LLine Patterns in Free Groups
CHRISTOPHER H. CASHEN AND NATAˇSA MACURA
Abstract.
We study line patterns in a free group by considering the topologyof the decomposition space, a quotient of the boundary at infinity of the freegroup related to the line pattern. We show that the group of quasi-isometriespreserving a line pattern in a free group acts by isometries on a related spaceif and only if there are no cut pairs in the decomposition space. Introduction
Given a finitely generated free group F of rank greater than one and a word w ∈ F , the w -line at g ∈ F is the set of elements { gw m } m ∈ Z . Up to translation andcoarse equivalence, we may assume that w is cyclically reduced and not a power ofanother element. A Cayley graph with respect to a free basis of F is a geometricmodel for F that is a tree, and in this case there is a unique geodesic in the treethat contains the vertices { gw m } m ∈ Z .The w -line at g is the same as the w line at h if and only if ¯ hg is a power of w ;the w -lines are the cosets of (cid:104) w (cid:105) in F .The line pattern generated by w is the collection of distinct w -lines. Similarly, ifwe take finitely many words w , as above, the line pattern generated by the collectionis the union of the patterns generated by the individual words. We will denote theline pattern L when we do not wish to specify generators.The main question is: Question 1.
Let F and F (cid:48) be finite rank free groups, possibly of different ranks.Consider collections of words { w , . . . , w m } ⊂ F and { w (cid:48) , . . . , w (cid:48) n } ⊂ F (cid:48) . Let L bethe line pattern in F generated by { w , . . . , w m } , and let L (cid:48) be defined similarly for F (cid:48) .Is there a quasi-isometry φ : F → F (cid:48) that preserves the patterns, in the sensethat there is some constant C so that for every line l ∈ L there is an l (cid:48) ∈ L (cid:48) suchthat the Hausdorff distance between φ ( l ) and l (cid:48) is at most C , and vice versa? A closely related question is:
Question 2.
Let F be a free group and L a line pattern in F . What is the group QI ( F, L ) of quasi-isometries of F that preserve the line pattern L ? In a pair of 1936 papers [19, 20], J. H. C. Whitehead gave an algorithm to answerthe following question:Given two finite (ordered) lists of words ( w , . . . , w k ) and ( w (cid:48) , . . . w (cid:48) k ) in a finiterank free group F , is there an automorphism φ of F such that for all i , φ ( w i ) = w (cid:48) i ? Mathematics Subject Classification.
Key words and phrases. free group, quasi-isometry, rigidity, line pattern, Whitehead graph,Whitehead’s Algorithm. a r X i v : . [ m a t h . G R ] J u l CHRISTOPHER H. CASHEN AND NATAˇSA MACURA
Questions 1 and 2 may be viewed as geometric versions of Whitehead’s question.To motivate the statement of our results, it is instructive to consider line patternsin a different setting.Line patterns in H n for n ≥ M be a compact hyperbolic orbifoldof dimension n ≥
3. Pick any collection of closed geodesics in M . The lifts of thesegeodesics to the universal cover H n are a line pattern; call it L . Theorem. [16, Theorem 1.1] QI ( H n , L ) ⊂ Isom( H n )This is an example of what we will call pattern rigidity . The hyperbolic orbifoldcase is special in that there is a canonical geometric model, H n , for π M . Forgettingthis for a moment, let Y be any geometric model for π M . For example, Y could bea Cayley graph of π M . We still get a line pattern L in Y , but it is not necessarilytrue that QI ( Y, L ) ⊂ Isom( Y ). However, there is a quasi-isometry φ : Y → H n .Each line in L gets sent to a line in H n , so we get a line pattern φ ( L ) in H n . Wehave: φ QI ( Y, L ) φ − = QI ( H n , φ ( L )) ⊂ Isom( H n )In the free group situation we do not have a canonical space to take the placeof H n that works for every line pattern. For a given line pattern we will try toconstruct a space X and a quasi-isometry φ : F → X such that pattern preservingquasi-isometries are conjugate into the isometry group of X : φ QI ( F, L ) φ − = QI ( X, φ ( L )) ⊂ Isom( X )A priori this would only give a quasi-action of QI ( F, L ) on X by maps boundeddistance from isometries. We actually prove something stronger. We will say a linepattern L in F is rigid if there is a space X , a quasi-isometry φ : F → X , andan isometric action of QI ( F, L ) on X that agrees with conjugation by φ , up tobounded distance.It is easy to see that not all patterns are rigid. A necessary condition is thatthe multiplicative quasi-isometry constants of QI ( F, L ) are bounded. Suppose L is contained in a proper free factor F (cid:48) of F , so that F = F (cid:48) ∗ F (cid:48)(cid:48) . Then QI ( F (cid:48)(cid:48) ) ⊂QI ( F, L ) contains a sequence of quasi-isometries with unbounded constants, so thepattern is not rigid.Another example where the lack of rigidity is apparent for algebraic reasons isthe pattern generated by the word ab ¯ a ¯ b in F = (cid:104) a, b (cid:105) . The automorphism groupof F preserves this line pattern, so again we have a sequence of pattern preservingquasi-isometries with unbounded constants.However, algebraic considerations do not fully determine which patterns arerigid. Consider the pattern in F generated by ab and a ¯ b . There is only a finite groupof outer automorphisms of F that preserve this pattern, so all pattern preservingautomorphisms are isometries, up to bounded distance. We might guess the patternis rigid, but in fact it is quasi-isometrically equivalent to the ab ¯ a ¯ b pattern, seeTheorem 6.2.Our main result shows that sufficiently complicated patterns are rigid. To makethis precise, we use a topological space that is a quotient of the boundary at infinityof a tree for F . This space is called the decomposition space associated to the linepattern. ine Patterns in Free Groups 3 Main Theorem.
Let L be a line pattern in a finitely generated, non-abelian freegroup, F . The following are equivalent: (1) The line pattern is rigid. (2)
The decomposition space has no cut pairs.Remark.
