is simple ] > . These cycles naturally lie in the symmetric group S n . However, because we won’t use this in anything thatfollows, we will just think of these cycles as homeless. MODEL FOR RANDOM THREE–MANIFOLDS 9
In particular, this implies that if ( P n ) n is a sequence of properties of M n and N n then as n → ∞ ,(2.1) if P [ N n has P n ] → P [ M n has P n ] → . Combining this with the theorem above, we get:
Corollary 2.2 (Topology of the boundary) . (a) We have lim n →∞ P [ M n has a single boundary component ] = 1 (b) We have g ( ∂M n ) ∼ n as n → ∞ in probability. The number of vertices.
We start by studying the number of vertices. Let us write V forthe number of vertices of T n . Note that V is also the number of boundary components of N n .As such the following proposition implies Theorem 2.1(a). Proposition 2.3.
We have P n [ V = 1] → , as n → ∞ .Proof. Let us write V small : Ω n → N for the random variable that counts the number of verticesof T n incident to most 2 n tetrahedra (with multiplicity – i.e. if a tetrahedron is incident to avertex in multiple corners, it is counted multiple times). We will prove that E n [ V small ] → , as n → ∞ . Note that this is sufficient to prove the proposition.We will write E n [ V small ] = (cid:88) a E n [ a ] . Here the sum runs over “labeled” vertices a . A labelled vertex a is the data of the gluings of allthe labeled faces incident to a given vertex. Here a : Ω n → { , } is the indicator for the eventthat a appears in ω . As such E [ a ] = 13 f (4 n − n − · · · (4 n − f + 1) , where f is the number of faces incident to the given vertex.Given a , write n , n , n , n for the number of tetrahedra that are incident in 1, 2, 3 and 4of their vertices to a respectively. Note that the number of tetrahedron faces involved in such agluing is given by 2 f = 3 n + 4( n + n + n )This implies that n must be even and that the number of gluings with the same selection ofvertices is 3 n +2 n (2 f )!! . The power of 3 comes from the fact that only the faces with three vertices adjacent to them canbe rotated. This number of faces is equal to n + 4 n . As such we obtain E n [ V small ] = (cid:88) A simple edge e where we write (cid:18) nn , n , n , n (cid:19) = n ! n ! · n ! · n ! · n ! · ( n − n − n − n − n )!counts the number of subsets of the n tetrahedra with the appropriate number of vertices inthem.We claim that this implies that E n [ V small ] = O (cid:0) n − (cid:1) . This follows by analyzing the terms inthe sum. The largest terms in this sum are when 2 f = 3 n + 4( n + n + n ) is smallest. Indeed,using that n + 2 n + 3 n + 4 n ≤ n , an elementary but tedious computation shows that a termdecreases when one of the n (cid:48) i s is increased. Given that the number of terms is quartic in n andthe number of terms that are larger than O (cid:0) n − (cid:1) is bounded, we obtain the estimate. (cid:3) The number of edges. The random variable that counts the number of edges that areincident to k tetrahedra will be denoted by E k : Ω n → N . In this variable, tetrahedra are counted with multiplicity. That is, if an edge appears multipletimes in the boundary of a given tetrahedron, this tetrahedron adds to its “length” each time.Note that (cid:88) k ≥ k · E k = 6 n. We will also write E = (cid:88) k ≥ E k : Ω n → N for the total number of edges. The goal of this section is to study the distribution of ( E k ) k as n → ∞ .We will also count edges that we will call simple . These are edges that neighbor each tetra-hedron at most once. Figure 2 shows an example. We will denote the number of simple edgesby E ◦ and the number of simple edges adjacent to k tetrahedra by E ◦ k . Note that E ◦ k = 0 for all k > n , which is not at all necessary for non-simple edges.Concretely, we will prove the following estimates When we write k = o ( f ( n )) for some function f : N → N , what we mean is that the statement holds for anyfunction k : N → N so that k ( n ) = o ( f ( n ). MODEL FOR RANDOM THREE–MANIFOLDS 11 Theorem 2.4 (Combinatorics of edges) . (a) We have E n [ E ] = 12 log( n ) + O (1) as n → ∞ .(b) For all k = o ( √ n ) we have E [ E ◦ k ] = 12 k · (1 + o (1)) as n → ∞ . Moreover, the error is uniform in k .(c) For all k = o ( n ) and k ≤ k ≤ k we have E [ E k ] ≤ k + O (cid:0) n − (cid:1) and E k (cid:88) l = k E l ≤ 12 log( k /k ) + O (1) as n → ∞ . In particular, for all k = o ( √ n ) and k ≤ k ≤ k we have E [ E k ] = 12 k · (1 + o (1)) , E k (cid:88) l = k E l = 12 log( k /k )(1 + o (1)) and E [ E k − E ◦ k ] = o (1) as n → ∞ .(d) For all K, L = o (cid:0) n / (cid:1) we have E [ E KL ] = o (1) as n → ∞ . Here E KL counts the number of pairs of edges of size ≤ K and ≤ L respectivelythat are incident to a common tetrahedron. Before we get to the proof of this theorem, let us briefly note how to derive the Euler chara-teristic of ∂N n from it. Proof of Theorem 2.1(b). Writing v , e and f for the number of vertices, edges and faces of thetriangulation on ∂N n , we have v = 2 E e = 6 n and f = 4 n. As such, Theorem 2.4(a) implies our claim. (cid:3) Peeling. In order to prove Theorem 2.4(a), (c) and (d) we will use peeling (see for instance[17]). Before we get to the proof, we need some preparation.The main idea behind peeling is to build our random cell complex T n in a specific order. Inparticular, we will describe a peeling algorithm that determines a sequence of cell complexes T (0) n , T (1) n , . . . , T (2 n ) n where T (0) n consists of n disjoint tetrahedra, T (2 n ) n = T n (in the sense that it has the samedistribution) and in general T ( i +1) n can be obtained from T ( i ) n by identifying exactly one pair offaces of T ( i ) n .We will use two different peeling algorithms. The first to prove part (a) and the second toprove parts (c) and (d). f ( t ) f f f e e e Figure 3. The face f ( t ) and its neighbors2.4.2. Algorithm 1. The first algorithm is very simple and closely resembles that of [10, Section8]. Peeling algorithm 1 :Initialisation:Objects: faces f (0) , f (1) , . . . , f (2 n ) , f (cid:48) (0) , f (cid:48) (1) , . . . , f (cid:48) (2 n ) and t ∈ Z .- Set t = 0.- Set T (0) n equal to a disjoint union of n tetrahedra.- Set f (0) equal to a face in ∂ T (0) n , picked uniformly at random.Iteration: while t < n , repeat the following steps:(1) Glue the face f ( t ) to a uniformly random face f (cid:48) ( t ) in ∂ T ( t ) n \ f ( t ) , with auniformly random gluing. Call the result T ( t +1) n .