aa r X i v : . [ m a t h . G R ] A p r COXETER GROUPS AND RANDOM GROUPS
DANNY CALEGARI
Abstract.
For every dimension d , there is an infinite family of convex co-compact reflection groups of isometries of hyperbolic d -space — the superideal (simplicial and cubical) reflection groups — with the property that a randomgroup at any density less than a half (or in the few relators model) containsquasiconvex subgroups commensurable with some member of the family, withoverwhelming probability. Contents
1. Introduction 12. Statements of the main theorems 23. Spines 44. Constructing spines 75. Acknowledgments 18References 181.
Introduction
The recent combined work of Wise [7], Ollivier-Wise [6] and Agol [1] shows thatrandom groups at density < / superideal hyperbolic polyhedron is obtained by taking a polyhedron P in realprojective space (of some dimension) and intersecting it with the region H boundedby a quadric (which is identified with hyperbolic space in the Klein model), so thatone obtains an infinite hyperbolic polyhedron P ∩ H . If P is a regular polyhedronand H is centered at the center of P , one obtains a regular superideal polyhedron P ∩ H , for which the symmetries of P are realized as hyperbolic isometries of P ∩ H .For judiciously chosen P , the dihedral angles of P ∩ H might be of the form 2 π/m for some integer m , and hyperbolic space can then be tiled by copies of P ∩ H . Thesymmetry group of this tiling is a superideal reflection group .In every dimension d , there are two interesting infinite families of convex cocom-pact hyperbolic reflection groups — the superideal simplex reflection group, andthe superideal cube reflection group. We denote the simplex groups by ∆( m, d ) andthe cube groups by ( m, d ), where the dihedral angles of the superideal regularpolyhedra in each case are 2 π/m . Date : August 27, 2018.
These reflection groups are Coxeter groups; for ∆( m, d ) the Coxeter diagram is r r r r r r m whereas for ( m, d ) the Coxeter diagram is r r r r r r m d + 1 nodes.The main result we prove in this paper is that for any d , a random finitelypresented group at any density less than a half (or in the few relators model)contains quasiconvex subgroups which are commensurable with some ∆( m, d ) forsome large m , and with some ( m, d ) for some large (possibly different) m , withoverwhelming probability.The two cases are treated in a similar way, but the combinatorial details of theargument are different in each case. The precise statements of our main theoremsare the Superideal Simplex Theorem and the Superideal Cube Theorem, stated in §
2. The proof of these theorems is carried out in § § Statements of the main theorems
We first discuss the superideal simplex groups ∆( m, d ). Example . Some low-dimensional examples are familiar: • In any dimension d , taking m = 3 gives the finite Coxeter group A d +1 ,which is just the symmetric group S d +2 . Similarly, in any dimension d ,taking m = 4 gives the finite Coxeter group BC d +1 , the symmetry groupof the ( d + 1)-dimensional cube (or equivalently of the ( d + 1)-dimensionalcross-polytope), known as the hyperoctahedral group . • In dimension d = 2 there is an (ordinary) hyperbolic simplex with angles2 π/m whenever m ≥
7. The groups ∆( m,
2) are all commensurable, andare commensurable with the fundamental groups of all closed surfaces ofgenus at least 2. • In dimension d = 3 the ideal regular simplex has m = 6; the group ∆(6 ,
3) iscommensurable with the fundamental group of the complement of the figure8 knot. For m ≥ m → ∞ the limit sets converge to the Apollonian gasket, withHausdorff dimension approximately 1 . • In dimension d ≥ m ≥ d , a “generic” finitelypresented group will contain some subgroup isomorphic to a finite index subgroupof some element of the ∆( m, d ) family. Explicitly, we show: Superideal Simplex Theorem.
