3-Manifolds Everywhere

The “generic” objects in the world of finitely presented groups are the random groups, in the sense of Gromov. There are two models of what one means by a random group, and we shall briefly discuss them both.

First, fix $k\geq 2$ and fix a free generating set $x_{1},x_{2},\cdots,x_{k}$ for $F_{k}$, a free group of rank $k$. A $k$-generator group $G$ can be given by a presentation

where the $r_{i}$ are cyclically reduced cyclic words in the $x_{i}$ and their inverses.

In the few relators model of a random group, one fixes $\ell\geq 1$, and then for a given integer $n$, selects the $r_{i}$ independently and randomly (with the uniform distribution) from the set of all reduced cyclic words of length $n$.

In the density model of a random group, one fixes $0<D<1$, and then for a given integer $n$, define $\ell=\lfloor(2k-1)^{Dn}\rfloor$ and select the $r_{i}$ independently and randomly (with the uniform distribution) from the set of all reduced cyclic words of length $n$.

Thus, the difference between the two models is how the number of relations ($\ell$) depends on their length ($n$). In the few relators model, the absolute number of relations is fixed, whereas in the density model, the (logarithmic) density of the relations among all words of the given length is fixed.

For fixed $k,\ell,n$ in the few relators model, or fixed $k,D,n$ in the density model, we obtain in this way a probability distribution on finitely presented groups (actually, on finite presentations). For some property of groups of interest, the property will hold for a random group with some probability depending on $n$. We say that the property holds for a random group with overwhelming probability if the probability goes to 1 as $n$ goes to infinity.

As remarked above, a “random group” really means a “random presentation”. Associated to a finite presentation of a group $G$ as above, one can build a 2-complex $K$ with one 0 cell, with one 1 cell for each generator $x_{i}$, and with one 2 cell for each relation $r_{j}$, so that $\pi_{1}(K)=G$. We are very interested in the geometry and combinatorics of $K$ (and its universal cover) in what follows.

All few-relator random groups are alike (with overwhelming probability). There is a phase transition in the behavior of density random groups, discovered by [Gromov(1993)] § 9: for $D>1/2$, a random group is either trivial or isomorphic to $\mathbb{Z}/2\mathbb{Z}$, whereas at any fixed density $0<D<1/2$, a random group is infinite, hyperbolic, and 1-ended, and the presentation determining the group is aspherical — i.e. the 2-complex $K$ defined from the presentation has contractible universal cover. Furthermore, [Dahmani–Guirardel–Przytycki(2011)] showed that a random group with density less than a half does not split, and has boundary homeomorphic to the Menger sponge.

The Menger sponge is obtained from the unit cube by subdividing it into 27 smaller cubes each with one third the side length, then removing the central cube and the six cubes centered on each face; and then inductively performing the same procedure with each of the remaining 20 smaller cubes; see Figure 1.

The Menger sponge has topological dimension 1, and is the universal compact space with this property, in the sense that any compact Hausdorff space of topological dimension 1 embeds in it.

It is important to understand what kinds of abstract groups $H$ arise as subgroups of a random group $G$. However, not all subgroups of a hyperbolic group are of equal importance: the most useful subgroups, and those that tell us the most about the geometry of $G$, are the subgroups $H$ with the following properties:

1. (1)

   the group $H$ itself is something whose intrinsic topological and geometric properties we understand very well; and

2. (2)

   the intrinsic geometry of $H$ can be uniformly compared with the extrinsic geometry of its embedding in $G$.

In other words, we are interested in well-understood groups $H$ which are quasi-isometrically embedded in $G$.

A finitely generated quasi-isometrically embedded subgroup $H$ of a hyperbolic group $G$ is itself hyperbolic, and therefore finitely presented. The inclusion of $H$ into $G$ induces an embedding of Gromov boundaries $\partial_{\infty}H\to\partial_{\infty}G$. Thus if $G$ is a random group, the boundary of $H$ has dimension at most 1, and by work of [Kapovich–Kleiner(2000)], there is a hierarchical description of all possible $\partial_{\infty}H$.