(2) If t < n : Pick a uniformly random face f ( t +1) ⊂ ∂ T ( t ) n (3) Add 1 to t .Note that the distribution of T (2 n ) n is the same as that of T n .2.4.3. Closing off edges. The reason for setting up the peeling algorithm is that we can nowcontrol the number of edges in T n by bounding the number of edges that are closed off – i.e.that disappear from the boundary – during each step of the process.As such, let us define random variables E ( t ) that count the number of edges that are closed offwhen T ( t ) n is created. This is the number of edges that lie in ∂ T ( t − n but not in ∂ T ( t ) n . Note that E = n (cid:88) t =1 E ( t ) . Moreover, since any edge that gets closed off at the t th step necessarily lies in f ( t ) , we have E ( t ) ≤ 3. One of the things we will argue below is that most of the time, we actually have E ( t ) ≤ 1. To this end, consider Figure 3. It shows a schematic overview of the situation aroundthe face f ( t ) at time t . Note that some of the faces f , f , f and f ( t ) may coincide. However, ifthey don’t, only one edge can be created at step t : e i is then closed off if and only if f ( t ) is gluedto f i with exactly one out of the three possible face identifications.So, in what follows, we will make a distinction between singular faces – faces that are theirown neighbor or of which some of the neighbors coincide – and regular faces – faces that are not MODEL FOR RANDOM THREE–MANIFOLDS 13 singular. We will write F ( t )sing for the random variable that counts the number of singular faces in ∂ T ( t ) n . Likewise, we define two sequences of random variables E ( t )sing , E ( t )reg where E ( t )sing = (cid:26) E ( t ) if f ( t ) is singular0 otherwise. and E ( t )reg = E ( t ) − E ( t )sing . Since there are 4 n − t faces left when the t th step starts, we have(2.2) E [ E ( t )reg | F ( t )sing ] = 4 n − t − F ( t )sing n − t n − t − 1) = 4 n − t − F ( t )sing (4 n − t )(4 n − t − f ( t ) ) are closed off during t th step and moreover thereare at most 3 choices for f (cid:48) ( t ) and 3 gluings per choice that result in an edge closure, we have:(2.3) E [ E ( t )sing | F ( t )sing ] ≤ F ( t )sing n − t · · n − t − 1) = 9 · F ( t )sing (4 n − t )(4 n − t − . So, we need to control F ( t )sing . Lemma 2.5. For all t ∈ N so that t < n . E n (cid:104) F ( t )sing (cid:105) ≤ · log (cid:18) n n − t (cid:19) + o (1) as n → ∞ . The implied error is independent of t Proof. Let us write ∆ F ( t )sing for the random variable that counts the number of singular faces thatis created at step t – since we are only interested in an upper bound, we will ignore singular facesthat disappear. Because F (0)sing = 0, we have F ( t )sing ≤ t (cid:88) s =1 ∆ F ( s )sing . So, let us try to control E [∆ F ( s ) ]. The only faces whose neighborhood changes during step s arethe neighbors of f ( s ) and f (cid:48) ( s ) . If such a neighbor of say f ( s ) is regular, it becomes singular onlyif one of its neighbors is also a neighbor of f (cid:48) ( s ) . Likewise, a regular neighbor of f (cid:48) ( s ) becomessingular only if it shares a neighbor with f ( s ) . In other words, ∆ F ( s ) can only be positive if thecombinatorial distance – the number of edges that needs to be traversed in ∂ T ( s ) n in order to movefrom one face to the other – between f ( s ) and f (cid:48) ( s ) is at most 3. Note that this is independent ofwhether or not f ( s ) and f (cid:48) ( s ) are regular.Since there are at most 3 · · f ( s ) , wehave E [∆ F ( s )sing | F ( s − ] = E [ f ( s ) ∆ F ( s )sing | F ( s − ] 124 n − s − . This implies that E n (cid:104) F ( t )sing (cid:105) ≤ (cid:88) ≤ t ≤ n − n − t − ≤ n − (cid:88) k =4 n − t − k = 12 log (cid:18) n n − t (cid:19) + o (1)as n → ∞ . (cid:3) The total edge count. We now have all the set up we need for the first part of Theorem2.4: Proof of Theorem 2.4(a). Fix any α ∈ (0 , E n [ E ( t ) ] = E n (cid:20) E ( t ) · F ( t )sing ≥ t α (cid:21) + E n (cid:20) E ( t ) · F ( t )sing 12 log( t ) n α n − t − , for all n large enough. Likewise, using Lemma 2.5, (2.3) and Markov’s inequality we obtain(2.4) 0 ≤ E n (cid:20) E ( t )reg · F ( t )sing ≥ t α (cid:21) ≤ 12 log( t ) n α n − t − . So we obtain n (cid:88) t =1 E n (cid:20) E ( t ) · F ( t )sing ≥ t α (cid:21) = O (cid:18) log( n ) n α (cid:19) as n → ∞ . In words: most edges are created when few singular faces are present.So, let us control this term. Again using (2.2), we have4 n − t − t α n − t n − t − ≤ E n (cid:20) E ( t )reg · F ( t )sing The proof of part (b) of our theorem – the count of the expected number ofsimple edges – will not use a peeling algorithm. Proof of Theorem 2.4(b). We write E n [ E ◦ k ] = (cid:88) ( c c ... c k ) E n [ ( c c ... c k ) ]where the sum runs over all cycles ( c c . . . c k ) of length k so that(i) c i is a corner in some tetrahedron, i.e. a pair faces(ii) and at most one corner of any given tetrahedron appears in the sequence.Finally, given ω ∈ Ω n , ( c c ... c k ) ( ω ) = (cid:26) c c . . . c k ) appears around an edge in ω MODEL FOR RANDOM THREE–MANIFOLDS 15 It follows from (ii) that every cycle ( c c . . . c k ) in the sum corresponds to an identificationof k pairs of faces. As such E n [ ( c c ... c k ) ] = 13 k (4 n − n − · · · (4 n − k + 1) . So, all that remains is counting the number of possible cycles ( c c . . . c k ) of corners. Writing l for the number of tetrahedra of which we use two corners, we have |{ ( c c . . . c k ) }| = 12 k (cid:18) nk (cid:19) · k · k · k ! . The reason for this expression is as follows. First we count sequences instead of cycles: • This gives a total of (cid:0) nk (cid:1) choices for the tetrathedra. • Per tetrahedron out of which a corner is used, we have a choice of 6 corners. This givesrise to a factor 6 k • Per corner that is used, we have two choices for the order in which the faces of that cornerappear. This leads to a factor 2 k • Finally, there are k ! ways to order the sequence.Since we don’t want to make a difference between sequences that differ by a cyclic permutationor are each others inverse, we divide by 2 k .So we get E n [ E ◦ k ] = 12 k k (cid:89) i =1 n − i + 44 n − i + 1= 12 k exp (cid:32) k (cid:88) i =1 log (cid:18) − i − n − i + 1 (cid:19)(cid:33) = 12 k exp (cid:32) − k (cid:88) i =1 i − n − i + 1 + O (cid:0) n − (cid:1)(cid:33) = 12 k exp( o (1))Where we used the fact that k = o ( √ n ) in the last line. (cid:3) Algorithm 2. The second algorithm is actually a collection of algorithms, tailored towardscounting the number of edges incendent to two edges in a fixed tetrahedron. As such, it startspeeling our random triangulation around a fixed starting edge e and then continuous to peelaround another fixed edge e (cid:48) once e is closed. Peeling algorithm 2 :Input:A labelled tetrahedron τ and two fixed labeled oriented edges e, e (cid:48) ⊂ τ .Initialisation:Objects: oriented edges e (0) , e (1) , . . . , e (2 n ) , faces f (0) , f (1) , . . . , f (2 n ) , f (cid:48) (0) , f (cid:48) (1) , . . . , f (cid:48) (2 n ) and t ∈ Z .- Set t = 0.- Set T (0) n equal to a disjoint union of n tetrahedra, containing τ .- Set e (0) = e .Iteration: while t < n , repeat the following steps:(1) Glue the face f ( t ) to the right of e ( t ) to a uniformly random face f (cid:48) ( t ) in ∂ T ( t ) n \ f , with a uniformly random gluing. Call the result T ( t +1) n .(2) - If t + 1 < n and e ( t ) (cid:42) ∂ T ( t +1) n : – If e (cid:48) ⊂ ∂ T ( t +1) , set e ( t +1) = e (cid:48) . – Else: pick a uniformly random edge e ( t +1) ⊂ ∂ T (0) n and orient itrandomly.