For any fixed d , a random group at any density < / or in the few relators model contains (with overwhelming probability) a subgroup OXETER GROUPS AND RANDOM GROUPS 3 commensurable with the Coxeter group ∆( m, d ) for some m ≥ , where ∆( m, d ) isthe superideal simplex group with Coxeter diagram r r r r r r m where there are d + 1 nodes.Example . Some special cases of this theorem were already known: • Taking d = 2, this theorem implies that a random group contains surfacesubgroups (with overwhelming probability). This is the main theorem ofCalegari-Walker in the paper [2]. • Taking d = 3, this theorem implies that a random group contains a sub-group isomorphic to the fundamental group of a hyperbolic 3-manifold withtotally geodesic boundary. This is the main theorem of Calegari-Wilton inthe paper [3]. In fact, the Commensurability Theorem proved in that paper,is exactly the statement of our main theorem in the case d = 3.We now discuss the superideal cube groups ( m, d ). Example . The low-dimensonal examples are again familiar:(1) In any dimension d , taking m = 3 gives the finite coxeter group BC d +1 ,the hyperoctahedral group. Taking m = 4 gives the hypercubic honeycomb ˜ C d +1 , which is virtually abelian, and acts cocompactly on Euclidean space.(2) In dimension d = 2 the group ( m,
2) is the symmetry group of the tilingby regular right-angled m -gons, whenever m ≥
5. These groups are allcommensurable with each other, and with the fundamental groups of allclosed surfaces of genus at least 2 (and with all ∆( m ′ , d = 3 the group (5 ,
3) is the symmetry group of the tiling byregular right-angled dodecahedra; this is commensurable with the (orbifold)fundamental group of the orbifold with underlying space the 3-sphere, andcone angle π singularities along the components of the Borromean rings,as made famous by Thurston. For m ≥ m,
3) are convexcocompact but not cocompact. Similarly, in dimension d = 4 the group(5 ,
4) is the symmetry group of the tiling by regular right-angled 120-cells, whereas ( m,
4) is convex cocompact but not cocompact for m ≥ d ≥ m, d ) are never cocompact when m ≥ Superideal Cube Theorem.
For any fixed d , a random group at any density < / or in the few relators model contains (with overwhelming probability) a subgroupcommensurable with the Coxeter group ( m, d ) for some m ≥ , where ( m, d ) isthe superideal cube group with Coxeter diagram r r r r r r m where there are d + 1 nodes. The case d = 2 reduces to the existence of closed surface subgroups, proved byCalegari-Walker in [2], but all other cases are new.We prove these theorems under the following two simplifying assumptions:(1) that the length n of the relations is divisible by a finite list of specificintegers (implicitly depending on the density D ); and DANNY CALEGARI (2) that the number k of free generators is sufficiently large depending on d ineither case; for the Superideal Simplex Theorem we assume 2 k − ≥ d + 1,whereas for the the Superideal Cube Theorem we assume 2 k − ≥ d + 1.These assumptions simplify the combinatorics, but they are superfluous and at theend of § Spines
The proof of our main theorems is entirely direct and constructive; it depends onbuilding certain 2-dimensional complexes subject to local and global combinatorialconstraints from pieces which correspond to relators in the random group (in aprecise way). In this section we describe these combinatorial constraints.3.1.
Sets as modules.
We frequently need to discuss finite sets together with anaction of a finite group. If a finite group H acts on a set A we say that A is an H -module . The cases of interest in this paper are: • the symmetric group S d acting on a d -element set by the standard permu-tation action; and • the hyperoctahedral group BC d acting on a 2 d -element set of d pairs by thestandard action, which permutes the pairs and acts on each pair as Z / Z .By abuse of notation, we refer to a set together with such structure as an S d -moduleor BC d -module respectively. If A is a 2 d element set with a BC d -module structure,and a is an element of A , the other element of the pair containing a is said to be antipodal to a .3.2. Graphs and 2-complexes.
Let G be a random group with k generators atdensity D < / n . That means G is the group defined by a presentation G := h x , · · · , x k | r , r , · · · , r ℓ i where ℓ = (2 k − nD , and where each r i is chosen randomly and independentlyfrom the set of all (cyclically) reduced words in the free group F k := h x , · · · , x k i of length n , with the uniform distribution. For an introduction to random groupssee [4] or [5].We let r = r , and let G r denote the random 1-relator group with presentation G r := h x , · · · , x k | r i The group G is the fundamental group of a 2-complex K , with one vertex, withone edge for each generator, and with one 2-cell for each relator. We denote the1-skeleton of K by X ; thus X is a k -fold rose. The group G r is the fundamentalgroup of a subcomplex K r , whose 1-skeleton is equal to X , and which is containsonly one 2-cell, the cell attached by the relation r = r . Definition 3.1. A graph over X is a graph Σ together with a simplicial immersionfrom Σ into X .A morphism over X between oriented graphs Σ → X , Σ ′ → X over X is asimplicial immersion Σ → Σ ′ such that the composition Σ → Σ ′ → X is the givenimmersion from Σ → X . OXETER GROUPS AND RANDOM GROUPS 5
If Σ is a graph over X , the oriented edges of Σ can be labeled by the x i and theirinverses, by pulling back the labels from the edges of X . Conversely, a graph withoriented edges labeled by the x i and their inverses, has the structure of a graphover X providing no adjacent oriented edges have inverse labels.Let L be a finite union of simplicial circles (i.e. circles subdivided into edges)with oriented edges labeled by generators or their inverses, in such a way that thecyclic word on each component of L is r . Then L is a graph over X .We will build a simplicial graph Σ over X together with a morphism L → Σ over X satisfying certain conditions. These conditions are different for the SuperidealSimplex Theorem and for the Superideal Cube Theorem, but they have commonfeatures, which we now describe. Definition 3.2.