First of all, if $\partial_{\infty}H$ is disconnected, then one knows by [Stallings(1968)] that $H$ splits over a finite group. Second of all, if $\partial_{\infty}H$ is connected and contains a local cut point, then one knows by [Bowditch(1998)] that either $H$ is virtually a surface group, or else $H$ splits over a cyclic group. Thus, apart from the Menger sponge itself, understanding hyperbolic groups with boundary of dimension at most 1 reduces (in some sense) to the case that $\partial_{\infty}H$ is a Cantor set, a circle, or a Sierpinski carpet — i.e. one of the “faces” of the Menger cube. The Sierpinski carpet is universal for 1-dimensional compact Hausdorff planar sets of topological dimension 1. Thus one is naturally led to the following fundamental question, which as far as we know was first asked explicitly by François Dahmani:

Combining the main theorem of this paper with the results of [Calegari–Walker(2015)] gives a complete answer to this question:

This is true in the few relators model with any positive number of relators, or the density model at any density less than a half. The situation is summarized as follows:

1. (1)

   $\partial_{\infty}H$ is a Cantor set if and only if $H$ is virtually free (of rank at least 2); we can thus take $H=\pi_{1}(\text{graph})$. The existence of free quasiconvex subgroups $H$ in arbitrary (nonelementary) hyperbolic groups $G$ is due to Klein, by the ping-pong argument.

2. (2)

   $\partial_{\infty}H$ is a circle if and only if $H$ is virtually a surface group (of genus at least 2); we can thus take $H=\pi_{1}(\text{surface})$. The existence of surface subgroups $H$ in random groups (with overwhelming probability) is proved by [Calegari–Walker(2015)], Theorem 6.4.1.

3. (3)

   $\partial_{\infty}H$ is a Sierpinski carpet if $H$ is virtually the fundamental group of a compact hyperbolic 3-manifold with (nonempty) totally geodesic boundary; and if the Cannon conjecture is true, this is if and only if — see [Kapovich–Kleiner(2000)]. The existence of such 3-manifold subgroups $H$ in random groups (with overwhelming probability) is Theorem 6.2.1 in this paper.

Explicitly, we prove:

Our theorem applies in particular in the few relators model where $\ell=1$. In fact, if one fixes $D<1/2$, and starts with a random 1-relator group $G=\langle F_{k}\;|\;r\rangle$, one can construct a quasiconvex 3-manifold subgroup $H$ (of the sort which is guaranteed by the theorem) which stays injective and quasiconvex (with overwhelming probability) as a further $(2k-1)^{Dn}$ relators are added.

Two groups are said to be commensurable if they have isomorphic subgroups of finite index. Commensurability is an equivalence relation, and it is natural to wonder what commensurability classes of 3-manifold groups arise in a random group.

We are able to put very strong constraints on the commensurability classes of the 3-manifold groups we construct. It is probably too much to hope to be able to construct a subgroup of a fixed commensurability class. But we can arrange for our 3-manifold groups to be commensurable with some element of a family of finitely generated groups given by presentations which differ only by varying the order of torsion of a specific element. Hence our 3-manifold subgroups are all commensurable with Kleinian groups of bounded convex covolume (i.e. the convex hulls of the quotients have uniformly bounded volume). Explicitly:

The Coxeter group $\Gamma(m)$ is commensurable with the group generated by reflections in the sides of a regular super ideal tetrahedron — one with vertices “outside” the sphere at infinity — and with dihedral angles $\pi/m$.

We now describe the outline of the paper.

As discussed above, our random groups $G$ come together with the data of a finite presentation

From such a presentation we can build in a canonical way a 2-complex $K$ with 1-skeleton $X$, where $X$ is a wedge of $k$ circles, and $K$ is obtained by attaching $\ell$ disks along loops in $X$ corresponding to the relators; so $\pi_{1}(K)=G$.

Our 3-manifold subgroups arise as the fundamental groups of 2-complexes $\overline{M}(Z)$ that come with immersions $\overline{M}(Z)\to K$ taking (open) cells of $\overline{M}(Z)$ homeomorphically to cells of $K$. Thus, for every 2-cell of $\overline{M}(Z)$, the attaching map of its boundary to the 1-skeleton $Z$ factors through a map onto one of the relators of the given presentation of $G$.

One way to obtain such a complex $\overline{M}(Z)$ is to build a 1-complex $Z$ as a quotient of a collection $L$ of circles together with an immersion $Z\to X$ where $X$ is the 1-skeleton of $K$, and the map $L\to X$ takes each component to the image of a relator. We call data of this kind a spine. In § 2 we describe the topology of spines and give sufficient combinatorial conditions on a spine to ensure that $\overline{M}(Z)$ is homotopic to a 3-manifold.