- Else: e ( t +1) = e ( t ) (3) Add 1 to t .Again note that the distribution of T (2 n ) n is the same as that of T n . Figure 4 shows what theinitial set up looks like. ee (cid:48) f (0) τ T (0) n Figure 4. The initial set up2.4.7. All edges.Proof of Theorem 2.4(c). Since we already have Theorem 2.4(b), we only need an upper boundon E [ E k ]. In order to prove such a bound, we will write E [ E k ] = 12 k (cid:88) e E [ ke ]where the sum runs over labelled oriented edges e in our collection of n tetrahedra (so the sumhas 12 n terms in total) and ke is the indicator for the event that e is incident to exactly k cornersin the complex T n . MODEL FOR RANDOM THREE–MANIFOLDS 17 In order to bound E [ ke ] from above, we use our second algorithm with e as input. The secondoriented edge e (cid:48) that the algorithm uses doesn’t play a role in this proof, so we pick an arbitraryedge. We will also only care what happens in the first k steps of the process.Just like before, if during every step of the process, f ( t ) has three distinct neighbors, none ofwhich is is f ( t ) itself – i.e. if f ( t ) is regular –, it is easy to control the probability that our edgecloses up in exactly k steps. So, just like before, we need to bound the probability that in thefirst k steps, our face becomes singular.With the same argument as in Lemma 2.5, we have E (cid:104) f ( t ) is singular (cid:105) ≤ n − t + 1 + E (cid:104) f ( t − is singular (cid:105) . In particular, E (cid:104) f ( k ) is singular (cid:105) ≤ k n − k + 1 . After k − n − k + 1 faces left, we obtain, and even if a singular face isinvolved in the k th gluing, there are at most 3 possible gluings that result in a closure. So we get: E [ ke ] = E (cid:104) ke f ( k ) is singular (cid:105) + E (cid:104) ke f ( k ) is regular (cid:105) ≤ n − k + 1) + 12 k n − k + 1 34 n − k + 1 . This means that E [ E k ] ≤ n k (cid:18) n − k + 1) + 12 k n − k + 1 34 n − k (cid:19) , which proves our claim. (cid:3) Proof of Theorem 2.4(d). Let k ≤ K and l ≤ L . Moreover, let E (cid:48) kl denote the number of pairs ofedges of sizes rexactly k and l respectively that are incident to a common tetrahedron. Just likein the proof above, we will write E [ E (cid:48) kl ] = 14 kl (cid:88) e,e (cid:48) E [ kle,e (cid:48) ]where the sum runs over pairs of labelled oriented edges e, e (cid:48) that are incident to a single tetra-hedron (so the sum has 12 n · 10 terms in total) and kle,e (cid:48) is the indicator for the event that e isincident to exactly k and e (cid:48) to l corners in the complex T n .We again write E [ kle,e (cid:48) ] = E (cid:104) kle,e (cid:48) f ( k ) is singular (cid:105) + E (cid:104) kle,e (cid:48) f ( k ) is regular and f ( k + l ) is singular (cid:105) + E (cid:104) kle,e (cid:48) f ( k ) and f ( k + l ) are regular (cid:105) . So we obtain, with exactly the same arguments as in the previous proof: E [ kle,e (cid:48) ] ≤ k n − k + 1 34 n − k + 1+ 13(4 n − k + 1) 12 l n − k − l + 1 34 n − k − l + 1+ 13(4 n − k + 1) 13(4 n − k − l + 1) So, multiplying this with 120 n and using that k, l = o ( n / ) uniformly gives E [ E (cid:48) kl ] = o ( n − / )as n → ∞ , where the implied constant is uniform over k, l . Now summing over k and l gives E [ E KL ] = o (1)as n → ∞ . (cid:3) Betti numbers. We give here the proof of our estimates on Betti numbers of M n (or N n )in Theorem 1.1(d). First, a generating family for H ( M n , ∂M n ) is given by images of the edgesof T n . Applying Markov’s inequality to 2.4,(a) we get that b ( M n , ∂M n ) = o ( θ ( n )) as n → ∞ for any function θ : N → R such that lim n →∞ θ ( n ) / log( n ) = + ∞ , establishing the first estimate.From the connectedness of the boundary 2.1,(a) it follows that with asymptotic probability 1 weget an exact sequence0 → H ( M n ) → H ( M n , ∂M n ) → H ( ∂M n ) → H ( M n ) → H ( M n , ∂M n ) → . with asymptotic probability 1. Now this exact sequence and Poincar´e duality (the “half lives,half dies” argument) imply that b ( M n ) = 12 b ( ∂M n ) + b ( M n , ∂M n )and together with the estimate for b ( M n , ∂M n ), 2.1,(b) we can conclude that with asymptoticprobability 1 we have | b ( M n ) − n | = o ( θ ( n ))for any function θ : N → R that grows super-logarithmically, which is the second estimate.2.6. Heegaard genus. Here we prove the estimates on Heegaard genus of the double D M n (or D N n ) of Theorem 1.1(c), following an argument of Nathan Dunfield. Recall that E is the numberof edges in the triangulation T n (equivalently the number of interior edges in the cellulation of M n ). We will first prove that(2.5) g ( D M n ) ≤ n + 1 + E which in view of Theorem 2.4 implies the upper bound we are after.To prove (2.5) we observe that M n minus regular neighbourhoods of its interior edges is ahandlebody of genus n + 1, as it is a regular neighbourhood of the dual graph to the cellulationof M n in truncated tetrahedra, which is a 4-valent graph on n vertices. For each edge e we writeits regular neighbourhood U e as D e × e where D e is a disc. We split it as D e = D e ∪ D e where D ie are half-discs and we put U ie = D ie × e ; note that M n ∪ (cid:83) e U ie is still a handlebody of genus n + 1. We consider the two copies M n , M n of M n in D M n (so D M n = M n ∪ M n ), for a subset W ⊂ M n we denote by W its image in M n , and we put H n = (cid:32) M n \ (cid:32)(cid:91) e U e (cid:33)(cid:33) ∪ (cid:32)(cid:91) e U e (cid:33) which is just M n ∪ (cid:83) e U e with 1-handles attached (one for each edge), so it is a handlebody ofgenus n + E + 1. Similarly H n = (cid:32) M n \ (cid:32)(cid:91) e U e (cid:33)(cid:33) ∪ (cid:32)(cid:91) e U e (cid:33) is a handlebody of the same genus. Now D M n = H n ∪ H n and this proves (2.5). MODEL FOR RANDOM THREE–MANIFOLDS 19 For the lower bound we observe that the long exact sequence associated with ∂M n → M n ∪ M n → D M n reduces to0 → H ( M n ) ⊕ H ( M n ) → H ( D M n ) → H ( ∂M n ) i −→ H ( M n ) ⊕ H ( M n ) → H ( D M n ) → . Since b ( D M n ) = b ( D M n ) (Poincar´e duality) and by our results on Betti numbers and genus ofthe boundary 2 b ( M n ) = b ( ∂M n ) up to super-logarithmic error, it follows that b ( D M n ) = rank( i ) + θ ( n )with θ ( n ) super-logarithmic. As i is diagonal embedding we have rank( i ) ≤ b ( M n ) = n up tosuper-logarithmic error, so we can conclude that b ( D M n ) ≥ n − θ ( n ) with θ super-logarithmic.On the other hand g ( D M n ) ≥ b ( D M n ) and this proves the lower bound.3. Geometry In this section we combine the combinatorial results from the previous section with hyperbolicgeometry. Recall that M n denotes the compact manifold with boundary associated to T n . Ourmain goal is to prove: Theorem 3.1 (Geometry) . We have lim n → + ∞ P [ M n carries a hyperbolic metric with totally geodesic boundary ] = 1 . This metric has the following properties:(a) The hyperbolic volume vol( M n ) of M n satisfies: vol( M n ) ∼ n · v O as n → ∞ in probability.