We use the phrase “has no cut pairs” inclusively to mean that the spaceis connected, has no cut points and no cut pairs.In Section 5.1 we show that when the decomposition space has cut pairs there isa sequence of pattern preserving quasi-isometries with unbounded quasi-isometryconstants, so the pattern is not rigid. We also show in this case that F is not finiteindex in QI ( F, L ).In the examples above, the pattern that is contained in a proper free factor wouldhave a disconnected decomposition space. For the other two, the decompositionspace is a circle.Determining if the decomposition space is connected is essentially Whitehead’sAlgorithm, which is discussed in Section 3. The idea is to build a graph, theWhitehead graph, associated to the line pattern. Connectivity of this graph isrelated to connectivity of the decomposition space, see Theorem 4.1.In Section 4 we use generalizations of the Whitehead graph to identify finite cutsets in the decomposition space. In particular, Theorem 4.15 allows us to tell ifthere are cut pairs in the decomposition space.The proof of the rigidity part of the theorem in Section 5 is similar in philosophyto the various geometric proofs of Stallings’ Theorem, see Dunwoody [4], Gromov[5], Niblo [11] or Kapovich [6]. The idea in these proofs is to use minimal surfaces,or a combinatorial approximation thereof, to cut up a space into pieces. One thenuses properties of the particular choice of surfaces to show that they are, or can bechosen to be, suitably independent, so that the complex dual to the cutting surfacesis a tree.We do something similar with small cut sets in the decomposition space. A novelfeature of our approach is that the argument takes place “at infinity”. The cut setswe use have more complicated interactions than those in Stallings’ Theorem, andin general the space dual to the collection of cut sets will not be a tree, it will be acube complex quasi-isometric to a tree.Working at infinity has the benefit that the cube complex we construct is canon-ical and inherits a canonical line pattern. QI ( F, L ) is conjugate to the group ofisometries of the cube complex that preserve the line pattern, see Theorem 5.5.This allows us to answer Questions 1 and 2 in the rigid case: Two line patternsin free groups are equivalent if and only if there is a pattern preserving isometrybetween the associated cube complexes. The free group F acts cocompactly bypattern preserving isometries on the cube complex, so QI ( F, L ) does as well. Thisallows us to give a description of QI ( F, L ) as a complex of groups. However, thevertex stabilizers will not, in general, be finitely generated groups.Consideration of line pattern preserving quasi-isometries arises naturally in Geo-metric Group Theory. Work of Papasoglu [14] shows that group splittings of finitelypresented groups over virtually cyclic subgroups are preserved by quasi-isometries.If a finitely presented, one-ended group has a non-trivial JSJ–decomposition overvirtually cyclic subgroups, then each vertex group of the decomposition has a linepattern coming from the incident edge groups. The equivalence classes of these line CHRISTOPHER H. CASHEN AND NATAˇSA MACURA patterns give quasi-isometry invariants for the group, and, in the rigid case, imposesevere restrictions on quasi-isometries of the group.In particular, the authors came upon this problem in the course of studyingmapping tori of free group automorphisms. In the case of a linearly growing auto-morphism, the mapping torus has a JSJ-decomposition with vertex groups F × Z .Understanding the line patterns in the free factors of the vertex groups is a keystep in the quasi-isometry classification of these mapping tori [3].2. Preliminaries
Cut Sets and Cubings. If X is a topological space, a cut set is a subset S ⊂ X such that X \ S = { x ∈ X | x / ∈ S } is disconnected. A single point that isa cut set is a cut point ; a pair of points that is a cut set is a cut pair , etc.A cut set S is minimal if no proper subset of S is a cut set of X .If S and S (cid:48) are cut sets of X we say S (cid:48) crosses S if S (cid:48) \ S has points in multiplecomponents of X \ S . This is not a symmetric relation, but it is if we assume that S and S (cid:48) are minimal.A cubing is a simply connected, non-positively curved cube complex. Cubingscan be used to encode the combinatorics of a collection of cut sets. Our treatmentof cubings is based on work of Sageev [15].Let { S i } i ∈ I be a collection of closed, minimal cut sets of X so that for each i , X \ S i has exactly two connected components, A i and A i . We will take thesuperscripts mod 2, so that the two components of X \ S i are A (cid:15)i and A (cid:15)i for (cid:15) ∈ { , } . Let Σ = { A i } i ∈ I ∪ { A i } i ∈ I Define a cube complex as follows. The vertices are the subsets V of Σ such that:(1) For all i ∈ I exactly one of A i or A i is in V .(2) If C ∈ V and C (cid:48) ∈ Σ with C ⊂ C (cid:48) then C (cid:48) ∈ V .Two vertices are connected by an edge if they differ by only one set in Σ.One can identify Σ with 2 I . The i -th “coordinate” is either 0 for A i or 1 for A i .Edges join vertices that differ in exactly one coordinate.The vertices are the elements of 2 I that are “consistent” with the cut set structurein the sense that if for some i and j we have A i ⊂ A j then we do not have anyvertices that are “1” in the i -th coordinate and “0” in the j -th coordinate. It is notconsistent to be simultaneously in A i and A j .Informally, having (cid:15) in the i -th coordinate corresponds to being in A (cid:15)i . There is asubtlety here, though. An element of 2 I might be consistent without being realizedas a component of X \ { S i } . It is possible that there are vertices such that (cid:15) i is thevalue of the i -th coordinate of the vertex, but ∩ i ∈ I A (cid:15) i i = ∅ . Remark.
There is a minor difference from Sageev’s contruction. In his notation wewould be considering A i = S i ∪ A i and A ci = A i . The nature of the cut sets we areinterested in would make it problematic to include them in one of the components.There is only one place where this requires us to change Sageev’s arguments, whichwe will point out shortly. Everywhere else, it is sufficient to replace a statementlike: A i ⊂ A j = ⇒ A cj ⊂ A ci with a statement like: A (cid:15) i i ⊂ A (cid:15) j j = ⇒ A (cid:15) j j ⊂ A (cid:15) i i ine Patterns in Free Groups 5 This statement follows easily from the fact that minimal cut sets are either mutuallycrossing or mutually non-crossing.Edges in the complex correspond to changing one coordinate from 0 to 1, or viceversa. However, to maintain consistency not every coordinate can be changed:
Lemma 2.1 ([15, Lemma 3.2]) . If V is a vertex and A (cid:15)i ∈ V then W = ( V \ { A (cid:15)i } ) ∪ { A (cid:15)i } is a vertex if and only if A (cid:15)i is minimal in V , in the sense that A (cid:15)i does not containany other A δj ∈ V . It turns out in general that there are still too many vertices. The graph thathas been constructed so far is not necessarily connected. This is where our con-struction differs from Sageev’s. For both his construction and ours, the idea is toselect a subcollection of the vertices, show that the subcollection belongs to a pathconnected subset of the graph, and then throw away everything not in that pathcomponent. Our construction will come later in Section 5. However, this is theonly place in which Sageev uses the special properties of his chosen collection Σ.The rest of his arguments go through unchanged in our setting.So assume that we have passed to a non-trivial path connected component ofthe original graph. Following Sageev again, one glues in one square (2 dimensionalcube) whenever one sees the boundary of a square in the graph. One proceeds byinduction to glue in an n –cube whenever one sees the boundary of an n –cube inthe ( n − e and e (cid:48) are equivalent if there is a finite sequence e = e , e , . . . , e k = e (cid:48) suchthat for each i , e i and e i +1 are opposite edges of some 2–cube, oriented in the samedirection.Equivalence classes of edges are called combinatorial hyperplanes . There is acorresponding idea of a geometric hyperplane . Consider an n –cube of the complex.It can be identified with a cube of side length 1 in R n where the vertices haveall coordinates in {± } . Consider the edges that correspond to changing the n -thcoordinate from − to . These edges belong to a combinatorial hyperplane. Thecorresponding portion of a geometric hyperplane is the intersection of the n –cubewith the coordinate hyperplane { ( x , . . . , x n ) ∈ R n | x n = 0 } . Such pieces are thenglued together for each cube with edges in the combinatorial hyperplane. Theorem 2.2. [15, Theorem 4.10]
Suppose J is a geometric hyperplane in a cubing Y . Then J does not intersect itself and partitions Y into two connected components. We take the metric on the cubing to be the path metric on the 1-skeleton. Thedistance between two vertices is the minimal number of edges in an edge pathjoining them, and such a minimal edge path is called a geodesic .A corollary of the preceding theorem is the following observation about geodesics:Let x and y be vertices in a cubing Y . If they are distance D apart, then a geodesicjoining them must cross D geometric hyperplanes, one through the midpoint ofeach edge of the path. Each of these hyperplanes disconnects Y , with x and y inopposite components. Therefore, any geodesic from x to y must cross the same D hyperplanes. Conversely, the distance between x and y in Y is the number ofhyperplanes separating them. CHRISTOPHER H. CASHEN AND NATAˇSA MACURA
Fix a hyperplane. There is an A (cid:15)i ∈ Σ such that every directed edge e in thehyperplane joins a vertex V e with a vertex ( V e \{ A (cid:15)i } ) ∪{ A (cid:15)i } . Furthermore, everyedge of this form belongs to the hyperplane [15, Lemma 3.9].Thus, we have a bijection between the set of geometric hyperplanes and thecollection { S i } of cut sets. This is how the cubing encodes the collection of cutsets. Cut sets of X correspond to hyperplanes of Y . Distance in Y corresponds tobeing separated by a given number of cut sets. An n –cube in Y corresponds to acollection of n distinct, pairwise crossing cut sets S i in X .2.2. Graphs and Complexes of Groups.