Since $X$ is a rose whose edges are endowed with a choice of orientation and labelling by the generators $x_{i}$, we will usually encode a map of graphs $\Gamma\to X$ by labelling (oriented) edges by $x_{i}^{\pm 1}$, in the spirit of [Stallings(1983)]. As usual, if an oriented edge $e$ of $\Gamma$ is labelled $x_{i}^{\pm 1}$ then the oriented edge $\bar{e}$ with the reverse orientation is labelled $x_{i}^{\mp 1}$. Note that there is a simple condition to ensure that such a map $\Gamma\to X$ is an immersion: one simply requires that no two oriented edges of $\Gamma$ incident at the same vertex have the same label. We call such a graph $\Gamma$ folded (also in the spirit of [Stallings(1983)]).

In § 3 we prove the Thin Spine Theorem, which says that we can build such a spine $L\to Z$, satisfying the desired combinatorial conditions, and such that every edge of the 1-skeleton $Z$ is long. Here we measure the length of edges of $Z$ by pulling back length from $X$ under the immersion $Z\to X$, where each edge of $X$ is normalized to have length $1$. In fact, if we let $G_{1}$ denote the 1-relator group

with associated 2-complex $K_{1}$ which comes with a tautological inclusion $K_{1}\to K$, then our thin spines have the property that the immersion $\overline{M}(Z)\to K$ factors through $\overline{M}(Z)\to K_{1}$.

For technical reasons, rather than working with a random relator $r_{1}$, we work instead with a relator which is merely sufficiently pseudorandom (a condition concerning equidistribution of subwords with controlled error on certain scales), and the theorem we prove is deterministic. Of course, the definition of pseudorandom is such that a random word will be pseudorandom with very high probability.

Explicitly, we prove:

This is by far the longest section in the paper, and it involves a complicated combinatorial argument with many interdependent steps. It should be remarked that one of the key ideas we exploit in this section is the method of random matching with correction: randomness (actually, pseudorandomness) is used to show that the desired combinatorial construction can be performed with very small error. In the process, we build a reservoir of small independent pieces which may be adjusted by various local moves in such a way as to “correct” the errors that arose at the random matching step. Similar ideas were also used by [Kahn–Markovic(2011)] in their proof of the Ehrenpreis Conjecture, by [Calegari–Walker(2015)] in their construction of surface subgroups in random groups, and by [Keevash(2014)] in his construction of General Steiner Systems and Designs. Evidently this method is extremely powerful, and its full potential is far from being exhausted.

The Thin Spine Theorem can be summarized by saying that as a graph, $Z$ has bounded valence, but very long edges. This means that the image of $\pi_{1}(Z)$ in $\pi_{1}(X)$ induced by the inclusion $Z\to X$ is very “sparse”, in the sense that the ball of radius $n$ in $\pi_{1}(X)$ contains $O(3^{n/\lambda})$ elements of $\pi_{1}(Z)$, where we can take $\lambda$ as big as we like. This has the following consequence: when we obtain $G_{1}$ as a quotient of $\pi_{1}(X)$ by adding $r_{1}$ as a relator, we should not kill any “accidental” elements of $\pi_{1}(Z)$, so that the image of $\pi_{1}(Z)$ will be isomorphic to $\pi_{1}(\overline{M}(Z))$, a 3-manifold group.

This idea is fleshed out in § 4, and shows that random 1-relator groups contain 3-manifold subgroups, although at this stage we have not yet shown that the 3-manifold is of the desired form. The argument in this section depends on a so-called bead decomposition, which is very closely analogous to the bead decomposition used to construct surface groups by [Calegari–Walker(2015)], and the proof is very similar.

In § 5 we show that the 3-manifold homotopic to the 2-complex $\overline{M}(Z)$ is acylindrical; equivalently, that it is homeomorphic to a compact hyperbolic 3-manifold with totally geodesic boundary. This is a step with no precise analog in [Calegari–Walker(2015)], but the argument is very similar to the argument showing that $\overline{M}(Z)$ is injective in the 1-relator group $G^{\prime}$. There are two kinds of annuli to rule out: those that use 2-cells of $\overline{M}(Z)$, and those that don’t. The annuli without 2-cells are ruled out by the combinatorics of the construction. Those that use 2-cells are ruled out by a small cancellation argument which uses the thinness of the spine. So at the end of this section, we have shown that random 1-relator groups contain subgroups isomorphic to the fundamental groups of compact hyperbolic 3-manifolds with totally geodesic boundary.

Finally, in § 6, we show that the subgroup $\pi_{1}(\overline{M}(Z))$ stays injective as the remaining $\ell-1$ random relators are added. The argument here stays extremely close to the analogous argument in [Calegari–Walker(2015)], and depends (as [Calegari–Walker(2015)] did) on a kind of small cancellation theory for random groups developed by [Ollivier(2007)]. This concludes the proof of the main theorem.