An ( m, d ) -regular simplicial spine or ( m, d ) -regular cubical spine is a graph Σ over X , together with a morphism L → Σ over X with the followingproperties:(1) The graph Σ is obtained by subdividing a regular graph. That is, all thevertices are either 2-valent (we call these internal ) or of the same fixedvalence > genuine ). In the simplex case, each genuinevertex is d + 1-valent; in the cube case, each genuine vertex is 2 d -valent andfurther admits the structure of a BC d -module. By abuse of notation, weomit the adjective “genuine” when discussing higher valence vertices unlessthe meaning would be ambiguous.(2) The map L → Σ is d to 1 in the simplex case, and 2( d −
1) to 1 in the cubecase; i.e. each edge of Σ is the preimage of exactly d (resp. 2( d − L . Furthermore, the set of preimages of each edge has the structure ofan S d (resp. BC d − )-module.(3) For each vertex v of Σ and each pair of distinct incident edges e, e ′ which arenot antipodal in the cube case, there is exactly one segment of L of length2 whose midpoint maps to v and whose adjacent edges map to e and e ′ . Inthe cubical case, if L ( e ) is the set of 2( d −
1) edges of L mapping to e withits BC d − -module structure, then the 2( d −
1) edges in L adjacent to L ( e )map bijectively to the 2( d −
1) edges of Σ not equal to e or its antipode; thislatter set also has a natural BC d − -module structure by restriction, and werequire that this map of BC d − -modules respect the module structure; seeFigure 1 for the local picture in the simplicial case and Figure 2 for thecubical case in dimensions 2 and 3.(4) Each component of L has exactly m vertices which map to genuine verticesof Σ.(5) Σ satisfies the cocycle condition (to be defined shortly).Let L → Σ be a spine satisfying the first three conditions to be ( m, d )-regular.The mapping cylinder M is a 2-complex which admits a canonical local embeddinginto R d in such a way that the (combinatorial) symmetries of the link of eachvertex or edge are realized by isometries of the ambient embedding (taking modulestructure into account). Taking a tubular neighborhood of this canonical thickeninggives rise to a D d − -bundle over M with a flat connection with holonomy in thesymmetric group S d − in the simplicial case, or BC d − in the cubical case, acting bythe standard representation. The restriction of this bundle to each component of L therefore determines a conjugacy class in S d − or BC d − respectively. The cocycle DANNY CALEGARI
Figure 1. local structure of simplicial spine near vertex in dimen-sions 2 and 3 condition is the condition that this conjugacy class is trivial, for each componentof L .Let L → Σ be an ( m, d )-regular spine. Form the mapping cylinder M and thenlet M be the 2-complex obtained from M by attaching a disk to each component of L . Since each component of L has the label r , the tautological immersion L → X extends to a cellular immersion M → K r . Figure 2. local structure of cubical spine near vertex in dimen-sions 2 and 3The defining properties of an ( m, d )-regular simplicial (resp. cubical) spine en-sure that the 2-complex M is locally isomorphic to the 2-dimensional spine S whichis dual to the tiling of d -dimensional hyperbolic space by regular superideal sim-plices (resp. cubes) with dihedral angles π/m . To see this, observe that the vertexstabilizers in the symmetry group of S are isomorphic to S d (resp. BC d − ) and theiraction on vertex links of S agrees with the module structure on the vertex links of OXETER GROUPS AND RANDOM GROUPS 7 M . There is therefore a developing map from the universal cover of M to S which isan immersion, and therefore an isomorphism. This realizes the fundamental groupof M as a subgroup of ∆( m, d ) (resp. ( m, d )) which is of finite index, since M iscompact.We will show in the next section that for r a sufficiently long random word, wecan construct an ( m, d )-regular spine (of either kind) in which all the maximal pathsin Σ containing only internal vertices in their interior (informally, the “topologicaledges” of Σ) are bigger than any specified constant λ . This is the main ingredientnecessary to show that the immersion M → K r → K is π -injective, and staysinjective when a further (2 k − nD − Constructing spines