In this section we give a brief accountof graphs and complexes of groups. The reader is referred to Bridson and Haefliger’sbook [2] for more detail.A graph of groups is a construction that builds a group by amalgamating smallergroups. Start with a finite connected graph Γ, and associate to each vertex or edge γ a local group G γ , along with injections φ e,v : G e → G v for each edge e and vertex v that is an endpoint of e .The fundamental group of the graph of groups is then obtained by taking asgenerators all the vertex groups as well as one generator g e for each edge e in thegraph. The relations are:(1) all the relations from the vertex groups,(2) for each edge e with endpoints v and v (cid:48) , and for each h ∈ G e , g e φ e,v ( h ) g − e = φ e,v (cid:48) ( h ) , (3) g e = 1 for each edge e in a chosen maximal subtree of Γ.The fundamental group does not depend on the choice of maximal subtree.Associated to a graph of groups there is a simplicial tree DΓ covering Γ calledthe Bass-Serre tree or the development of the graph of groups. The fundamentalgroup of the graph of groups acts by isometries on DΓ , with vertex stabilizers equalto conjugates of the vertex groups in the graph of groups, and edge stabilizers equalto conjugates of the edge groups.Conversely, given a cocompact isometric action of a group G on a simplicialtree we get a graph of groups decomposition for G by taking the graph to be thequotient of the tree by the G action and choosing local groups to be vertex andedge stabilizers.A complex of groups is generalization of the graph of groups to higher dimensionalcomplexes. In particular, a group acting cocompactly by isometries on a polyhedralcomplex can be given a complex of groups structure by associating to each cell inthe quotient a group isomorphic to the stabilizer of the cell in the original complex.Unlike in the graph of groups case, not every complex of groups is developable.That is, starting with a complex of groups Γ, there may not exist a complex X so that the fundamental group of the complex of groups acts on X with quotientΓ. However, if you start with a group acting on a polyhedral complex, then theresulting graph of groups is developable, the development is just the polyhedralcomplex that you started with.A developable complex of groups is faithful if no non-trivial element of the fun-damental group of the complex of groups acts trivially on the development.To insure that the quotient is still a polyhedral complex, one should assume thatif an element of the group leaves a cell invariant, then it fixes it pointwise. This is ine Patterns in Free Groups 7 called an action without inversions . If this is not the case, it can be achieved bysubdividing cells.Lim and Thomas have worked out a covering theory for complexes of groups [7].A particular result that will be of interest to us is: Theorem. [7, Theorem 4]
Let X be a simply connected polyhedral complex, and let G be a subgroup of Aut( X ) (acting without inversions) that induces a complex ofgroups Γ . Then there is a bijection between the set of subgroups of Aut( X ) (acingwithout inversions) that contain G , an the set of isomorphism classes of coveringsof faithful, developable complexes of groups by Γ . If G acts cocompactly on X then so does any subgroup H of Aut( X ) containing G , and we get a covering of the compact quotient complexes. If the complex ofgroups coming from the H action has finite local groups then we get finite covering,so G is a finite index subgroup of H .2.3. Coarse Geometry.
In this section and the next we establish the languageand basic ideas of coarse geometry and trees. Again, see Bridson and Haefliger’sbook [2] for more detail.Let (
X, d X ) and ( Y, d Y ) be metric spaces. Let A and B be subsets of X .The (open) r –neighborhood of A is the set N r ( A ) = { x ∈ X | d X ( x, A ) < r } .The Hausdorff distance between A and B is: d H ( A, B ) = inf { r | A ⊂ N r ( B ) and B ⊂ N r ( A ) } We will use the common convention that some object is r – [adjective] if it hasthe property for the specified r , and is [adjective] if there exists some r such thatthe object is r –[adjective]. A and B are r – coarsely equivalent if d H ( A, B ) ≤ r . A is r – coarsely dense in X if A is r –coarsely equivalent to X .A map φ : X → Y is a ( λ, (cid:15) )- quasi-isometric embedding if there exist λ ≥ (cid:15) ≥ x, x (cid:48) ∈ X :1 λ d X ( x, x (cid:48) ) − (cid:15) ≤ d y ( φ ( x ) , φ ( x (cid:48) )) ≤ λd X ( x, x (cid:48) ) + (cid:15) If, in addition, the image of φ is (cid:15) -coarsely dense in Y , then φ is a ( λ, (cid:15) )- quasi-isometry .Maps φ and ψ from X to Y are r – coarsely equivalent , or are equivalent up to r –bounded distance , if for all x ∈ X , d Y ( φ ( x ) , ψ ( x )) ≤ r . QI ( X, Y ) is the set of quasi-isometries from X to Y modulo coarse equivalence.Suppose A is r –coarsely dense in X and φ is a pseudo-map that assigns to each a ∈ A a subset φ ( a ) in Y of diameter at most R . Suppose there are λ ≥ (cid:15) ≥ a and a (cid:48) in A :1 λ d X ( a, a (cid:48) ) − (cid:15) − R ≤ inf { d Y ( y, y (cid:48) ) | y ∈ φ ( a ) , y (cid:48) ∈ φ ( a (cid:48) ) } and sup { d Y ( y, y (cid:48) ) | y ∈ φ ( a ) , y (cid:48) ∈ φ ( a (cid:48) ) } ≤ λd x ( a, a (cid:48) ) + (cid:15) + R Then the pseudo-map φ determines a unique (up to coarse equivalence) extensionto a ( λ, λr + (cid:15) + R )–quasi-isometric embedding Φ : X → Y such that for all a ∈ A ,Φ( a ) ∈ φ ( a ). For each x ∈ X choose a closest a ∈ A and choose any Φ( x ) ∈ φ ( a ). CHRISTOPHER H. CASHEN AND NATAˇSA MACURA
Suppose for some x we define Φ (cid:48) ( x ) by choosing a different closest a (cid:48) ∈ A andΦ (cid:48) ( x ) ∈ φ ( a (cid:48) ). Then d Y (Φ( x ) , Φ (cid:48) ( x )) ≤ sup { d Y ( y, y (cid:48) ) | y ∈ φ ( a ) , y (cid:48) ∈ φ ( a (cid:48) ) }≤ λd X ( a, a (cid:48) ) + (cid:15) + R ≤ λ · r + (cid:15) + R, so Φ and Φ (cid:48) are coarsely equivalent.The fact that Φ is a quasi-isometric embedding follows easily.If φ : X → Y is a ( λ, (cid:15) ) quasi-isometry, consider the inverse pseudo-map thattakes a point in φ ( X ) to its preimage in X . This preimage has diameter at most (cid:15) ,and the image of φ is (cid:15) -coarsely dense in Y . We can therefore extend this pseudo-map to a ( λ, (cid:15) ( λ + (cid:15) ))–quasi-isometry ¯ φ : Y → X . The compositions φ ◦ ¯ φ and¯ φ ◦ φ are coarsely equivalent to the identity maps in Y and X , respectively. We call¯ φ a coarse inverse of φ .With this notion of inverse, the set QI ( X ) of quasi-isometries from X to itself,modulo coarse equivalence, becomes a group, the quasi-isometry group of X .Let G be a finitely generated group and let B be a finite generating set. The wordmetric on G with respect to B is defined by setting | g | to be the minimum lengthof a word equal to g in G written in terms of generators in B or their inverses.The Cayley graph of G with respect to B is the graph with one vertex for eachelement of G and an edge [ g, g (cid:48) ] connecting vertex g to vertex g (cid:48) if g (cid:48) = gb forsome b ∈ B . Make this a metric graph by assuming that each edge has length one.The distance between two vertices g and g (cid:48) is the length of the shortest edge pathjoining them. Thus, the distance from g to the identity vertex is the same as | g | inthe word metric. G acts on the Cayley graph by isometries via left multiplication.While the Cayley graph depends on the choice of finite generating set, differentchoices yield quasi-isometric graphs. More generally, if G acts properly and cocom-pactly by isometries on a length space X , then X is quasi-isometric to G with (any)word metric. We call such a space X a geometric model of G .2.4. Free Groups and Trees.