(b) There exists a constant c λ > so that the first discrete Laplacian eigenvalue λ ( M n ) of M n satisfies lim n → + ∞ P [ λ ( M n ) > c λ ] = 1 . (c) There exists a constant c d > such that the diameter diam( M n ) of M n satisfies: lim n → + ∞ P [diam( M n ) < c d log(vol( M n ))] = 1 (d) There exists a constant c s > such that the systole sys( M n ) of M n satisfies: lim n → + ∞ P [sys( M n ) > c s ] = 1 (e) For every ε > , lim n → + ∞ P (cid:20) sys( D M n ) < n / − ε (cid:21) = 1 . The same holds for the minimal length among arcs in M n that are homotopically non-trivial relative to ∂M n . We will prove this theorem in multiple steps. The first is hyperbolisation, which follows fromLemmas 3.6 and 3.7). The asymptotic behaviour of the volume is then determined in Proposition3.10 and the spectral gap is proven in Proposition 3.8. We prove the bounds on the diameter andsystole in Proposition 3.11.Finally, we will also prove that the Benjamini-Schramm limit of the sequence ( M n ) n is theoctatree. T n M n Y n Figure 5. The three building blocks for T n , M n and Y n respectively. The facesthat the gluing is performed along are shaded.3.1. Random models for hyperbolic manifolds. We will first describe two manifolds associ-ated to an element ω ∈ Ω n in our probability space Ω n . The first is a cusped hyperbolic manifold Y n and the second is the manifold M n that we saw in the previous section, but now viewed as aDehn filling of Y n .Figure 5 gives a topological picture of what is going on. We already associated a manifold M n to T n by truncating all the tetrahedra involved at their vertices. If we now contract the edges inthe interior of this manifold and remove the resulting vertices, we obtain a new manifold Y n thatis built out of a gluing of octahedra. The link of the octahedra’s ideal vertices in this manifoldare annuli, and we can fill them with cylinders to go back to the compact manifold.In what follows we describe this in some more detail.3.1.1. Manifolds with cusps and boundary. Let O be the ideal regular octahedron in H (it canbe realised as the convex hull of the vertices of a regular octahedron on the boundary at infinity S ). Its dihedral angles are right angles and its faces are ideal triangles. We orient each face withits outward normal.We take n copies of O which we label as follows: for each copy we attribute a label in { , . . . , n } to four of its faces so that no two of them are adjacent (and we ask that the labeling map beinjective). Each of the unlabeled faces is then determined by the labels of the three faces adjacentto it and since it is orianted we can identify it with a 3-cycle on their labels.This setting is similar to that of 2.1 and we can perform the same random construction fromit: we partition the non-labeled faces uniformly randomly into pairs, and we glue the two facesin a pair in a uniformly randomly choses orientation-reversing way.The resulting octahedral complex is a non-compact manifold with boundary, and by endowingeach O with its hyperbolic structure we obtain that the result is a complete orientable hyperbolicmanifold with totally geodesic boundary. We denote by X n the random hyperbolic manifold withboundary we constructed. We will also consider Y n where we condition on there not being anyloops or bigons in the graph dual to the tesselation by octahedra. We record the hyperbolicstructure in a lemma in order to be able to refer to it later on. Lemma 3.2. The manifolds X n and Y n carry complete hyperbolic metrics of finite volume withtotally geodesic boundary. We denote by Θ n the (finite) set of all hyperbolic manifolds obtained by gluing n octagons inthis fashion. Let Y be in some Θ n . Each cusp c of Y is tesselated by squares. Let (cid:96) ( c ) be theirnumber (we will also call this the length of c ) and for k ∈ N let B k ( Y ) be the number of cusps of Y with (cid:96) = k . Lemma 3.3. The random variable C k := B k ( Y n ) has the same distribution as the variable E k introduced at the beginning of 2.1. MODEL FOR RANDOM THREE–MANIFOLDS 21 Compact manifolds. Recall from Section 2.1 that N n is a manifold with boundary obtainedfrom randomly gluing truncated tetrahedra along their faces. Moreover, M n is a random manifoldthat has the distribution of N n , conditioned on the dual graph G n not having any loops andmultiple edges.Let us now describe how M n (and N n respectively) can be obtained from Y n (and X n respec-tively) via Dehn filling.Let Y ∈ Θ n . Its boundary S is a hyperbolic surface with cusps. Moreover there is a pairingon the cusps where we associate two cusps of S if they are asymptotic in Y . If we remove ahorospherical neighbourhood of each cusp we obtain a compact manifold Y whose boundary ismade up of S together with closed annuli linking paired cusps, and by the thick-thin decomposition Y is homeomorphic to Y minus the annuli. We can then perform surgery on Y as follows: toeach annulus we glue a cylinder D × [0 , 1] along the boundary [0 , × ∂D . We obtain a compactmanifold M with boundary S . We denote by Ξ n the set of such manifolds M obtained from Y ∈ Θ n .Note that M n (and N n respectively) has the same distribution as the Dehn filling of Y n (and X n respectively) described above. We again record this in a proposition to be able to refer to itlater: Proposition 3.4. The variables M n (respectively N n ) and the Dehn filling of Y n (respectively X n ) described above have the same distribution. Bounds for Dehn surgeries and hyperbolicity. In this section we prove that M n ishyperbolic with asymptotic probability 1 (this is part of the statement of Lemma 3.7) and wegive precise bounds for the variation in geometry between Y and M .Our proof goes in two steps. First we use Andreev’s theorem to control what happens when“small” cusps of Y are filled and after this we use recent results by Futer–Purcell–Schleimer tocontrol the change in geometry when the “large” cusps are filled.3.2.1. Andreev’s Theorem. To construct explicit hyperbolisations we will need Andreev’s theoremdescribing acute-angles polyhedra in H . We refer to [39] for a proof of this result. Before givingthe statement we recall that given a combinatorial 2-polyhedron P , with dual graph H , a circuitin H is said to be prismatic if for any edge in the circuit, the endpoints of the corresponding edgeof P are distinct. The following is a combination of Theorem 1.4 and Proposition 1.5 in [39]: Theorem 3.5 (Andreev’s Theorem) . Let P be an abstract polyhedron with at least six faces, and α a function from edges of P to (0 , π/ . Then there exists a realisation of P in H which is offinite volume and whose dihedral angles are given by α if and only if the following conditions aresatisfied. • For any three edges e , e , e meeting in one vertex we have (cid:80) α ( e i ) ≥ π (equility ocurringif and only if the vertex is ideal). • If ( e , . . . , e k ) is a prismatic k -circuit with k = 3 , then (cid:80) α ( e i ) < ( k − π . Filling small cusps. We start by filling the small cusps of Y n . These will be cusps of length ≤ n / . Note that the resuling manifold, that we call Z n is rigid – i.e. if we can find a completehyperbolic metric of finite volume on it, it’s unique up to isometry – by Mostow–Prasad rigidityof its double. Another way to describe it is that it is the restriction to Y of the unique Dehn surgery on the double D Y whichis equivariant with respect to the reflection in ∂Y . Recall that this is the graph on 2-dimensional faces of P with an edge between two faces for each edge theyshare. Lemma 3.6. There exists J > such that the following holds for any Margulis constant δ > and any ε > . For any Y ∈ Θ n , • let c , . . . , c m be the cusps of Y of length at most n / , • let Z be the manifold obtained by filling c , . . . , c m , • let Z be the union of all octahedra of Y containing one of the c i and Z its complement.Then with probability at least − ε in the model Y n for n large enough, we have that: • The δ -thick part of the image of Z in Z is J -bilipshitz to that of Z ; • The image of Z in Z σ is isometric to Z .Proof. This is the (only) part of our proof of hyperbolisation that will use the assumption thatthe