We start with L , a finite union of simplicial circles each labeled with the relator r . The spine Σ is obtained from L by identifying oriented edges of L in groups of d or 2( d − L → Σ i limiting to Σ, where each L → Σ i is a morphism over X . Thus, each intermediateΣ i should immerse in X . We call an identification of subgraphs of Σ i − with thesame labels, giving rise to a quotient Σ i − → Σ i over X a legal quotient if Σ i isimmersed in X . We assume in the sequel that our quotients are always legal.In our intermediate quotients Σ i , some edges are in the image of d (resp. 2( d − L , and some are in the image of a unique edge of L . We call edgesof the first kind glued , and edges of the second kind free .In order to simplify notation, in the sequel we let d denote either d in the sim-plicial case, or 2( d −
1) in the cubical case.We choose a big number λ divisible by d , and an even number N with N ≫ λ ,and assume for the sake of convenience that λ · N divides n , the length of the relator r (the size of λ that we need will ultimately depend only on the density D , butnot on the length n of the relators). Then we subdivide each component of L intoconsecutive segments of length λ .4.1. Creating beachballs. A block in L is a sequence of N consecutive segments.Since we assume that λ · N divides n , each component of L can be subdivided into n/ ( λ · N ) consecutive blocks. Each block is made up of alternating odd segments and even segments ; by abuse of notation we refer to the segment immediately beforethe first segment in a block (which is contained in the previous block) as the “firsteven segment”. We say that a d -tuple of blocks is compatible if their odd segmentscan all be identified in groups of d (in the order in which they appear in the blocks)in such a way that the resulting quotient is legal. This means that each d -tupleof even segments at the same location in the d blocks must start and end with different letters, and the same must be true for the last d -tuple of even segments inthe blocks immediately preceding the given collection of blocks. The existence of acompatible collection of blocks requires 2 k − ≥ d .When d blocks are glued together along their odd segments, each collectionof even segments except for the first and last one is identified at their vertices,producing a beachball — a graph with two vertices and d edges, each joining onevertex to the other. See Figure 3. DANNY CALEGARI
Figure 3.
A beachball for d = 9We say that a ( d + 1)-tuple of blocks is supercompatible if each sub d -tuple iscompatible. Evidently, the existence of supercompatible tuples requires 2 k − ≥ d + 1, which is our second simplifying assumption.Recall that d = d in the simplicial case, and d = 2( d −
1) in the cubical case. Sup-pose r is a random reduced word of length n . Since r is random, if n is sufficientlylong it is possible to partition a proportion (1 − ǫ ) of the blocks into compatible d -tuples, and glue their odd segments, for ǫ depending only on n , and therefore1 /ǫ ≫ N, λ , with probability 1 − O ( e − n c ). This leaves a proportion ǫ of the blocksleft unglued. For each of these unglued blocks, find a collection of d glued blocksso that the d + 1 tuple is supercompatible. If we unglue the collection of d gluedblocks, and then take d + 1 copies of our graph, we can glue the resulting d ( d + 1)blocks in compatible groups of d .In the cubical case, in order to give each collection of edges the natural structureof a BC d − -module, it is convenient to first take some finite number of disjointcopies of L and partition them into 2( d −
1) subsets of equal size. The group BC d − acts on each set of 2( d −
1) copies of L as a standard representation, givingthese components the structure of a BC d − -module. When we partition blocks into d -tuples (where d = 2( d −
1) in the cubical case), each tuple should contain exactlyone block from each subset, so that the elements of the block inherit a natural BC d − action.Thus at the end of this step, we have constructed Σ which consists of collectionsof glued edges of length λ (each the image of d edges of L ), a reservoir of beachballswith every edge of length λ and all possible edge labels in almost uniform distri-bution of possible edge labels, and a remainder consisting of a d -valent graph inwhich every edge has length λ . Note that the remainder is obtained as the unionof the first and last (even) segments from each block, so the mass of the remainderis of order O (1 /N ), where by assumption we have chosen N ≫ λ . Furthermore inthe cubical case, for each component X of the reservoir or remainder, the edges of X incident to each vertex inherit a natural BC d − -module structure. OXETER GROUPS AND RANDOM GROUPS 9
Covering move.