A further section § 7 proves the Commensurability Theorem. The proof is straightforward given the technology developed in the earlier sections.

Before introducing spines, we first motivate them by describing the analogous, but simpler, theory of fatgraphs.

A fatgraph $Y$ is a (simplicial) graph together with a cyclic ordering of the edges incident to each vertex. This cyclic ordering can be used to canonically “fatten” the graph $Y$ so that it embeds in an oriented surface $S(Y)$ with boundary, in such a way that $S(Y)$ deformation retracts down to $Y$. Under this deformation retraction, the boundary $\partial S(Y)$ maps to $Y$ in such a way that the preimage of each edge $e$ of $Y$ consists of two intervals $e^{\pm}$ in $\partial S(Y)$, each mapping homeomorphically to $e$, with opposite orientations.

Abstractly, the data of a fatgraph can be given by an ordinary graph $Y$, a 1-manifold $L$, and a locally injective simplicial map $f:L\to Y$ of (geometric) “degree 2”; i.e. such that each edge of $Y$ is in the image of two intervals in $L$. The surface $S(Y)$ arises as the mapping cone of $f$. If one orients $L$ and insists that the preimages of each edge have opposite orientations, the result is a fatgraph and an oriented surface as above. If one does not insist on the orientation condition, the mapping cone need not be orientable. Attaching a disk along its boundary to each component of $L$ produces a closed surface, which we denote $\overline{S}(Y)$.

We would like to discuss a more complicated object called a spine, for which the analog of $\overline{S}(Y)$ is a 2-complex homotopy equivalent to a compact 3-manifold with boundary. The 2-complex will arise by gluing 2-dimensional disks onto the components of a 1-manifold $L$, and then attaching these disks to the mapping cone of an immersion $f:L\to Z$ where $Z$ is a 4-valent graph, and the map $f$ is subject to certain local combinatorial constraints.

The first combinatorial constraint is that the map $f:L\to Z$ should be “degree 3”; that is, the preimage of each edge of $L$ should consist of three disjoint intervals in $L$, each mapped homeomorphically by $f$.

Since $Z$ is 4-valent, at each vertex $v$ of $Z$ we have 12 intervals in $L$ that map to the incident edges; these 12 intervals should be obtained by subdividing 6 disjoint intervals in $L$, where the dividing point maps to $v$. Since $L\to Z$ is an immersion, near each dividing point the given interval in $L$ runs locally from one edge incident to $v$ to a different one. There are three local models (up to symmetry) of how six edges of $L$ can locally run over a 4-valent vertex $v$ of $Z$ so that they run over each incident edge (in $Z$) to $v$ three times (this notion of “local model” is frequently called a Whitehead graph in the literature). These three local models are illustrated in Figure 2.

The third local model is distinguished by the property that for each pair of edges of $Z$ adjacent to $v$, there is exactly one interval in $L$ running over $v$ from one edge to the other. We say that such a local model is good. The second combinatorial constraint is that the local model at every vertex of $Z$ is good.

If $f:L\to Z$ is good, and $M(Z)$ is the mapping cone of $f$, then the 2-complex $M(Z)$ can be canonically thickened to a 3-manifold, since a neighborhood of the mapping cone near a vertex $v$ embeds in $\mathbb{R}^{3}$ in such a way that the tetrahedral symmetry of the combinatorics is realized by symmetries of the embedding. Similarly, along each edge the dihedral symmetry is realized by the symmetry of the embedding. The restriction of this thickening to each component of $L$ is the total space of an $I$-bundle over the circle; we say that $f:L\to Z$ is co-oriented if each of these $I$-bundles is trivial. The third combinatorial constraint is that $f:L\to Z$ is co-oriented.

By abuse of notation, we call $\overline{M}(Z)$ the thickening of $Z$.

Now let’s fix $k\geq 2$ and a free group $F_{k}$ on $k$ fixed generators. Let $X$ be a rose for $F_{k}$; i.e. a wedge of $k$ (oriented) circles, with a given labeling by the generators of $F_{k}$. Let $G$ be a random group (in whatever model) at length $n$. Each relator $r_{i}$ is a cyclically reduced word in $F_{k}$, and is realized geometrically by an immersion of an oriented circle $\iota_{i}:S^{1}_{i}\to X$. Attaching a disk along each such circle gives rise to the 2-complex $K$ described in the introduction with $\pi_{1}(K)=G$.