dual graph G n is simple.Let O , . . . , O n be the octahedra tesselating Y and O i , 1 ≤ i ≤ k those containing a cusp c i with (cid:96) ( c i ) ≤ n / and O k +1 , . . . , O n the remaining ones. By Theorem 2.4(d) we may assume thateach O i contains exactly one such cusp. We have Z = O ∪ · · · ∪ O k and Z = O k +1 ∪ · · · ∪ O n .Then the part of boundary of Z and Z along which they are glued is a disjoint union of idealregular squares.To construct the hyperbolic structure on the filled manifold we replace O , . . . , O k by polyhedraconstructed as follows. First we assume that (cid:96) ( c i ) ≥ 4. Consider the following polyhedron, whichis an octahedron on which 1 vertex has been replaced by an edge (marked red in the picture):There are no prismatic 3- or 4-circuits in the dual graph so it follows from Andreev’s Theorem(Theorem 3.5) that for l ≥ 4, this has the structure of an hyperbolic polyhedron P l with rightangles at all edges except the red one which has angle 2 π/l (we need l ≥ (cid:96) ( c i ) = 3 then we can still construct P as follows: the combinatorial polyhedron has asymmetry along the red edge, which decomposes it as the double of the following polyhedronalong the blue-colored face : We view Y as the complement of the core arcs in Z . MODEL FOR RANDOM THREE–MANIFOLDS 23 and the latter has no prismatic circuits, so it admits a hyperbolic structure with right angles onthe black edges and π/ l ≥ Q l be the hyperbolic manifold obtained by gluing l copies of O in a circularpattern, along disjoint faces sharing an ideal vertex. The faces opposite to the glued faces forma union of disjoint ideal triangles in Q l . Dehn surgery on Q l amounts to replacing each copy of O by a copy of the polyhedron P l (the edge with angle 2 π/l replacing the ideal vertex on whichsurgery is done), to obtain a polyhedron Q (cid:48) l whose boundary is two l -gons, l regular ideal squaresand 2 l ideal triangles meeting at right angles. This is illustrated in the following figure, wherethese are colored blue and the central edge red :Now if M is generic in the sense of 2.4(d) and furthermore all edges of length at most K ( n ) (where K ( n ) is any o ( n / )) are simple (which is generic by 2.4(b),(c)) then Z is a disjoint union of Q l sand filling the small cusps amounts to replacing each of these with a Q (cid:48) l . In particular we can gluethe rest of the octahedra in the pattern given by G to obtain Z . This proves that the image of We could also have realised the surgery by explicit polygons for non-simple edges but we found this argumentto be simpler. Z in Z is isometric to Z . Since P l converges in Gromov–Hausdorff topology (pointed anywherein the thick part) to O the δ -thick parts of P l are uniformly (independently of l ) bilipschitz tothe δ -thick part of O , and it follows immediately that Z is uniformly (independently of generic G ) bilipschitz to its image in Z . (cid:3) Filling large cusps. In order to fill the cusps remaining in Z , we rely on results of Futer–Purcell–Schleimer. Again using Z to denote the manifold obtained from Y by filling its msallcusps, we will think of Y ⊂ Z ⊂ M . We have: Lemma 3.7. Let Y, Z be as in Lemma 3.6. Then, for any δ > (smaller than the Margulisconstant for H ) and any η > , with probability for the Y n model converging to 1 as n → + ∞ the following holds: • There exists Riemannian metrics g , g π on M such that ( Y, g ) is the complete hyperbolicstructure and the completion of ( M, g π ) is a compact hyperbolic manifold with totallygeodesic boundary which is diffeomorphic to the Dehn filling of Y . • Moreover Z ≥ δ/ ⊂ M ≥ δ ⊂ Z ≥ δ/ and these inclusions are (1 + η ) -lipschitz.Proof. Let c m +1 , . . . , c h be all remaining cusps in Z (recall that c , . . . , c m were the cusps of lengthat most n / ). Realising them as arbitrary horosphere quotients in Z these cusps have an area a j and a length of the vertical curve l j . Let L j = l j / √ a j ; this does not depend on the arbitrarychoice of horospheres (as long as their quotients are homeomorphic to 2-tori). Following [22,Definition 1.3] we define L > L = h (cid:88) j = m +1 L j . For m + 1 ≤ j ≤ h let k j be the number of cusps c i , 1 ≤ i ≤ m , which share an octahedron with c j – in other words, the number of small cusps that share an octahedron with c j . We claim thatwith probability tending to 1 we have max( k j ) (cid:28) log( n ) n / (cid:96) ( c j ).To prove this we separate two cases: first, when n / ≤ (cid:96) ( c j ) ≤ n / (note 7 / 24 = 1 / − / 24 =1 / / 24) we have by 2.4(d) that with probability tending to 1 we have k j = 0 for all these j .In the remaining cases we have that k j (cid:28) h · n / and since h = E (the number of edges in theoriginal triangulation) and E (cid:28) log( n ) by 2.4(a) it follows that if (cid:96) ( c j ) ≥ n / k j (cid:28) log( n ) n / ≤ log( n ) n / (cid:96) ( c j )which finishes the proof of the claim.Now it follows from Lemma 3.6 if m + 1 ≤ j ≤ h we have at least (cid:96) ( c j ) − k j regular squares inthe tesselation of c j in Z , and somax m +1 ≤ j ≤ h (cid:32) L j (cid:33) ≤ max m +1 ≤ j ≤ h (cid:18) (cid:96) ( c j ) − k j (cid:19) (cid:28) n / . Using 2.4(a) again we get that with asymptotic probability 1 we have h (cid:88) j = m +1 L j ≤ α log( n ) n / in particular L ≥ n / with asymptotic probability 1. With this our lemma is an immediateconsequence of [22, Theorem 9.28] as this implies that for any fixed δ , with asymptotic probablity1 we have that Z satisfies the hypothesis (9.30) in this statement (with (cid:15) = δ ). (cid:3) MODEL FOR RANDOM THREE–MANIFOLDS 25 Expansion. For a non-necessarily compact Riemannian manifold V we denote by λ ( V )the bottom of the discrete spectrum of the Laplace–Beltrami operator of V (if V has non-emptyboundary we take it to be the minimum between the spectra with Neumann or Dirichlet condi-tions). Using the results from the preceding section and comparison results in spectral geometrydue to Mantuano and Hamenst¨adt we prove the following. Proposition 3.8. There exists c λ > such that: lim n → + ∞ P [ λ ( M n ) ≥ c λ ] = 1 . Proof. One way to prove Proposition 3.8 would be a minor modification of the argument in [9,Section 4], based on the Cheeger constant of M (see also the appendix to [8]). We will insteadwork with the double N = D M to be able to apply directly the results by Hamenst¨adt andMantuano.In this proof we work with a M ∈ Ξ n which carries a hyperbolic metric with totally geodesicboundary (which we proved happens with asymptotic probability 1).Let N = D M be the double of M along its (totally geodesic) boundary, which is a closedhyperbolic manifold. The space L ( D M ) decomposes into the direct sum of ± ∂M and these spaces correspond to spaces of functions on M satisfying Neumannor Dirichlet conditions on ∂M . So we have that λ ( D M ) = λ ( M ) and in the rest of the proofwe will be concerned with establishing that λ ( D X ) is bounded away from 0.We fix a Margulis constant δ . By [26, Theorem 1] we have that λ ( N ) > λ ( N ≥ δ ) / λ ( N )is uniformly bounded away from zero, in which case we are finished). So we must bound λ ( N ≥ δ )from below.To do so we will use the following result which is an immediate application of [36, Theorem3.7]: • let V be a compact hyperbolic 3–manifold with inj( V ) ≥ δ (for example the δ -thick partof a manifold of finite volume if its boundary is smooth), • Let X be a maximal δ/ V , on which we put the graph structurewhere there is an edge between x, y ∈ X if they are at distance at most 2 δ from eachother in V ( X is called a discretisation of V , and δ its mesh).Then there is c > δ such that λ ( V ) ≥ cλ ( X ).We record the following well-known facts which we will use to compare between discretisationsof our manifolds Y, Z and M Lemma 3.9. Let E , E be two metric geodesic spaces and X i a discretisation of E i .