The next move is called the covering move , and its effectis to undo some of the gluing of some previous step, and then to reglue in sucha way that the net effect is to transform some collection of beachballs into morecomplicated graphs, each of which is a finite cover of a beachball, with prescribedtopology and edge labels. This move does not necessarily preserve the total massof beachballs, and can be used to adjust the total mass of beachballs of any specifickind, at the cost of transforming some other (prescribed) set of beachballs intocovers.To describe this move, we first take some consecutive strings of beachballs andarrange them in a “matrix” form, where each string of beachballs is a column ofthe matrix, and the different strings make up the rows of the matrix. See Figure 4for a matrix consisting of 7 columns of 3 consecutive beachballs.
Figure 4.
A matrix of 7 columns of 3 consecutive beachballs. Inthis and subsequent figures, glued edges are in red and free edgesare in black.The covering move takes as input a matrix of s columns of r consecutive beach-balls with compatible labels on corresponding glued segments. This means thefollowing. In each column there are r + 1 consecutive glued segments interspersedwith the r beachballs. Thus between each pair of consecutive rows of beachballsthere is a row of glued segments, each consisting of d glued edges with some label.We require that all the labels on glued edges in the same row are equal.In the cubical case, the edges in each column have a natural structure of a BC d − -module. Thus it is possible to “trivialize” the module structure on theunion of edges in each row, giving them the structure of a product of a standard BC d − -module with an s -element set. The covering move pulls apart each interior row of glued segments (i.e. the rowsbetween a pair of consecutive row of beachballs) into d s edges, all with the samelabel, then permutes them into s new sets of d elements and reglues them. Morespecifically, if we identify the set of d s edges with the product { , · · · , s }×{ , · · · , d } where BC d − acts on the second factor, we permute each { , · · · , s } × i factor bysome element of S s . Thus the move is described on each of the rows by an element p j ∈ S d s , the product of d copies of the symmetric group S s , for 0 ≤ j ≤ r where p and p r are the identity permutation.For each 1 ≤ j ≤ r the s beachballs in the j th row are rearranged by this moveinto a graph which is a (possibly disconnected) covering space of a beachball ofdegree s ; which covering space is determined by the permutations p j − and p j . If p j − = p j it is a trivial covering space, in the sense that it consists just of s disjointbeachballs. However, even if it is trivial as a covering space, if p j − and p j are notthe identity, the edges making up each beachball might be different from the edgesmaking up the beachballs before the move. We allow two possibilities for the labelson the new beachball covers in each row:(1) the beachball cover in the given row can have arbitrary topology, and thelabels on the edges are legal; or(2) the beachball cover in the given row is trivial (i.e. p j − = p j ) and on eachbeachball the edges all have the same label.In the former case we do nothing else to the row. In the latter case, since allthe edge labels are equal (and the gluing is not legal as it stands), we collapse thebeachball to the interior of a glued segment of length 3 λ . So the net effect of thecovering move is to transform some set of beachballs into covering spaces of thesame mass, while possibly eliminating some set of beachballs of comparable mass.Two kinds of beachball covering spaces are especially important in what follows;these are • a 2-fold cover of a beachball called a barrel ; and • a d -fold cover of a beachball called a bipart , whose underlying graph is thecomplete bipartite graph K d , d .See Figure 5. Figure 5.
A barrel and a bipart for d = 4We now describe two especially useful examples of covering moves which will beused in the sequel. OXETER GROUPS AND RANDOM GROUPS 11
Example . The elimination move trades 2 d beachballs for 2 bi-parts, and eliminates a collection of d beachballs with the same labels. It is con-venient to perform this move in such a way that the biparts created have labels of covering type . This means that the edge labels on the bipart are pulled back fromsome legal edge labeling on a beachball under the covering projection. See Figure 6. Figure 6.
The elimination move for d = 4 Example . The rolling move trades 4 beachballs for 2 barrels.This move is self-explanatory. See Figure 7.
Figure 7.
The rolling move for d = 44.3. Tearing the remainder.