Let F be the free group of rank n , with free gen-erating set (free basis) B = { a , . . . , a n } . For g ∈ F , let ¯ g denote g − .Let T = C B ( F ) be the Cayley graph of F with respect to B . Since we havechosen a free generating set, T is a tree, a graph with no loops.The tree has a boundary at infinity ∂ T that is a Cantor set. Adding the boundarycompactifies the tree; T = T ∪ ∂ T is a compact topological space whose topologyagrees with the metric topology on T . For any two points t and t (cid:48) in T there is aunique geodesic [ t, t (cid:48) ] joining them.Let v and w be vertices in T . Define:shadow v ( w ) = { x ∈ T | w ∈ [ v, x ] } Let shadow v ∞ ( w ) = shadow v ( w ) ∩ ∂ T .If ξ ∈ ∂ T and v ∈ T let v = v , v , . . . be the vertices along [ v, ξ ]. The setsshadow v ( v i ) give a neighborhood basis for ξ . The topology on T is independent ofthe choice of v .Since T is hyperbolic, any quasi-isometry φ : T → T (cid:48) extends to a homeomor-phism ∂φ : ∂ T → ∂ T (cid:48) . ine Patterns in Free Groups 9 Line Patterns and the Decomposition Space.
Suppose l = { gw m } m ∈ Z is a line in the pattern. The line l has distinct endpoints at infinity: l + = gw ∞ = lim i →∞ gw i and l − = gw −∞ = lim i →−∞ gw i The lines in the pattern never have endpoints in common, so we can decompose ∂ T into disjoint subsets that are either the pair of endpoints of a line from thepattern or a boundary point that is not the endpoint of a line.Define the decomposition space D L (or just D when L is understood) associ-ated to a line pattern L to be the space that has one point for each set in thedecomposition of ∂ T , with the quotient topology.Let q : ∂ T → D be the quotient map. For x ∈ D , q − ( x ) is either a single pointthat is not the endpoint of any line in L , or q − ( x ) = { l + , l − } for some l ∈ L . Theformer we call bad points , the later, good points .The quotient map q induces a bijection between L and the good points of D ,which we denote by q ∗ .If S ⊂ D we will use the notation ˆ S = q − ( S ) ⊂ ∂ T . Further, if S consists ofgood points we will use ˜ S to be the collection of lines of L given by q − ∗ ( S ).The decomposition space is a perfect, compact, Hausdorff topological space.A quasi-isometry φ from T to T (cid:48) extends to a homeomorphism ∂φ : ∂ T → ∂ T (cid:48) .In particular, if there are line patterns L in T and L (cid:48) ∈ T (cid:48) , and if φ is a patternpreserving quasi-isometry, then the homeomorphism ∂φ : ∂ T → ∂ T (cid:48) descends to ahomeomorphism of the corresponding decomposition spaces.3. Whitehead’s Algorithm
Since Whitehead’s original work [19, 20], a number of authors have refined White-head’s Algorithm and applied it to related algebraic questions. Section I.4 of thebook of Lyndon and Schupp [8] gives a version of Whitehead’s Algorithm and someof the classical applications.More recently, Stallings [17] and Stong [18] gave 3–manifold versions of White-head’s Algorithm. In each of these papers the aim was to show that a version ofWhitehead’s Algorithm could be used to determine if, given a finite list of words( w , . . . w k ) in F , there is a free splitting of F such that every w i is conjugate intoone of the free factors. Stallings calls this “algebraically separable”. This algebraicquestion is then shown to be equivalent to a geometric question about whetheror not a collection of curves in a handlebody has a property that Stallings calls“geometrically separable” and Stong calls “disk-busting”.In this section we review Whitehead’s Algorithm. Our language is similar tothat of Stallings and Stong, except that our group actions are on the left and pathconcatenations are on the right, while they use the opposite convention.3.1. Whitehead Graphs.
Let w ∈ F be a cyclically reduced word. Let B = { a , . . . , a n } be a free basis of F . The Whitehead Graph of w with respect to B ,Wh B ( ∗ ) { w } , is the graph with 2 n vertices labeled a , . . . , a n , ¯ a , . . . ¯ a n , and an edgebetween vertices v and v (cid:48) for each occurrence of ¯ vv (cid:48) in w (as a cyclic word). Thegraph depends on the choice of B , and, of course, on w , but we will write Wh( ∗ )when these are clear. Remark.
At present the ( ∗ ) may be ignored; it will be explained in the next section.For example, if F = (cid:104) a, b (cid:105) Figures 1-4 show some Whitehead Graphs. b ¯ b a ¯ a Figure 1.
Wh( ∗ ) { a } b ¯ b a ¯ a Figure 2.
Wh( ∗ ) { ab } b ¯ b a ¯ a Figure 3.
Wh( ∗ ) { ab ¯ a ¯ b } b ¯ b a ¯ a Figure 4.
Wh( ∗ ) { a ba ¯ b } Notice that Wh( ∗ ) { a } is disconnected; the vertices b and ¯ b are isolated.Wh( ∗ ) { ab } is connected but becomes disconnect if the vertex b (or ¯ b ) is deleted; b and ¯ b are cut vertices .Wh( ∗ ) { ab ¯ a ¯ b } is connected and has no cut vertices.Wh( ∗ ) { a ba ¯ b } is also connected with no cut vertices, and has multiple edgesbetween vertices a and ¯ a .More generally, one can make a Whitehead graph representing finitely manywords w , . . . , w m . We call this Whitehead graph Wh( ∗ ) { w , . . . , w m } or Wh( ∗ ) {L} ,where L is the line pattern generated by { w , . . . , w m } .3.2. Whitehead Automorphisms.
Let φ be an automorphism of F . Apply-ing φ changes the Whitehead graph Wh B ( ∗ ) { w , . . . , w m } to the Whitehead graphWh B ( ∗ ) { [ φ ( w )] , . . . , [ φ ( w m )] } , where [ φ ( w i )] means choose a cyclically reducedword in the conjugacy class of φ ( w i ).An automorphism that permutes B or swaps a generator with its inverse givesan isomorphic Whitehead graph. Definition 3.1. A Whitehead automorphism is an automorphism of the followingform: Pick x ∈ B ∪ ¯ B , a generator or the inverse of a generator. Pick Z ⊂ B ∪ ¯ B such that x ∈ Z and ¯ x / ∈ Z . ine Patterns in Free Groups 11 Define an automorphism φ x,Z by defining φ x,Z ( x ) = x and for the rest of thegenerators y ∈ B : φ x,Z ( y ) = y if y / ∈ Z and ¯ y / ∈ Zxy if y ∈ Z and ¯ y / ∈ Zy ¯ x if y / ∈ Z and ¯ y ∈ Zxy ¯ x if y ∈ Z and ¯ y ∈ Z We say that the automorphism φ x,Z is the Whitehead automorphism that pushes Z through x .To visualize what is happening, consider the rose with one vertex and one ori-ented loop for each element of B . The fundamental group is F . The Whiteheadautomorphism φ x,Z is the automorphism of the fundamental group induced by thehomotopy equivalence that pushes one or both ends of the y –loop around the x –loop according to whether y or ¯ y or both are in Z , or leaves the y –loop alone ifneither y nor ¯ y are in Z . See also Section 4.2.Define the complexity of the collection w , . . . , w n to be the number of edges ofWh( ∗ ) { w , . . . , w m } . This is equivalent to the sum of the lengths of the w i , andalso half the sum of the valences of the vertices.Comparing Wh B ( ∗ ) { w , . . . , w m } to Wh B ( ∗ ) { [ φ x,Z ( w )] , . . . , [ φ x,Z ( w m )] } we seethat the valences of vertices other than x and ¯ x do not change. The new valenceof x and ¯ x is equal to the number of edges that go between Z and Z c . Thus, theWhitehead automorphism reduces the complexity of the Whitehead graph exactlywhen there are fewer edges joining Z and Z c than the valence of x . Theorem.