(1) The inclusion X i ⊂ E i is a quasi-isometry with constants depending only on the mesh.(2) If ϕ is a quasi-isometry from E to E and q is a nearest-neighbour projection from E to X then q ◦ ϕ induces a quasi-isometry from X to X , whose quasi-isometry constantsdepend only on those of ϕ and on the meshes of X , X .Proof. To prove the “quasi-isometric embedding” part of the first statement take x, x (cid:48) ∈ X i , thenthe nerve of a cover of a geodesic in E i between x, x (cid:48) by δ -balls (where δ is the mesh) centered in X i gives a path in X i with length at most 2 δ − d E i ( x, x (cid:48) ); the reverse inequailty is immediate. Itis also immediate to check that a quasi-inverse is given by any nearest-point projection, which isa quasi-isometry whose constants also depend only on the mesh. The second point immediatelyfollows. (cid:3) Let G be a discretisation of N ≥ δ with mesh δ/ 2. Let ϕ : N ≥ δ → D Y ≥ δ be defined as follows:by Lemma 3.7 we have M ≥ δ ⊃ Z ≥ δ/ , so we can define a retraction π : M ≥ δ → Z ≥ δ/ byfollowing the geodesic flow in the direction orthogonal to the boundary ∂Z ≥ δ/ —if there aremultiple possible directions to follow, i.e. if we are on a core geodesic, we choose one arbitrarily.We extend this to N ≥ δ = D M ≥ δ by symmetry. By the rest of the statement of the lemma,this is (1 + η )-bilipschitz on N ≥ δ and since M ≥ δ ⊂ Z ≥ δ/ , for x ∈ N ≥ δ \ D Z ≥ δ/ we have d M ( x, D Z ≥ δ/ ) ≤ aδ for an absolute a . It follows that π is a (1 + η, bηδ )-quasi-isometry, for someabsolute b . Now we extend by symmetry the J -bilipschitz map ψ from Z ≥ δ/ to Y ≥ δ/ given byLemma 3.6 and put ϕ = ψ ◦ π , which from what we said is a quasi-isometry from N ≥ δ to D Y ≥ δ/ with constants depending only on δ, η .Applying the lemma to ϕ we get that G and an arbitrary discretisation G of Y ≥ δ/ with mesh δ/ δ . Let DG be the graphdual to the tesselation of N by octahedra; it is obtained by taking two copies of G and adding fouredges between every pair of corresponding vertices. On G we can define a map to DG by mappingall vertices in a given octahedron of M to the center of that octahedron (we choose arbitarily forvertices on the boundary between two faces). This is a quasi-isometry with constants dependingonly on δ (via the diameter of O ≥ δ ). Composing ϕ with this we get a quasi-isometry from G to G . By [36, Theorem 2.1] it follows that λ ( G ) ≥ c (cid:48) λ ( DG ) where c depends only on δ . As G is a discretisation of D X ≥ δ , by loc. cit., Theorem 3.7 (see the statement at the beginning of thesection) it finally follows that λ ( D X ≥ δ ) ≥ c (cid:48)(cid:48) λ ( DG ). It is well known that G n is an expanderasymptotically almost surely; the sharpest bounds on its spectral gap are due to Friedman [21].The double DG is quasi-isometric to G with uniform constants via the inclusion, so loc. cit.,Theorem 2.1 gives us that it is also an expander a.a.s. We conclude that λ ( N ≥ δ ), and hence also λ ( N ), is bounded away from zero. (cid:3) Volumes. If the manifold X ∈ Ξ n is hyperbolic then it has a hyperbolic volume vol( X ).Otherwise we take vol( X ) = 0. Recall that v O denotes the volume of the right-angled hyperbolicoctahedron. Proposition 3.10. We have vol( M n ) ∼ n · v O as n → ∞ in probability.Proof. If M ∈ Ξ n is hyperbolic then it is a Dehn surgery on a manifold Y ∈ Θ n . The latter isa union of n copies of the octahedron O . As the hyperbolic volume decreases under hyperbolicDehn surgery we get that vol( X ) ≤ nv O .All statements in the following paragraph hold asymptotically almost surely. By Lemma 3.6we have that(3.1) vol( Z ) ≥ vol( Y ) − O ( n / log n ) = n vol( O ) − O ( n / log n )(at most 4 n / log n octahedra are changed from Z to Y since this is an upper bound for thenumber of squares in the small cusps in a generic Y by Theorem 2.4(a)). By Lemma 3.7 wehave that for any positive δ and η we have, since if two riemannian metrics on a manifold are η (cid:48) -blilipschitz to each other ther the volume forms are O ( η ) pointwise close to each other, that:vol( M ) ≥ (1 − cη ) vol( Z ≥ δ/ )for some c > δ, η . The thin part of Z is made of O (log( n )) tubes coming from theDehn filling of small cusps, so constibuting a volume O (log n ), and the rest is cusps. For a cusp MODEL FOR RANDOM THREE–MANIFOLDS 27 C we have vol( C ) = Area( ∂C ), and the boundary of the cusps of Z ≥ δ/ is made of n − O (log n )euclidean squares with edge length O ( δ ). It follows that(3.2) vol( M ) ≥ (1 − cη ) vol( Z ≥ δ/ ) ≥ (1 − cη ) vol( Z ) − δO ( n ) . Taking δ and η to 0 we get the statement we want from (3.1) and (3.2). (cid:3) Diameter and systole. Lemmas 3.6 and 3.7 above together with our combinatorial bounds(Theorem 2.4) and results by Futer–Purcell–Schleimer and Bollob´as–Fernandez-de-la-Vega implythe following bounds on the diameter and systole of M n and D M n : Proposition 3.11. (a) There exists a constant c d > such that the diameter diam( M n ) of M n satisfies: lim n → + ∞ P [diam( M n ) < c d log(vol( M n ))] = 1 (b) There exists a constant c s > such that the systole sys( M n ) of M n satisfies: lim n → + ∞ P [sys( M n ) > c s ] = 1 (c) For every ε > , lim n → + ∞ P (cid:20) sys( D M n ) < n / − ε (cid:21) = 1 . The same holds for the minimal length among arcs in M n that are homotopically non-trivial relative to ∂M n .Proof. We start with item (a). Bollob´as–Fernandez-de-la-Vega [5] proved that the diameter (inthe graph distance) of a random 4-regular graph G n on n vertices satisfiesdiam( G n ) ≤ log ( n ) + o (log( n ))in probability.Again, using results from graph theory [7, 43], we may assume that G n is conditioned to nothave loops or multiple edges, so that G n is uniformly quasi-isometric to the δ -thick part of Y n (with constants that only depend on δ ).Now we pick δ > H . Using Lemmas 3.6 and 3.7, plusthe fact that the polytopes P l descibed in the proof of Lemma 3.6 converge to O , this implies thatthe δ -thick part ( M n ) ≥ δ of M n is uniformly quasi-isometric to G n . Hence, there exists a constant C δ > M n ) ≥ δ ) ≤ C δ log( n )asymptotically almost surely.In order to control the diameter of the thin parts of M n , it’s easier to think in terms of Margulistubes, so we will consider the double D M n as a Dehn filling of D Y n . The Margulis Lemma tellsus that the thin part of D M n consists of standard tubes (see for instance [4, Chapter D]) of theform T r = { x ∈ D M n : d( x, γ ) < r } where γ is a simple closed geodesic. As such, the diameter of such a tube is at most 2 r + (cid:96) ( γ ) ≤ r + δ .The length of a meridian on the boundary torus of a standard tube is 2 π sinh( r ). In Y n , thelength of the meridian is a constant multiple of