The existence of the simplicial homeomorphisms $g_{i}$ lets us label the components $L_{i}$ by the corresponding relators in such a way that the map $f:L\to Z$ has the property that the preimages of each edge of $Z$ get the same labels, at least if we choose orientations correctly, and use the convention that changing the orientation of an edge replaces the label by its inverse. So we can equivalently think of the labels as living on the oriented edges of $Z$. Notice that the maps $g_{i}$, if they exist at all, are uniquely determined by $g,f,\iota_{j}$ (at least if the presentation is not redundant, so that no relator is equal to a conjugate of another relator or its inverse).

Evidently, if $f:L\to Z$, $g:Z\to X$ is a spine over $K$, the immersion $g:Z\to X$ extends to an immersion of the thickening $\overline{g}:\overline{M}(Z)\to K$, that we call the tautological immersion.

Our strategy is to construct a spine over $K$ for which $\overline{M}(Z)$ is homotopy equivalent to a compact hyperbolic 3-manifold with geodesic boundary, and for which the tautological immersion induces a quasi-isometric embedding on $\pi_{1}$.

Instead of working directly with random chains, we use a deterministic variant called pseudorandomness.

For any $T,\epsilon$, a random reduced word of length $N$ will be $(T,\epsilon)$-pseudorandom with probability $1-O(e^{-N^{c}})$ for a suitable constant $c(T,\epsilon)$. This follows immediately from the standard Chernoff inequality for the stationary Markov process that produces a random reduced word in a free group (cf. [Calegari–Walker(2015), Lemma 3.2.2]).

With this definition in place, the statement of the Thin Spine Theorem is:

The strange appearance of the number 648 (or 5,832 for $k=2$) in the statement of this theorem reflects the method of proof. First of all, observe that if $f:L\to Z$ is any spine, then since $f$ has degree 3, the total length of $L$ is divisible by 3. If this spine is over $K$, then each component of $L$ has length $|r|$, and if we make no assumptions about the value of $|r|$ mod 3, then it will be necessary in general for the number of components of $L$ to be divisible by 3.

Our argument is to gradually glue up more and more of $L$, constructing $Z$ as we go. At an intermediate stage, the remainder to be glued up consists of a collection of disjoint segments from $L$, and the power of our method is precisely that this lets us reduce the gluing problem to a collection of independent subproblems of uniformly bounded size. But each of these subproblems must involve a subset of $L$ of total length divisible by 3 or 6, and therefore it is necessary to “clear denominators” (by taking 2 or 3 disjoint copies of the result of the partial construction) several times to complete the construction (in the case of rank 2 one extra move might require a further factor of 9).

Finally, at the last step of the construction, we take 2 copies of $L$ and perform a final adjustment to satisfy the co-orientation condition.

The remainder of this section is devoted to the proof of Theorem 3.1.2.

Let $L$ be a labeled graph consisting of 648 disjoint cycles (or 5,832 disjoint cycles if $k=2$), each labeled by $r$. We will build the spine $Z$ and the map $f:L\to Z$ in stages. We think of $Z$ as a quotient space of $L$, obtained by identifying segments in $L$ with the same labels. So the construction of $Z$ proceeds by inductively identifying more and more segments of $L$, so that at each stage some portion of $L$ has been “glued up” to form part of the graph $Z$, and some remains still unglued.

We introduce the following notation and terminology. Let $\Lambda_{0}$ be a single cycle labeled $r$. At the $i$th stage of our construction, we deal with a partially glued graph $\Lambda_{i}$, constructed from a certain number of copies of $\Lambda_{i-1}$ via certain ungluing and gluing moves. At each stage, $\Lambda_{i}$ is equipped with a labelling, defining a map $\Lambda_{i}\to X$. We will always be careful to ensure that $\Lambda_{i}$ is folded, i.e. that the map $\Lambda_{i}\to X$ is an immersion. The glued subgraph of $\Lambda_{i}$ is denoted by $\Gamma_{i}$, and the unglued subgraph by $\Upsilon_{i}$. Shortly, the unglued subgraph $\Upsilon_{i}$ will be expressed as the disjoint union of two subgraphs: the remainder $\Delta_{i}$ and the reservoir $\Omega_{i}$.

Each of these graphs are thought of as metric graphs, whose edges have lengths equal to the length of the words that label them. The mass of a metric graph $\Gamma$ is its total length, denoted by $m(\Gamma)$. The type of a graph refers to the collection of edge labels (which are reduced words in $F_{k}$) associated to each edge. A distribution on a certain set of types of graphs is a map that assigns a non-negative number to each type; it is integral if it assigns an integer to each type. We use this terminology without comment in the sequel.