Aut( F ) is generated by: (1) exchanges of a generator with its inverse (2) permutations of the generators (3) Whitehead automorphisms
This is clear since this set of automorphisms contains the Nielsen generators forAut( F ) [12].Whitehead’s Algorithm is as follows: First, check if any Whitehead automor-phisms reduce the complexity of the Whitehead graph. Repeat. Once you havereduced to minimal complexity, there are only finitely many graphs to consider.Build a graph with one vertex for each possible Whitehead graph with the givencomplexity, and an edge between two vertices if one of the given generators of theautomorphism groups takes one graph to the other. One can then show that thedesired automorphism exists if and only if the reduced Whitehead graphs for thetwo lists of words lie in the same connected component of this graph.If { w , . . . , w m } is a subset of a free basis then the minimal complexity Whiteheadgraph should have m disjoint edges.If there is a free splitting F = F (cid:48) ∗ F (cid:48)(cid:48) with every w i in F (cid:48) or F (cid:48)(cid:48) then the minimalcomplexity Whitehead graph should be disconnected.The presence of a cut vertex in the Whitehead graph indicates that the graphis not reduced. If x is a cut vertex, let Z be the union of { x } and the verticesof a connected component of Wh( ∗ ) {L} \ { x } not containing ¯ x . The Whiteheadautomorphism φ x,Z reduces complexity.One application of Whitehead’s Algorithm is that a word w is an element of afree basis of F = F n if and only if the minimal complexity Whitehead graph for w consists of a single edge and 2( n −
1) isolated vertices.
More generally, the width of an element w is the rank of the smallest free factorof F containing w . The minimal complexity Whitehead graph for an element ofwidth m in F = F n consists of 2( n − m ) isolated vertices and a connected graphwithout cut vertices on the remaining vertices.4. Whitehead Graphs and the Topology of the Decomposition Space
The decomposition space associated to a line pattern first appears in the litera-ture in work of Otal [13], who proves that the decomposition space is connected ifand only if there exists a basis B of F such that Wh B ( ∗ ) is connected without cutvertices.A similar theorem appears in the thesis of Reiner Martin [10], who referencesnotes of Bestvina. Theorem 4.1. [10, Theorem 49]
For any w ∈ F −{ } , the following are equivalent: (1) w is contained in a proper free factor of F . (2) The width of w is strictly less than the rank of F . (3) There exists a disconnected Whitehead graph of w . (4) The decomposition space associated to the pattern generated by w is discon-nected. (5) Every Whitehead graph for w with no cut vertices is disconnected. The goal of this section is to further explore the relationship between general-izations of the Whitehead graph and the topology of the decomposition space. Inparticular, we are interested in finite cut sets in the case that the decompositionspace is connected.
Remark.
The theorem stated in [10] has an additional equivalent condition: forany basis there exists a generalized Whitehead graph that is disconnected. We willnot make use of this. In our notation, Martin’s generalized Whitehead graph isWh B ( N r ( ∗ )) { w } (see below).4.1. Geometric Interpretation and Generalizations.
Fix a free basis B for F n , and let T be the corresponding Cayley graph.Let X be a closed, connected subset of T . Consider the connected componentsof T \ X . Take these components as the vertices of a graph. Connect vertices v , v by an edge if there is a line in l ∈ L with one endpoint in the componentcorresponding to v and the other in the component corresponding to v . Call thisgraph Wh B ( X ) {L} , and notice that when X = ∗ is a single vertex this graph isexactly Wh B ( ∗ ) {L} . Since L is equivariant we get the same graph for any choiceof vertex.We will also give a combinatorial construction of our generalization of the White-head graph. However, the intuition that informs our arguments comes from theabove geometric interpretation. One should visualize the Whitehead graph as theportion of the line pattern that passes through a subset of a tree, rather than as anabstract graph. Where appropriate, as in Figure 5 and Figure 6, we have includedthe relevant portions of the tree to aid in this visualization.If X is a finite connected subset of T we can build up Whitehead graphsWh( X ) {L} in a combinatorial way by splicing together copies of Wh( ∗ ) {L} foreach of the vertices of X . Splicing is a method of combining graphs. The term wascoined by Manning in [9] where he uses splicing to construct Whitehead graphs offinite covers of a handlebody from the Whitehead graph of the base handlebody. ine Patterns in Free Groups 13 Let v be a vertex of a graph Γ and let v (cid:48) be a vertex of a graph Γ (cid:48) of valenceequal to the valence of v . Given a bijection between edges of Γ incident to v andedges of Γ (cid:48) incident to v (cid:48) , the splicing map , form a new graph whose vertices arethe vertices of Γ and Γ (cid:48) minus the vertices v and v (cid:48) . Edges not incident to v or v (cid:48) are retained in the new graph. Finally, for each pair of edges [ w, v ] in Γ and [ w (cid:48) , v (cid:48) ]in Γ (cid:48) identified by the splicing map, add an edge [ w, w (cid:48) ] in the new graph.In other words, we have deleted v and v (cid:48) , leaving the edges incident to thosevertices with “loose ends”. The splicing map tells us how to splice a loose end at v to a loose end at v (cid:48) to get an edge in the new graph.For Whitehead graphs the splicing map is determined by the line pattern. Sup-pose we have adjacent vertices g and ga in a Cayley graph, and correspondingWhitehead graphs Wh( g ) and Wh( ga ). We splice them together to build the White-head graph Wh([ g, ga ]). The g vertex and ga vertex in T are adjacent across an a –edge, so the splicing vertices are the a –vertex of Wh( g ) and the ¯ a –vertex ofWh( ga ). Each edge in Wh( g ) incident to a corresponds to a length two subwordof one of the generators of the line pattern of the form xa or ¯ ax . Suppose an edgecorresponds to a subword xa , and suppose the next letter is y , so there is a lengththree subword xay . We define the splicing map to identify the edge correspondingto this particular instance of the subword xa to the edge in Wh( ga ) (incident to ¯ a )corresponding to this particular instance of the subword ay .We can make the splicing easier to visualize if we draw the Whitehead graphswith loose ends at the vertices. Figure 5 shows the Whitehead graph for the patterngenerated by the words ab and a ¯ b in F = F = (cid:104) a, b (cid:105) , along with the underlyingtree. The word ab will contribute an edge from ¯ a to b and an edge from ¯ b to a . Thetwists in the graph indicate the splicing maps. ∗ a ¯ a b ¯ b Figure 5.
Wh( ∗ ) { ab, a ¯ b } Let ∗ be the identity vertex. Take a copies of this graph at ∗ and at a and splicethem together. We get the splicing map by considering the words. There is an ab –line at ∗ . If the first letter is a , the previous letter was b , so we see an edge from¯ b to a in the Whitehead graph at ∗ . The next letter is b , so in the Whitehead graphat a we see an edge from ¯ a to b , and the twist in the graph indicates that these twoedges should be spliced together. Similarly, there is an a ¯ b –line at ∗ . It contributes an edge from b to a in theWhitehead graph at ∗ , and this continues on to an edge from ¯ a to ¯ b in the Whiteheadgraph at a . Note.
Unless noted otherwise, figures are drawn so that the splicing map is achievedby an orientation preserving isometry of the page. ∗ a ¯ a b ¯ b aba ¯ b a Figure 6.
Wh( ∗ ) { ab, a ¯ b } and Wh( { a } ) { ab, a ¯ b } Figure 7.
Wh([ ∗ , a ]) { ab, a ¯ b } The geometric and splicing constructions produce the same graph for sets X with finitely many vertices. We could try to take limits of the spliced graphs when X is infinite, but if X ⊂ T contains endpoints of some l ∈ L , then splicing does notactually produce a graph.If both endpoints of l are in X then after finitely many splices there is an edgecorresponding to l , but in the limit the edge grows to be an open interval notincident to any vertices; the vertices escape to infinity. This line does not occur ifwe follow the geometric definition, because it is not joining two different componentsof the complement of X in T . Similarly, if only one endpoint of l is in X then splicingproduces a graph G with a half line attached. If we throw out these “non-closededges” we get the graph Wh( X ) {L} of the geometric definition. Remark.