the combinatorial length of the correspondingcusp (and hence bounded by 6 n ).We want to estimate the lengths of meridians in M n in terms of those in Y n . To do so we firstobserve that these lengths are the same between Y n and Z n by Lemma 3.6. Then using Lemma3.7 in the same way that we used to construct a retraction in the proof of Proposition 3.8 we see that there is a bilipschitz map (with constants independent of n ) between the boundaries of M ≥ δ and Z ≥ δ , which sends meridian to meridian. It follows that there exists a constant D δ > D M n is at most D δ · n . This in turn implies that theradius of each such tube can be bounded by E δ log( n ) for some constant E δ > 0, depending on δ only. Combining this with our estimate on the diameter of the thick part and the estimates onvolume from Proposition 3.10 this implies item (a).We proceed to item (b). We observe that, for δ below the Margulis constant in H , the δ -thin part of M n is simply connected. In particular, any closed geodesic that passes through the( δ/ M n has length at least d(( M n ) ≥ δ , ( M n ) <δ/ ), which is uniformly bounded frombelow (by applying Lemmas 3.6 and 3.7) to the ( δ/ M n ). The ( δ/ M n is bilipschitz to the (cid:101) δ -thick part of Y n for some uniform (cid:101) δ > O ≥ (cid:101) δ ,which gives us a lower bound on the systole of the ( δ/ M n . Together with theuniform bound on the length of geodesics that pass through the thin part, this implies a lowerbound on sys( M n ).For item (c) we use our combinatorial bounds again. Recall from the proof of Lemma 3.7 that,with probability tending to 1 as n → ∞ , the total cusp length L satisfies L > n / − o (1) . [22,Corollary 6.13] now immediatley implies the result. (cid:3) Benjamini–Schramm convergence. Coxeter groups. Let T be ideal terahedron obtained by cutting O along all of its medianhyperplanes. Let Γ T be the associated reflection group. It is a Coxeter group with presentationΓ T = (cid:104) σ, τ , τ , τ | τ i , σ , [ τ i , τ j ] , ( στ j ) (cid:105) This group is useful for us because of the following lemma. Lemma 3.12. Any X ∈ Θ n is an orbifold cover of T .Proof. Let X ∈ Θ n and G the graph dual to its tesselation by octahedra. Then X is an orbifoldcover of O if and only if G is 4-edge-colourable, which is not always the case.However if we replace each vertex of G by a cube with four outgoing edges placed at pairwisenon-adjacent vertices we get a graph G (cid:48) . We colour its edges as follows : all edges between cubesare coloured with σ , and inside the cube we choose the unique colouring corresponding to thelabels (in Z / Z )—this makes sense since the edge corresponds to a face of O and the adjacentedges of O are each specified by a τ i . This specifies a unique map X → T which is an orbifoldcover. (cid:3) Invariant random subgroups. Let G be a Lie group (we will only consider G = PGL ( C )).We recall from [1] that an invariant random subgroup of G is a Borel probability measure on theChabauty space of closed subgroups of G (a compact Hausdorff topological space the definitionof which can be found in loc. cit.) which is invariant under the action of G on this space byconjugation.An important constructions of such is the following: if Λ ≤ G is a subgroup whose normaliseris a lattice Γ in G then the closure of the conjugacy class of Λ supports a unique invariant randomsubgroup (the image of Haar measure on G/ Γ). We denote this by µ Λ .Using this we can associate an invariant random subgroup to the random variable M n asfollows: let Ξ hyp n be the subset of manifolds in Ξ n which support a complete hyperbolic structurewith totally geodesic boundary. For M ∈ Ξ hyp n we consider the hyperbolic orbifold on M whosesingular locus is its boundary ∂M (a mirror) and the hyperbolic structure on the interior is that of MODEL FOR RANDOM THREE–MANIFOLDS 29 M . This is a compact hyperbolic orbifold and we choose an arbitrary monodromy group Γ M ≤ G for it and let µ M = µ Γ M . If M (cid:54)∈ Ξ hyp n we take µ M to be the Dirac mass at the trivial subgroup. We put:(3.3) µ n = (cid:88) M ∈ Ξ n P ( M n = M ) µ M . We also need to define some other invariant random subgroups which will play a role in whatfollows. Consider the ideal octahedron O as a complete hyperbolic orbifold (all faces beingmirrors). Let Γ O be its orbifold fundamental group, which is generated by the reflections on thesides of O . Let Q be the group generated by the rotations of angle 2 π/ P = Q \ O is an hyperbolic orbifold; let Γ P be its orbifold fundamental group, which we view asa lattice in G (we need this larger group because not every manifold in Ξ n is an orbifold cover of O ).Since O is right-angled, mapping four reflections on nonadjacent faces to the identity gives amap π : Γ O → ∗ i =1 Z / Z (each remaining face maps to the generator of one of the free factors), and the latter is isomorphicto D ∞ ∗ D ∞ where D ∞ = Z / Z ∗ Z / Z is the infinite dihedral group. Let O ∞ be the associatedcover; it is the infinite hyperbolic polyhedron obtained by gluing copies of O in a 4-valent treepattern, along non-adjacent faces ( D ∞ ∗ D ∞ acts via its action on the 4-valent tree). Note thatif we view the mirrors as a totally geodesic boundary O ∞ is the universal cover of any manifoldin some Θ n .Since Q respects the colouring of the faces of O we have that ker( π ) is a normal subgroupin Γ P . We let µ O ∞ be the invariant random subgroup of G associated to the normal subgroupker( π ) ≤ Γ P . Our main result in this section is then the following. Proposition 3.13. The invariant random subgroup µ n converges to µ O ∞ Proof. Let µ (cid:48) n be the invariant random subgroup associated to the random variable Y n . We willfirst prove that the sequence µ (cid:48) n converges to µ O ∞ . Let Γ P be the group defined above andΛ = ker( π ) ≤ Γ P . Every M ∈ Ξ n admits a (possibly non-continuous) picewise isometric map to O (by mapping its marked octahedra to O ). The non-continuity comes from the rotations madewhen gluing faces so the composition M → O → P is continuous and hence a covering map. LetΓ n ≤ Γ P be the invariant random index-12 n subgroup corresponding to M n .The Schreier graph of Γ P / Γ n with loops removed is obtained from the graph dual to thetesselation of M n by replacing each vertex with a fixed graph Q . The dual graphs follow thesame distribution as the configuration model, and as this model of graphs BS-converges to thetree (this follows from [6]) we get that the random variable Γ n converges in distribution to Λ(since the Schreier graph of Γ P / Λ is obtained from the tree by replacing vertices with Q ). Now µ n is the IRS obtained by induction of Γ n from Γ to PGL ( C ), and µ O ∞ by induction of Λ (see[2, 11.1] for the definition of induction). As induction is continuous we get that µ (cid:48) n converges to µ O ∞ .Now if µ (cid:48)(cid:48) n is the IRS associated to Z it follows immediately from the convergence of µ (cid:48) n togetherwith Lemma 3.6 that we also have lim( µ (cid:48)(cid:48) n ) = µ O ∞ .We pass to the larger space of random pointed metric spaces with Benjamini–Schramm topology(see [24, Section 5]). If µ is an IRS of PGL ( C ) we denote by µ ≥ δ the random pointed compactmanifold with boundary which comes from conditioning the point to be in the thick part (note that doing so we lose all invariance properties). It follows from the previous paragraph that( µ (cid:48)(cid:48) n ) ≥ δ converges to µ ≥ δO ∞ and from Lemma 3.7 that µ ≥ δn also does.Now the map ( M, x ) (cid:55)→ ( M ≥ δ , x ) is an homeomorphism onto its image: it is continuous (im-mediate) and injective (the boundary of the thin part determines the complex length of the coregeodesic if it is a tube, and the isometry class of the cross-section if it is a torus), and the spaceof hyperbolic manifolds pointed in their thick part is compact. We can thus conclude that µ n converges to µ O ∞ . (cid:3) References [1] Miklos Abert, Nicolas Bergeron, Ian Biringer, Tsachik Gelander, Nikolay Nikolov, Jean Raimbault, and IddoSamet. On the growth of L -invariants for sequences of lattices in Lie groups. Ann. of Math. (2) , 185(3):711–790, 2017.