The following properties will remain true at every stage of our construction. The branch vertices of $\Lambda_{i}$ (those of valence greater than two) have valence four. The length of each edge will always be at least $\lambda$. We will be careful to ensure that any branch vertex in the interior of the glued part $\Gamma_{i}$ is good in the sense of Section 2. Vertices in the intersection of the glued and unglued parts, $\Gamma_{i}\cap\Upsilon_{i}$, will always be branch vertices, and will be such that one adjacent edge is in $\Gamma_{i}$, and the remaining three adjacent edges in $\Upsilon_{i}$ have distinct labels.

In particular, we start with $\Lambda_{0}=\Upsilon_{0}$ and $\Gamma_{0}=\varnothing$. At the last stage of our construction we will have $\Lambda_{8}$ constructed from 324 copies of $\Lambda_{0}$ (or 2,916 if $k=2$), which is completely glued up; that is, $\Gamma_{8}=\Lambda_{8}$ and $\Upsilon_{8}=\varnothing$. Finally, the modification in Section 3.12 doubles the mass of $\Lambda_{8}$ in order to ensure that the co-orientation condition is satisfied. Taking $Z$ to be the result of this construction and $f$ to be the quotient map $L\to Z$ proves the theorem.

We will regard $k$ and $\lambda$ as constants. The first step of the construction is to pick some very big constant $N\gg\lambda$ where still $T\gg N$ (we will explain in the sequel how to choose $T$ and $N$ big enough) so that $N$ is odd.

Let $s$ be the remainder when $|r|-3\lambda$ is divided by $3(N+1)\lambda$. By pseudorandomness, we may find three subsegments in $r$ of reduced form

where $a_{i},b_{i},c_{i}$ are single edges, such that the labels $a_{1},b_{1},c_{1}$ are all distinct and $a_{2},b_{2},c_{2}$ are all distinct, and the length of $x$ is $s$. We take three copies of $\Lambda_{0}$, fix one of the above subsegments in each copy, and glue the parts of these subsegments labeled $x$ together to obtain $\Lambda_{1}$. Note that the requirement that the labels $a_{i},b_{i},c_{i}$ are distinct ensures that $\Lambda_{1}$ remains folded. We summarize this in the following lemma.

We next decompose the unglued subgraph $\Upsilon_{1}$ into disjoint segments of length $3N\lambda$ separated by segments of length $3\lambda$. Call the segments of length $3N\lambda$ long strips and the segments of length $3\lambda$ short strips. Now further decompose each long strip into alternating segments of length $3\lambda$; we call the odd numbered segments sticky and the even numbered segments free.

We will usually denote a long strip by

where the $x_{i}$ are sticky, the $a_{i}$ (or $b_{i}$ etc) are free, and all are of length $3\lambda$. When we also need to include the neighboring short strips, we will usually extend this notation to

where $a_{0}$ and $a_{N+1}$ (or $b_{0},b_{N+1}$ etc) denote the neighboring short strips.

A compatible triple of long strips can be bunched — i.e. the sticky segments can be glued together in threes, creating $(N-1)/2$ football bubbles, each football consisting of the three segments $a_{i},b_{i},c_{i}$ (for some $i$) arranged as the edges of a theta graph. See Figure 3.

By pseudorandomness, a proportion of approximately $(1-\epsilon)$ of the long strips in $\Gamma_{1}$ can be partitioned into compatible triples. Then each compatible triple can be bunched, creating a reservoir of footballs (i.e. the theta graphs appearing as bubbles) and a remainder, consisting of the union of the unglued pieces (except for the footballs). In what follows, we will denote the remainder in $\Upsilon_{i}$ by $\Delta_{i}$ and the reservoir by $\Omega_{i}$.

We summarize this observation in the next lemma. By an extended long strip, we mean a long strip, together with half the adjacent short strips.

At this point, we have bunched all but $\epsilon$ of the long strips into triples. In particular, for sufficiently small $\epsilon$, the number of bunched triples is far larger than the number of unbunched long strips. We will now argue that we may, in fact, adjust the construction so that every long strip is bunched. The advantage of this is that after this step, the unglued part consists entirely of the reservoir (a union of football bubbles) and a remainder consisting of a trivalent graph in which every edge has length exactly $3\lambda$. The key to this operation is the idea a super-compatible 4-tuple of long segments.