Stong [18] defines a generalized Whitehead graph that coincides with ourdefinition, but, like Martin [10], only makes use of Wh B ( N r ( ∗ )) { w } . ine Patterns in Free Groups 15 Whitehead Automorphisms Revisited.
Let us consider how applicationof a Whitehead automorphism changes the line pattern.Suppose x , y and z are in B∪ ¯ B with y (cid:54) = z . Consider a Whitehead automorphism φ = φ x,Z (recall Definition 3.1). Let l ∈ L be a line that goes through vertices y , ∗ and z , where ∗ is the identity vertex. The line l is the geodesic that goes throughvertices of the form { y (¯ yzu ) m } m ∈ Z , where u is some word in F that does not beginwith ¯ z or end with y .First suppose that y, z ∈ Z , y, z (cid:54) = x and ¯ y, ¯ z / ∈ Z , so that φ ( x ) = x , φ ( y ) = xy and φ ( z ) = xz . Then φ ( l ) is the line that includes vertices of the form: { φ ( y (¯ yzu ) m ) } m ∈ Z = { xy (¯ y ¯ xxzφ ( u )) m } m ∈ Z = { xy (¯ yzφ ( u )) m } m ∈ Z Since u does not begin with ¯ z or end in y , the same is true for φ ( u ). Therefore, φ ( l ) goes through vertices xy , x and xz . The line l that went through ∗ has been“pushed through” the x edge to a line φ ( l ) that goes through x and not through ∗ .Using similar arguments one can show:(1) φ ( l ) goes through x and not through ∗ if y and z are in Z .(2) φ ( l ) goes through ∗ and not through x if y and z are in Z c .(3) φ ( l ) goes through both ∗ and x if exactly one of y or z is in Z .4.3. Cut Sets in the Decomposition Space.
The next two lemmas use es-sentially the same ideas that go into the proofs of Theorem 4.1 in [10] and [13,Proposition 2.1].
Lemma 4.2.
If for some choice of basis
Wh( ∗ ) is disconnected, then D is discon-nected.Proof. Let ∗ be the identity vertex in T . Let B be a free basis of F such thatWh B ( ∗ ) is not connected. Vertices of Wh( ∗ ) are in bijection with B ∪ ¯ B . There issome partition of B ∪ ¯ B into subsets A and A (cid:48) so that no lines of L connect A to A (cid:48) . Let ˆ A = ∪ a ∈A shadow ∗∞ ( a )The sets ˆ A and ˆ A c ⊂ ∂ T are both nonempty clopens, sets that are both open andclosed. Since there are no lines of L with one endpoint in ˆ A and one in ˆ A c , theirimages in D are disjoint nonempty clopens, so D is disconnected. (cid:3) Lemma 4.3.
Suppose there exists a free basis B of F such that Wh B ( ∗ ) is connectedwithout cut vertices. Let T be the Cayley graph of F corresponding to B . Pick anyedge e in T . Let ∗ and v be the endpoints of e . Let ˆ A = shadow ∗∞ ( v ) . That is, ˆ A is the “half ” of ∂ T on the “ v -side” of e . The set A = q ( ˆ A ) is connected in D .Proof. Suppose there are open sets B and C of D such that A ⊂ B ∪ C and A ∩ B ∩ C = ∅ . The set ˆ A is open in ∂ T , so A (cid:48) = ˆ A ∩ q − ( B ) and A (cid:48)(cid:48) = ˆ A ∩ q − ( C )are open. Assuming that A (cid:48) is nonempty, we will show that A (cid:48)(cid:48) must be empty,which implies A is connected.ˆ A is closed in ∂ T , so A (cid:48) and A (cid:48)(cid:48) are closed. Compactness of ∂ T implies A (cid:48) and A (cid:48)(cid:48) are compact clopens. Since A (cid:48) is compact and open, there are finitely manyvertices x , . . . , x a so that A (cid:48) = ∪ ai =1 shadow ∗∞ ( x i )There is a similar finite collection y , . . . , y b that determines A (cid:48)(cid:48) .Consider the convex hull H of { x i } ai =1 ∪ { y j } bj =1 ∪ { v } ; it is a finite tree. Callthe vertices of H other than { x i } ai =1 ∪ { y j } bj =1 ∪ { v } the “interior vertices”. Since ˆ Ae ∗ v Figure 8.
The boundary of the tree split into “halves”. A (cid:48) ∪ A (cid:48)(cid:48) = A , H includes all edges incident to its interior vertices. Let X be theunion of the set of interior vertices with { v } .Construct the Whitehead graph Wh( X ). It has a + b + 1 vertices correspondingto the x i and y j and the edge e of T .The graph is connected without cut points, since it can be constructed by splicingtogether finitely many copies of Wh( ∗ ), which is connected without cut points. Inparticular, the vertex e is not a cut vertex.Assume A (cid:48) is nonempty. If x = v then A (cid:48) = A , so A (cid:48)(cid:48) = ∅ , and we are done.Otherwise, x is a vertex of Wh( X ) \ v . An edge of Wh( X ) \ v incident to x corresponds to a line l ∈ L with one endpoint in the shadow of x and the otherendpoint in the shadow of z for some z ∈ { x i } ai =2 ∪ { y j } bj =1 . In the decompositionspace these two endpoints are identified, and we already know that the image of thefirst endpoint is in B . This means that z must be in { x , . . . , x a } . Since Wh( X ) \ v is connected we conclude that all the vertices of Wh( X ) \ v belong to { x , . . . , x n } ,so A (cid:48)(cid:48) = ∅ . Thus, A is connected in D . (cid:3) Thus, if Wh( ∗ ) is connected without cut vertices, then for any edge e in T theboundaries of the two connected components of T \ e correspond to connected setsin the decomposition space. Since Wh( ∗ ) is connected there is also at least one linein L crossing e . This means that these two connected sets in the decompositionspace have a point in common. Corollary 4.4.
Suppose
Wh( ∗ ) has no cut vertices. Then the decomposition spaceis connected if and only if Wh( ∗ ) is connected.Proof of Theorem 4.1. Conditions (1) and (2) are equivalent by definition. Theequivalence of (1) and (3) is a consequence of Whitehead’s Algorithm.(3) = ⇒ (4) is Lemma 4.2.Corollary 4.4 implies the contrapositive of (4) = ⇒ (5).It is always possible to eliminate cut vertices with Whitehead automorphisms,so (5) = ⇒ (3). (cid:3) Here is another corollary of Lemma 4.3:
Corollary 4.5.
Suppose
Wh( ∗ ) is connected without cut vertices. Pick any edge e in T . Let ˜ S be the collection of (finitely many) lines of L that cross e . Then S = q ∗ ( ˜ S ) is a cut set in D . We will call such a set S coming from all the lines crossing an edge an edge cutset . ine Patterns in Free Groups 17 From now on, unless otherwise noted, we will assume that any Whitehead graphWh( ∗ ) is connected without cut vertices. Thus, the decomposition space is con-nected. Our goal is to identify finite minimal cut sets.We have an easy sufficient condition to see that a set ˜ S = { l , . . . , l k } ⊂ L givesa cut set S = q ∗ ( ˜ S ) of D . Proposition 4.6.
Let ˜ S = { l , . . . , l k } be a finite collection of lines in L . Let X be any compact, connected set in T . In Wh( X ) , delete the interior of any edgecorresponding to one of the lines l i . If the resulting graph Wh( X ) \ ˜ S is disconnectedthen S = q ∗ ( ˜ S ) is a cut set. The proof of this proposition is similar to Corollary 4.5, but it will also be aspecial case of the next proposition. Before moving on, though, let us consider anexample that shows that this proposition does not give a necessary condition for S to be a cut set.Consider the pattern L generated by the pair of words b and ab ¯ a ¯ b in F = (cid:104) a, b (cid:105) .The Whitehead graph (with loose ends) for this pattern is shown in Figure 9. Figure 9.