[2] Miklos Abert, Nicolas Bergeron, Ian Biringer, Tsachik Gelander, Nikolay Nikolov, Jean Raimbault, and IddoSamet. On the Growth of L -Invariants of Locally Symmetric Spaces, II: Exotic Invariant Random Subgroupsin Rank One. International Mathematics Research Notices , 05 2018.[3] Hyungryul Baik, David Bauer, Ilya Gekhtman, Ursula Hamenst¨adt, Sebastian Hensel, Thorben Kastenholz,Bram Petri, and Daniel Valenzuela. Exponential torsion growth for random 3-manifolds. Int. Math. Res. Not.IMRN , (21):6497–6534, 2018.[4] Riccardo Benedetti and Carlo Petronio. Lectures on hyperbolic geometry . Universitext. Springer-Verlag, Berlin,1992.[5] B. Bollob´as and W. Fernandez de la Vega. The diameter of random regular graphs. Combinatorica , 2(2):125–134, 1982.[6] B´ela Bollob´as. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. , 1(4):311–316, 1980.[7] B´ela Bollob´as. Random graphs , volume 73 of Cambridge Studies in Advanced Mathematics . Cambridge Uni-versity Press, Cambridge, second edition, 2001.[8] Emmanuel Breuillard. Expander graphs, property ( τ ) and approximate groups. In Geometric group theory ,volume 21 of IAS/Park City Math. Ser. , pages 325–377. Amer. Math. Soc., Providence, RI, 2014.[9] Robert Brooks. The spectral geometry of the Apollonian packing. Comm. Pure Appl. Math. , 38(4):359–366,1985.[10] Robert Brooks and Eran Makover. Random construction of Riemann surfaces. J. Differential Geom. , 68(1):121–157, 2004.[11] Thomas Budzinski, Nicolas Curien, and Bram Petri. On the minimal diameter of closed hyperbolic surfaces.Preprint, arXiv:1909.12283, 2019.[12] Thomas Budzinski, Nicolas Curien, and Bram Petri. Universality for random surfaces in unconstrained genus. Electron. J. Combin. , 26(4):Paper 4.2, 34, 2019.[13] Peter Buser. A note on the isoperimetric constant. Ann. Sci. ´Ecole Norm. Sup. (4) , 15(2):213–230, 1982.[14] Guillaume Chapuy and Guillem Perarnau. On the number of coloured triangulations of d-manifolds. DiscreteComput. Geom., to appear , 2020+.[15] Sergei Chmutov and Boris Pittel. On a surface formed by randomly gluing together polygonal discs. Adv. inAppl. Math. , 73:23–42, 2016.[16] Fran¸cois Costantino, Roberto Frigerio, Bruno Martelli, and Carlo Petronio. Triangulations of 3-manifolds,hyperbolic relative handlebodies, and Dehn filling. Comment. Math. Helv. ∼ curien/enseignement.html, 2020.[18] Kelly Delp, Diane Hoffoss, and Jason Fox Manning. Problems in groups, geometry, and three-manifolds.Preprint, arXiv:1512.04620, 2015.[19] Nathan M. Dunfield and William P. Thurston. Finite covers of random 3-manifolds. Invent. Math. , 166(3):457–521, 2006.[20] Peter Feller, Alessandro Sisto, and Gabriele Viaggi. Uniform models and short curves for random 3-manifolds.Preprint, arXiv: 1910.09486, 2020.[21] Joel Friedman. A proof of Alon’s second eigenvalue conjecture and related problems. Mem. Amer. Math. Soc. ,195(910):viii+100, 2008.[22] David Futer, Jessica S. Purcell, and Saul Schleimer. Effective bilipschitz bounds on drilling and filling, 2019.[23] Alex Gamburd. Poisson-Dirichlet distribution for random Belyi surfaces. Ann. Probab. , 34(5):1827–1848, 2006. MODEL FOR RANDOM THREE–MANIFOLDS 31 [24] T Gelander. A lecture on invariant random subgroups. In New Directions in Locally Compact Groups , pages186–204. Cambridge Univ. Press, 2018.[25] Larry Guth, Hugo Parlier, and Robert Young. Pants decompositions of random surfaces. Geom. Funct. Anal. ,21(5):1069–1090, 2011.[26] Ursula Hamenst¨adt. Small eigenvalues and thick-thin decomposition in negative curvature. Ann. Inst. Fourier ,to appear.[27] Ursula Hamenst¨adt and Gabriele Viaggi. Small eigenvalues of random 3-manifolds. Preprint, arXiv:1903.08031,2019.[28] Holger Kammeyer. Introduction to (cid:96) -invariants , volume 2247 of Lecture Notes in Mathematics . Springer,Cham, 2019.[29] Emmanuel Kowalski. Crible en expansion. Number 348, pages Exp. No. 1028, vii, 17–64. 2012. S´eminaireBourbaki: Vol. 2010/2011. Expos´es 1027–1042.[30] Marc Lackenby. Heegaard splittings, the virtually Haken conjecture and property ( τ ). Invent. Math. ,164(2):317–359, 2006.[31] Thang T. Q. Lˆe. Growth of homology torsion in finite coverings and hyperbolic volume. Annales de l’InstitutFourier , 68(2):611–645, 2018.[32] Alexander Lubotzky, Joseph Maher, and Conan Wu. Random methods in 3-manifold theory. Tr. Mat. Inst.Steklova , 292(Algebra, Geometriya i Teoriya Chisel):124–148, 2016.[33] Michael Magee, Fr´ed´eric Naud, and Doron Puder. A random cover of a compact hyperbolic surface has relativespectral gap − ε . Preprint, arXiv:2003.10911, 2020.[34] Joseph Maher. Random Heegaard splittings. J. Topol. , 3(4):997–1025, 2010.[35] Joseph Maher. Random walks on the mapping class group. Duke Math. J. , 156(3):429–468, 2011.[36] Tatiana Mantuano. Discretization of compact Riemannian manifolds applied to the spectrum of Laplacian. Ann. Global Anal. Geom. , 27(1):33–46, 2005.[37] Maryam Mirzakhani. Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus. J.Differential Geom. , 94(2):267–300, 2013.[38] Jean Raimbault. G´eom´etrie et topologie des vari´et´es hyperboliques de grand volume. Actes du s´eminarie deTh´eorie spectrale et g´eom´etrie , 31:163–195, 2014.[39] Roland K. W. Roeder, John H. Hubbard, and William D. Dunbar. Andreev’s theorem on hyperbolic polyhedra. Ann. Inst. Fourier (Grenoble) , 57(3):825–882, 2007.[40] Martin Scharlemann. Heegaard splittings of compact 3-manifolds. In Handbook of geometric topology , pages921–953. North-Holland, Amsterdam, 2002.[41] Alessandro Sisto and Samuel J. Taylor. Largest projections for random walks and shortest curves in randommapping tori. Math. Res. Lett. , 26(1):293–321, 2019.[42] Gabriele Viaggi. Volumes of random 3-manifolds. Preprint, arXiv:1905.04935, 2019.[43] N. C. Wormald. Models of random regular graphs. In Surveys in combinatorics, 1999 (Canterbury) , volume267 of London Math. Soc. Lecture Note Ser. , pages 239–298. Cambridge Univ. Press, Cambridge, 1999.[44] Alex Wright. A tour through Mirzakhani’s work on Riemann surfaces. Bull. Amer. Math. Soc. to appear, 2020. Institut de Math´ematiques de Jussieu–Paris Rive Gauche ; UMR7586, Sorbonne Universit´e - Cam-pus Pierre et Marie Curie, 4, place Jussieu, 75252 Paris Cedex 05, France E-mail address : [email protected] Institut de Math´ematiques de Toulouse ; UMR5219, Universit´e de Toulouse ; CNRS, UPS IMT,F-31062 Toulouse Cedex 9, France E-mail address ::