We now describe a modification of the gluing whichreplaces the remainder with a new remainder of similar mass, but composed entirelyof barrels. In the process, a set of beachballs of similar mass are transformed intobiparts.
The tear move is applied to two vertices v, v ′ in a component Y of the remainder.It uses up the two beachballs in the reservoir which are adjacent to v and v ′ , anda further 3( d −
1) beachballs drawn from the middle of the reservoir in consecutivegroups of three. The result of the move “tears” Y apart at v and v ′ , producing d copies of each vertex v i , v ′ i for 1 ≤ i ≤ d , each v i joined to v ′ i by d − d ( d −
1) edges coming from ( d −
1) of the beachballs), and transformsthe remaining 2 d beachballs into two biparts. In the process, 2 d glued edges areunglued, and then reglued in a different configuration; thus it is necessary for thelabels on each set of d edges to agree. The tear move is illustrated in Figure 8; asin § BC d − -module structure on edges in the cubical case. Figure 8.
The tear move for d = 4.Applying a tear move at a pair of vertices v, v ′ in Y creates a collection of half-barrels i.e. a beachball with one edge replaced by a pair of free edges joined tonearby free vertices ; see Figure 9. Figure 9.
A half-barrel for d = 4.We refer to the non-free edges in a half-barrel as barrel edges , and observe that thenumber of non-barrel edges in Y is unchanged by a tear move. Thus after applyingthe tear move once to create several half-barrels, we can apply the tear move againto the pair of free vertices at the end of one of the half-barrels, in the process OXETER GROUPS AND RANDOM GROUPS 13 creating an honest barrel which may be transferred to the reservoir, reducing thetotal number of non-barrel edges in Y . We can use total number of non-barreledges of Y as a measure of complexity, and observe that after applying O (1 /N )tear moves, we can tranform the entire remainder to a mass of O (1 /N ) barrelsand biparts, while the reservoir still contains almost exactly the same number ofbeachballs in almost exactly the uniform distribution.4.4. Hypercube gluing.
At this stage of the construction we have a reservoirconsisting of an almost equidistributed collection of beachballs, and a relativelysmall mass of pieces consisting of biparts and barrels. In this section and thenext we explain how to glue up beachballs. There are two different (but related)constructions depending on whether we are in the simplicial or cubical case. In thesimplicial case we use the hypercube gluing .A finite cover of the hypercube gluing can be used to glue up all the barrelsand biparts produced by the tear moves, together with some other complementarycollection of pieces produced by the covering move. The existence of such a com-binatorial cover in which the pieces from the remainder can be included follows byLERF for free groups, and it is straightforward to use the covering move to producecomplementary pieces of the correct topology and labels. An explicit descriptionof the combinatorics of the necessary covers in dimension 3 is given in [3], § Figure 10.
Beachball immersed in the 1-skeleton of a cube indimensions 2, 3 and 4.It remains to describe the hypercube gluing (recall that we are in the simplicialcase). Let C denote the 1-skeleton of the d -dimensional cube, and with each edge oflength λ/d . Let e , · · · , e d denote vectors of length λ/d aligned positively along the d coordinate axes in order. There is a path of length λ from the vertex (0 , , · · · ,
0) tothe vertex ( λ/d, λ/d, · · · , λ/d ) of C , obtained by concatenating straight segments oflength λ/d in the order e , e , · · · , e d . Taking d cyclic permutations of this sequenceof paths gives d paths of length λ between extremal vertices of C , whose union isa beachball. This is illustrated in Figure 10 in dimensions 2, 3 and 4.There are 2 d − pairs of extremal vertices of C , and the union of 2 d − suitablylabeled beachballs can be arranged along the 1-skeleton of C as above. Gluing the beachballs in this pattern is legal, and the resulting graph locally satisfies the firstthree conditions to be a ( m, d )-regular simplicial spine.4.5. Lens gluing.
The analog of the hypercube gluing for cubical spines is the lens gluing . This is the most complicated move in our construction, and we firstdescribe this move in low dimensions before giving the definition in generality.In dimension 2 a beachball has degree 2 (i.e. it has 2 edges); BC = Z / Z actsas the full permutation group. Two beachballs with suitable labels can be identifiedalong their boundaries, laid out along a circle; see Figure 11. Figure 11.