Wh( ∗ ) { a, ab ¯ a ¯ b } (loose)This graph is connected without cut points, so the decomposition space is con-nected. We claim that the endpoints of any b –line give a cut point in the decom-position space. For instance, the b –line through the identity vertex has endpoints b ∞ , b −∞ in ∂ T . Let A = q ( b ∞ ) = q ( b −∞ ).Let ∗ be the identity vertex. Letˆ B = ∪ m ∈ Z shadow ∗∞ ( b m a ) , that is, ˆ B consists of all the boundary points ξ of T such that the first occurrenceof a or ¯ a in the geodesic from the identity to ξ is an a .Now ˆ B is open, and every line of L with one endpoint in ˆ B has both endpointsin ˆ B . Let B = q ( ˆ B ); the preimage is ˆ B = q − ( B ), so B is open in D . Similarly, let B (cid:48) be the image in D of the boundary points of T such that the first occurrence of a or ¯ a in the geodesic from the identity is an ¯ a . D = A ∪ B ∪ B (cid:48) , and A = B \ B = B (cid:48) \ B (cid:48) , so A is a cut point.For any compact, connected X , Wh( X ) looks like a circle with a number ofdisjoint chords (see Figure 10).The edges of the circle correspond to ab ¯ a ¯ b –lines, and the chords correspond to b –lines. This graph has no cut points, so deleting the interior of an edge does notdisconnect it. Figure 10.
Wh( N ( ∗ ))This example has shown that to decide if ˜ S gives a cut set, it is not enough todelete the interiors of edges in a Whitehead graph.There will be several different notions of deleting parts of Whitehead graphs, solet us standardize notation. Let X ⊂ Y ⊂ T , and let l ∈ L and ˜ S ⊂ L . Let e be anedge of T incident to exactly one vertex of X .The edge e corresponds to a vertex in Wh( X ). The graph Wh( X ) \ e is obtainedfrom Wh( X ) by deleting this vertex, but retaining the incident edges as loose endsat e .If v is a vertex of T that is distance 1 from X , then there is a unique edge e withone endpoint equal to v and the other in X . Define Wh( X ) \ v = Wh( X ) \ e .Similarly, Wh( X ) \ Y is obtained from Wh( X ) by deleting each vertex of Wh( X )that corresponds to an edge in Y . Visualizing Whitehead graphs in the tree,Wh( X ) \ Y is the portion of Wh( Y ) that passes through the set X .Wh( X ) \ l is obtained from Wh( X ) by deleting the interior of the edge corre-sponding to l , if such an edge exists. Similarly, obtain Wh( X ) \ ˜ S by deleting theinteriors of any edges corresponding to a line in ˜ S .Wh( X ) \ ¨ l is obtained from Wh( X ) by deleting the interior of the edge corre-sponding to l as well as the two vertices that are its endpoints, retaining loose endsat these vertices.Consider the line pattern L generated by ab ¯ a ¯ b and b . Let l be the b –line throughthe identity vertex ∗ . Let X = ∗ and let Y = [¯ b, b ]. Figures 11–14 illustrate ourdifferent notions of deleting from Wh( X ). Lemma 4.7 (Hull Determines Connectivity) . Let S be a nonempty, finite subsetof D that is not just a single bad point. Let H be the convex hull of q − ( S ) . Thereis a bijection between connected components of Wh( H ) and connected componentsof D \ S .Proof. Components A i of T \ H are the kind of sets in Lemma 4.3. Therefore, q ( ∂ A i ) is connected in D . The set ∂ A i is open in ∂ T . For a subcollection {A i j } corresponding to a connected component of Wh( H ), we have that q ( (cid:83) j ∂ A i j ) is an ine Patterns in Free Groups 19 a ¯ a b ¯ b Figure 11.
Wh( ∗ ) a ¯ a b ¯ b Figure 12.
Wh( ∗ ) \ l a ¯ a b ¯ b Figure 13.
Wh( ∗ ) \ ¨ l a ¯ a b ¯ b Figure 14.
Wh( ∗ ) \ [¯ b, b ]open connected set in D \ S , as in the proof of Lemma 4.3. The complement of thisset in D \ S is either empty or is a union of sets of a similar form, correspondingto other connected components of H . Thus, q ( (cid:83) j ∂A i j ) is closed, and is thereforea connected component of D \ S . (cid:3) Pick any vertex ∗ ∈ T . If ξ ∈ D is a bad point, the previous argument appliesif we take H to be the ray [ ∗ , ξ ]. If Wh( ∗ ) is connected without cut points thenWh( H ) is connected. Therefore: Corollary 4.8.
No bad point of D is a cut point. In a sense, Lemma 4.7 achieves our goal of relating the topology of the decompo-sition space to generalizations of the Whitehead graph. However, this generalizedWhitehead graph is infinite. In the next two sections we show that the same infor-mation can be obtained from a finite portion of this Whitehead graph.4.4.
Identifying Cut Points and Cut Pairs.
The previous corollary tells usthat any cut point is a good point, so its preimage in ∂ T is a pair of points. Wehave a similar situation if there is a cut pair consisting of two bad points; thepreimage of such a set in ∂ T is a pair of points. In both cases, the convex hull is aline.Suppose g ∈ F \ { } is cyclically reduced with H + = g ∞ and H − = g −∞ . Let H be the convex hull of these two points. Let X = [ ∗ , g ) be the segment joining theidentity vertex to the g vertex in T . We know, by Lemma 4.7, that the connected components of D \ q ( {H − , H + } )are in bijection with components of Wh( H ). We can construct Wh( H ) by splicingtogether g -translates of Wh( X ) \ H .Wh( X ) \ H is Wh( X ) \ { e, ge } for some edge e incident to ∗ and ge incident to g = g ∗ , so Wh( X ) \ H has a collection of loose ends at e and at ge . The actionof g identifies Wh( X ) \ H with Wh( g X ) \ H , which has loose ends at ge and g e .The line pattern determines for us a splicing map for splicing the loose ends ofWh( X ) \ H at ge to the loose ends of Wh( g X ) \ H at ge .It is an easy consequence of the hypothesis that Wh( ∗ ) is connected withoutcut vertices that for any segment [ ∗ , g k ) ⊂ H , every component of Wh([ ∗ , g k )) \ H contains a loose end at e (and a loose end at g k e ). Thus, the number of componentsof Wh( H ) is bounded above by the number of components of Wh( ∗ ) \ H . To boundthe number of connected components of Wh( H ) below we need to know if distinctconnected components of Wh( X ) \ H become connected when we splice on moretranslates.Let P be a partition of the loose ends of Wh( X ) \ H at e that is at least as coarseas connectivity in Wh( X ) \ H , ie, if two loose ends belong to the same connectedcomponent of Wh( X ) \ H then they belong in the same subset of the partition.Let | P | be the number of subsets in the partition; P is nontrivial if | P | > P is at least as coarse as connectivity, every vertex and edge in Wh( X ) \ H is connected to loose ends in exactly one subset of P . Let P (cid:48) be the partition ofthe loose ends of Wh( X ) \ H at ge such that two loose ends are in the same subsetof the partition if and only if they are connected to loose ends at e in a commonsubset of the partition P .The g action determines a partition gP of the loose ends of Wh( g X ) \ H at ge by pushing forward the partition P .We say the partition P is compatible with the splicing map if there is a bijectionbetween subsets of the partitions of P (cid:48) and gP and the splicing map splices edgesin a subset of P (cid:48) to edges in the corresponding subset of gP .The trivial partition is always compatible with the splicing map, but this givesus no information. Another obvious partition to consider would be the partitionthat comes from connectivity in Wh( X ) \ H . The is the partition in which twoloose ends of Wh( X ) \ H at e belong to the same subset of the partition if andonly if they belong to the same connected component of Wh( X ) \ H . Suppose thispartition is compatible with the splicing map. This would mean that two looseends of Wh( X ) \ H at ge in the same connected component of Wh( X ) \ H mustsplice to two loose ends of Wh( g X ) \ H at ge in the same connected componentof Wh( g X ) \ H , so splicing introduces no new connectivity. In this case it followsthat for all k ≥ ∗ , g k )) \ H is equalto the number of connected components of Wh( X ) \ H .However, this is not always the case. Splicing may introduce new connectivity.Compatibility of the partition controls how much new connectivity is introduced.If we have a partition compatible with the splicing map, then, after splicing, thepartition P is at least as coarse as connectivity in Wh([ ∗ , g )) \ H . Moreover, P willstill be compatible with the splicing map at g e , so we may continue by inductionto show: ine Patterns in Free Groups 21 Proposition 4.9.