By the previous lemma, we may assign to each unglued long strip $S_{0}$ in the remainder $\Delta_{2}$ a glued triple $(S_{1},S_{2},S_{3})$ such that $(S_{0},S_{1},S_{2},S_{3})$ is super-compatible. We may re-bunch these into the four possible compatible triples that are subsets of our super-compatible 4-tuple, viz:

The result of performing this operation on every long strip in the remainder $\Delta_{2}$ yields the new partially glued graph $\Lambda_{3}$.

As their name indicates, the bubbles in the reservoir will be held in reserve until a later stage of the construction to glue up the remainder. Some intermediate operations on the remainder have “boundary effects”, which might disturb the neighboring bubbles in the reservoir in a predictable way. So it is important to insulate the remainder with a collar of bubbles which we do not disturb accidentally in subsequent operations.

Fix a constant $0<\theta_{n}<1$ and divide each long strip in $\Lambda_{n}$ into two parts: and inner reservoir, consisting of an innermost sequence of consecutive bubbles of length $1-\theta_{n}$ times the length of the long strip, and an outer reservoir, consisting of two outer sequences of consecutive bubbles of length $\theta_{n}/2$. The number $\theta_{n}$ is called the slack. Boundary effects associated to each step that we perform will use up bubbles from the outer reservoir and at most halve the slack. Since the number of steps we perform is uniformly bounded, it follows that — provided $N$ is sufficiently large — even at the end of the construction we will still have a significant outer reservoir with which to work.

After collecting long strips into compatible triples, the collection of football bubbles in the reservoir is “almost equidistributed”, in the sense that the mass of any two different types of football is almost equal (up to an additive error of order $\epsilon$). However, it is useful to be able to adjust the pattern of gluing in order to make the distribution of football bubbles conform to some other specified distribution (again, up to an additive error of order $\epsilon$).

This operation has an unpredictable effect on the remainder, transforming it into some new 3-valent graph (of some possibly very different combinatorial type); however, it preserves the essential features of the remainder that are known to hold at this stage of the construction: every edge of the remainder (after the operation) has length exactly $3\lambda$; and the total mass of the remainder before and after the operation is unchanged (so that it is still very small compared to the mass of any given football type).

Let $\mu$ be a probability measure on the set of all football types, with full support — i.e. so that $\mu$ is strictly positive on every football type. (In the sequel, $\mu$ will be the cube distribution described below, but that is not important at this stage.) Suppose we have three long strips of the form

so that the result of the gluing produces $(N+1)/2$ bunches each with the label $x_{i}$ (for $i$ odd), and $(N-1)/2$ football bubbles each with the label $(a_{i},b_{i},c_{i})$ (for $i$ even). We think of the $(a_{i},b_{i},c_{i})$ as unordered triples — i.e. we only think of the underlying football as an abstract graph with edge labels up to isomorphism. A given sequence of labels $x_{i}$ and (unordered!) football types $(a_{i},b_{i},c_{i})$ might arise from three long strips $s,t,u$ in $6^{(N+3)/2}$ ways, since there are 6 ways to order each triple $a_{i},b_{i},c_{i}$.

Let $\rho$ denote the uniform probability measure on football types, and let $\mu^{\prime}$ be chosen to be a multiple of $\mu$ such that $\rho>\mu^{\prime}$ for all types. Fix $\theta_{4}>0$ such that $\min\mu^{\prime}/\rho>\theta_{4}$. As the notation hints, $\theta_{4}$ will turn out to be the slack in the partially glued graph $\Lambda_{4}$, and we accordingly partition each long strip of $\Lambda_{3}$ into an inner reservoir, of proportional length $(1-\theta_{4})$, and an outer reservoir consisting of two strips of proportional length $\theta_{4}/2$.

We put all possible triples of long strips labeled as above into a bucket. Next, color each even index $i$ in $[N\theta_{4}/2,N(1-\theta_{4}/2)]$ (i.e. those contained in the inner reservoir) black with probability

and color all the remaining indices white, where $\mu^{\prime}(i)$ is short for the $\mu^{\prime}$ measure of the football type $(a_{i},b_{i},c_{i})$ (note that our choice of $\theta_{4}$ guarantees that the assigned probabilities are never greater than 1).