Lens gluing d = 2In dimension 3 a beachball has degree 4; BC = D acts by the dihedral group,preserving or reversing a circular order on the 4 edges. Eight beachballs withsuitable labels can be glued up along a K , graph in S , where each edge of eachbeachball runs along a segment of length 2 in the K , graph, as in Figure 12 (whereone of the vertices is at “infinity” in the figure):The lens gluing in dimension d is closely related to the hypercube gluing indimension d −
1. Recall the definition of the hypercube gluing move from § d − beachballs of degree d − d − d − C denote the1-skeleton of the cube, and denote the immersion of a beachball b into C by b → C .For any graph L , the spherical graph associated to L is the graph with one S (i.e. two vertices) for each vertex in L , and one S ∗ S for each edge in L . If v is a vertex of L , we denote the corresponding pair of vertices in S ( L ) by v v
1. Similarly, if e is an edge of L , we denote the corresponding edges in S ( L ) by e e e
10 and e
11 (in fact this construction generalizes in an obvious way tosimplicial complexes of arbitrary dimension). There is a canonical simplicial mapfrom S ( L ) → L which forgets the 01 labels.Now we apply this construction to C ; the lens gluing will map 4 × d − beach-balls of degree 2( d −
1) into S ( C ) in such a way that the projection to C will mapeach 4 beachballs onto each degree ( d −
1) beachball mapping into C in the hyper-cube gluing, with each edge of the beachball downstairs in the image of two edgesupstairs.If b → C is a beachball, and b ′ → S ( C ) is one of the four beachballs upstairsmapping to it, each edge of b maps to a path in C of length d −
1, which is covered
OXETER GROUPS AND RANDOM GROUPS 15
Figure 12.
Lens gluing d = 3by two paths in S ( C ) of length d −
1. Each such path is determined (given its imagein C ) by a word of length d in the alphabet 0 ,
1. Our beachballs upstairs have theproperty that the pair of lifts of each edge of b are described by the same pair of 01words. Moreover, the collection of 4 pairs of 01 words describing the lifts of b arethe same for all b → C . So to describe the lens move we just need to give 4 sets ofpairs of 01 words of length d . We call these height pairs . An explicit formula for d > (00 d − , d − d − , d − d − x, d − x )(1(1001) d − (1 − x ) , d − (1 − x ))where x is 0 or 1 depending on the parity of d , and where an expression like w p/q for w a word of length q means the initial string of length p of the word w ∞ := · · · .The meaning of this formula is best explained by a picture; see Figure 13.The case d = 2 is degenerate; the case d = 3 gives rise to the height pairs(000 , , (001 , , (111 , , (110 , d = 3 describedabove (the other four beachballs are in the preimage of the other beachball in thehypercube gluing). Figure 13.
Height graphs for d = 9Note that there is an ambiguity in our choice of labels of each pair of verticesof S ( C ) over v by v v
1; thus the group Z / Z vert( C ) acts by automorphismsof S ( C ) over C . Thus although this is not evident in the formulae, under thissymmetry group all four height pairs are in the same orbit.The case d = 4 is hard to draw without the diagram becoming cluttered; Fig-ure 14 shows the 4 beachballs of degree 6 in the d = 4 lens gluing which project toone beachball of degree 3 in the d = 3 hypercube gluing. Figure 14.
Four beachballs in the d = 4 lens gluing which projectto one beachball in the d = 3 hypercube gluing.4.6. Regularity.
The third condition for ( m, d )-regularity is that each componentof L has exactly m vertices which map to high-valent vertices of Σ. The only movewhich adjusts the number of such vertices on each component of L is the eliminationmove. This move takes pieces which may be drawn from any component L ; so wesimply take enough disjoint copies of L , and symmetrize the components from whichthe moves are drawn, so that we apply this move the same number of times to everycomponent. This will ensure the third condition for ( m, d )-regularity.4.7. Cocycle condition.
The cocycle is sensitive to more combinatorial data thanwe have been using so far; it depends not only on the set of labels appearing as theedges in a beachball, but the way in which these labels come from consecutive evensegments on each of d blocks in a compatible d -tuple. Gluing different compatible d -tuples gives rise to the same beachball collections but with different cocycles(relative to a trivialization along the odd segments of the blocks). If the originalgluing was done randomly, there are pairs of compatible segments whose gluingsgive rise to the same set of beachballs, but with cocycles differing by any elementin the symmetric group S d − in the simplicial case, or any element of BC d − inthe cubical case; adjusting the gluing by interchanging elements of these pairs gives OXETER GROUPS AND RANDOM GROUPS 17 the same collection of beachballs and the same remainder (and therefore the samegluing problem) but with the cocycle adjusted in the desired away. Boundedlymany moves of this kind for each component produces an ( m, d )-regular simplicialor cubical spine with all topological edges of length at least λ .4.8. Few generators.