Now pull apart all the triples of long strips, and match them into new triples $s,t,u$ according to the following rule: if a given index $i$ is white, the corresponding labels $s_{i},t_{i},u_{i}$ should all be different, and equal to $a_{i},b_{i},c_{i}$ (in some order); if a given index $i$ is black, the corresponding labels $s_{i},t_{i},u_{i}$ should all be the same, and equal to exactly one of $a_{i},b_{i},c_{i}$. Then we can glue up $s,t,u$ to produce footballs $(a_{i},b_{i},c_{i})$ exactly for the white labels, and treat the black labels as part of the neighboring sticky segments, so that they are entirely glued up.

We do this operation for each bucket (i.e. for each collection of triples with a given sequence of sticky types $x_{i}$ and football types $(a_{i},b_{i},c_{i})$). The net effect is to eliminate a fraction of approximately $(\rho-\mu^{\prime}(i))/\rho$ of the footballs with label $i$; thus, at the end of this operation, the distribution of footballs is proportional to $\mu$, with error of order $\epsilon$.

Although this adjustment operation can achieve any desired distribution $\mu$, in practice we will set $\mu$ equal to the cube distribution, to be described in the sequel. In any case, for a fixed choice of distribution $\mu$, the slack $\theta_{4}$ only depends on $k$ and $\lambda$, and therefore can be treated as a constant.

We summarize this in the following lemma.

At this stage the remainder consists of a 3-valent graph in which every edge has length exactly $3\lambda$. The total mass of the remainder is very small compared to the mass of the reservoir, but it is large compared to the size of a single long strip. Furthermore, there is no a priori bound on the combinatorial complexity of a component of the remainder.

We explain how to modify the gluing by a certain local move called a tear¹¹1“tear” in the sense of: “There were tears in her big brown overcoat”, which (inductively) reduces the combinatorial complexity of the remainder (which a priori is arbitrarily complicated) until it consists of a disjoint collection of simple pieces. These pieces come in three kinds:

1. (1)

   football bubbles;

2. (2)

   bizenes: these are graphs with 6 edges and 4 vertices, obtained from a square by doubling two (non-adjacent) edges; and

3. (3)

   bicrowns: these are complete bipartite graphs $K_{3,3}$.

The bubbles, bizenes and bicrowns all have edges of length exactly $3\lambda$. They are depicted in Figure 4. Note that bizenes doubly cover footballs and bicrowns triply cover footballs. If the labels on a bizene or bicrown happen to be pulled back from the labels on a football bubble via the covering map, then we say that the bizene or bicrown is of covering type.

We now describe the operation of tearing. Take two copies of $\Lambda_{4}$, denoted by $\Lambda_{4}$ and $\Lambda^{\prime}_{4}$ (with vertices, edges and subgraphs of $\Lambda^{\prime}_{4}$ denoted with primes in the obvious manner) and let $p$ and $p^{\prime}$ be branch vertices of $\Delta_{4}$ and $\Delta^{\prime}_{4}$ respectively. These vertices are the ends of disjoint sequences of alternating bunched triples and football bubbles. The tearing operation uses up two (appropriately labeled) sequences of alternating bunched triples and football bubbles, each of length 3. The precise definition of this operation is best given by example, and is illustrated in Figure 5.

On the left of the figure we have the vertices $p$ and $p^{\prime}$ of $\Lambda_{4}$ and $\Lambda^{\prime}_{4}$, together with two strings of $3$ bubbles. These strings are pulled apart and reglued according to the combinatorics indicated in the figure. Thus, the labels on the bunched triples at each horizontal level should all agree, and the labels on the footballs should be such that the result of the gluing is still folded. The existence of strings of football bubbles with these properties is guaranteed by pseudorandomness and the definition of the long strips.

Thus the operation of tearing uses up 8 footballs (as in the figure), and it has several effects on the remainder. First, $\Lambda_{4}\cup\Lambda^{\prime}_{4}$ is pulled apart at $p$ and $p^{\prime}$, producing six new vertices $p_{1},p_{2},p_{3}$ and $p_{1}^{\prime},p_{2}^{\prime},p_{3}^{\prime}$, and adding two new edges $x_{i}$ and $x_{i}^{\prime}$ (from the footballs separating the adjacent bunched triples) joining $p_{i}$ to $p_{i}^{\prime}$. Second, two new bicrowns are created, assembled from the pieces of three identically labeled footballs. Third, the slack at $p$ and $p^{\prime}$ is reduced to approximately half of its previous value. If the strings of three football segments are taken from the inner half of the outer reservoir, it will reduce the slack at the vertices of $\Lambda_{4}\cup\Lambda^{\prime}_{4}$ at the end of these strips; but the size of the slack at these vertices will stay large.