We now indicate how to modify our arguments and con-structions so that they hold without our simplifying assumptions.The first assumption — that the length n of the relators is divisible by some bigfixed constant λ · N (where also d divides λ ) — is easy to dispense with. We let ρ be the remainder when dividing n by λ · N , then find d copies of a subword of r of length λ · N + ρ so that it is legal to glue these d words in d distinct copies of r ; what is left after this step is a graph with edges whose lengths are divisible by λ · N , and the rest of the construction can go through as before.The second assumption — that the number k of free generators satisfies 2 k − ≥ d + 1 — can be finessed by looking more carefully at the construction. It can beverified that we do not really use the hypothesis that the word r is random withrespect to the uniform measure on reduced words of length n ; instead we can makedo with a much a weaker hypothesis, namely that we generate random words by any stationary ergodic Markov process, whose subword distribution of any fixedlength has enough symmetry, and such that there are at least d + 1 letters that mayfollow any legal substring.The relevance of this is as follows: a random reduced word of length n in a freegroup of rank k can be thought of as a random reduced word of length n/s in a freegroup of rank k (2 k − s − (whose symmetric generating set is the set of reducedwords in F k of length s ) generated by a certain stationary ergodic Markov process.We can perform matching of subwords and build spines in this new “alphabet”; theresult will not immerse in X , since there might be folding of subtrees of diameterat most 2 s . But the spine will still have the crucial properties that the fundamentalgroup of the 2-complex M is commensurable with ∆( m, d ), and that the numberof immersed paths of length ν grows like (2 k − sν/λ where λ is as big as we like.Taking s big enough to “simulate” a group of rank at least d + 1, we can then take λ much bigger so that s/λ is arbitrarily small. This small exponential growth rateof subpaths is the key to proving that π ( M ) maps injectively with quasiconveximage.4.9. Conclusion of the theorem.
The remainder of the argument is almost in-dentical to the arguments in [2] and [3], and makes use in the same way of themesoscopic small cancellation theory for random groups developed by Ollivier [5].It depends on two ingredients:(1) a bead decomposition , exactly analogous to that in [2], § §
4; and(2) a mesoscopic small cancellation argument , which applies techniques of Ol-livier as in [2], § § m, d )-regular simplicial or cubical spine Σ whichis made up of O ( n δ ) pieces of size O ( n − δ ) arranged in a circle, with each pieceseparated from the next by a neck (an unusually long glued segment) of length C log n . We can construct a spine as in [3], § constant δ and write r as a product r = r s r s · · · r m s m where each r i has length approximately n − δ and each s i has length approximately n δ , and we choose lengths so that m is divisible by d . We then fix a small positiveconstant C < δ/ log(2 k −
1) and look for a common subword x of length C log n in s i , s i + m/ d , · · · , s i +( d − m/ d with indices taken mod m , and then glue these distinctcopies of x into unusually long segments (called lips ) of what will become the spineΣ. The lips partition the remainder of L into subsets b i each with mass O ( n − δ ),and we can extend the partial gluing along the lips into an ( m, d )-regular simplicialor cubical spine by gluing intervals in each b i independently. As in [2], § § β , any immersedsegment γ → Σ with length βn whose image in X lifts to r or r − , already lifts to L (i.e. it appears in the boundary of a disk of M ). From this and the fact thata random 1-relator group is C ′ ( µ ) for any positive µ it follows that M → K r is π -injective and its image is quasiconvex.Finally, the fact that the valence of Σ is uniformly bounded and the length ofevery segment is at least λ where we may choose λ as big as we like, implies that M → K r stays π -injective and quasiconvex when we attach the disks correspondingto the remaining (2 k − nD − § § Acknowledgments
Danny Calegari was supported by NSF grant DMS 1405466. The L A TEX macrofor the symbol was copied from pmtmacros20.sty by F. J. Yndur´ain. I wouldlike to thank the anonymous referee for their helpful comments.
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Department of Mathematics, University of Chicago, Chicago, Illinois, 60637
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