Suppose P is a partition that is compatible with the splicingmap. Then for any segment Y = [ ∗ , g k ) of H , the number of connected componentsof Wh( Y ) \ H is greater than or equal to | P | . Given a compatible partition, there are two cases to consider. If for some Y we have no free edges in Wh( Y ) \ H then the number of connected components ofWh( H ) is greater than or equal to | P | . In particular, if | P | ≥
2, then q ( {H + , H − } )is a bad cut pair in D .Any particular line l ∈ L overlaps with H for distance at most | g | × (2 + maximum number of consecutive g ’s in a generating word for L . )This is infinite if and only if g is a generator of the line pattern. Thus, g isa generator of the line pattern if and only if in every Y there is a free edge inWh( Y ) \ H , since in this case H = l where l ∈ L is the g –line through the identity.In this case we could have chosen the partition P so that one of the subsets isthe singleton consisting of the loose end of the free edge. The partition P (cid:48) also hasa subset that is a singleton, consisting of the other loose end of the free edge. Sucha partition has a segregated free edge .We do not see the free edge in Wh( H ), so in general we can only concludethat Wh( H ) has at least | P | − | P | − ≥ q ( {H + , H − } ) = q ∗ ( l ) is a cut point in D . Proposition 4.10 ( q ( { g ∞ , g −∞ } ) Cut Set Criterion) . Let g ∈ F \ { } be anelement of the free group. With notation as above, let P be the finest partitionthat is compatible with the splicing map and at least as coarse as connectivity in Wh( X ) \ H . Then: (1) If P is trivial then q ( { g ∞ , g −∞ } ) is not a cut set. (2) If P is nontrivial and has no segregated free edge then q ( { g ∞ , g −∞ } ) is abad cut pair. (3) If P has a segregated free edge and | P | = 2 then q ( { g ∞ , g −∞ } ) is not a cutset. (4) If P has a segregated free edge and | P | > then q ( { g ∞ , g −∞ } ) is a cutpoint.Proof. If P is trivial then Wh([ ∗ , g )) \ H is connected, so q ( { g ∞ , g −∞ } ) is not acut set. Similarly, if | P | = 2 and there is a segregated free edge then H = l for l ∈ L and Wh([ ∗ , g ]) \ l is connected, so q ( { g ∞ , g −∞ } ) is not a cut set.In the other cases, Wh( H ) has multiple components, so q ( { g ∞ , g −∞ } ) is a cutset. (cid:3) The proposition tells us that given a g we can decide if q ( { g ∞ , g −∞ } ) is a cutset. We call this a periodic cut set. Next we show that if there are cut points orcut pairs then there are periodic cut sets: Proposition 4.11. If D has cut points or cut pairs then there is some R dependingon L and some g with | g | ≤ R such that q ( { g ∞ , g −∞ } ) is a cut set. To identify cut points we just need to apply Proposition 4.10 to the generators of L , so in this case it is sufficient to take R to be the length of the longest generatorof L . The work of proving Proposition 4.11 lies in finding an R that works for thecut pair case: Lemma 4.12. If q ( {H + , H − } ) is a cut pair then there is some R depending on L and some g ∈ F \ { } with | g | ≤ R such that q ( { g ∞ , g −∞ } ) is a cut set. Note that q ( { g ∞ , g −∞ } ) is either a cut point or a bad cut pair. Proof.
Let H be the convex hull of {H + , H − } . We may assume that H containsthe identity vertex ∗ .Use ∗ ) \ H contains an edge, so the number ofcomponents is at most the complexity of Wh( ∗ ).For any segment X of H we have:2 ≤ H ) ≤ X ) \ H ) ≤ ∗ ) \ H ) ≤ complexity of Wh( ∗ )Number the vertices of H consecutively with integers with ∗ = v and indexincreasing in the H + direction. Number the edges of H so that e i is incident to v i − and v i . We consider these edges oriented in the direction of increasing index.An oriented edge of T comes with a label that is a generator or inverse of a generatorof F .The function f ( i ) = H −∞ , v i ]) \H ) is nonincreasing and, for high enough i , stabilizes at H ). Since we started with a cut pair, for high enough i there isno free edge in Wh([ H −∞ , v i ]) \ H . After changing by an isometry and relabeling,if necessary, we may assume that i = 0 is “high enough” in the previous twostatements.Fix a numbering from 1 to c = H ) on the components of Wh( H ). Ateach v i we get a partition P i into c subsets of the loose ends of Wh( v i ) \ H at e i by connectivity in Wh( H ). Similarly, we get a partition P (cid:48) i of the loose endsof Wh( v i ) \ H at e i +1 . These partitions are at least as coarse as connectivity inWh([ v i , v j ]) \ H for any j ≥ i .By construction, the splicing map at e i +1 connecting loose ends of Wh( v i ) \ H at e i +1 to loose ends of Wh( v i +1 ) \ H at e i +1 is compatible with the partitions P (cid:48) i and P i +1 .For each edge pair ( e i , e i +1 ) there is a corresponding label pair L i that gives anontrivial word of length two in F . There are 2 n (2 n −
1) such words.Let m be the number of partitions of (complexity of Wh( ∗ )) things into c nonempty subsets. Consider the segment [ v , v R ], where R = 2 n (2 n − m . Somelabel pair appears at least m times. Let { i j } mj =1 be a set of indices such that the L i j are the same.Let g j,k be the element of F that takes v i j to v i k .If we fix P i we get a map of the elements g ,k into the set of all possible partitionsby g ,k (cid:55)→ g ,k P i , so for some 1 ≤ j < k we have g ,j and g .k mapping to the samepartition. Therefore, g j,k P i j = P i k . g = g i,j is then the desired element. (cid:3) Remark.
In the preceding proof we found an element g such that the g –actionpreserved a partition. We did not insist that the g –action also fixed the numberingof components of Wh( H ); these may be permuted. The proof may easily be modifiedto fix the numbering, at the expense of a larger bound on | g | . Corollary 4.13.
Existence of a cut pair implies existence of a cut point or bad cutpair. ine Patterns in Free Groups 23
Corollary 4.14.
With R as in the previous proposition, for any pair of points {H + , H − } ⊂ ∂ T , if X is a segment of the convex hull H of {H + , H − } of lengthgreater than R , and if there are no cut pairs in the decomposition space, then oneof the following is true: (1) Wh( X ) \ H is connected. (2) Wh( X ) \ H has two components, one of which is a free edge. Theorem 4.15 (Detecting Cut Pairs) . There is a finite algorithm for detecting cutpairs in the decomposition spaceProof.
Given a list of words generating a line pattern, apply Whitehead automor-phisms, if necessary, to eliminate cut vertices. If the graph becomes disconnected,stop; the decomposition space is disconnected by Corollary 4.4.If it is possible to disconnect the Whitehead graph by deleting the interiors of twoedges, stop; these two edges correspond to a cut pair. In particular, this happensif the Whitehead graph has any valence two vertices.Use Proposition 4.10 to check if any of the generators of the line pattern give acut point in the decomposition space. If so, stop.Let R be the constant from Lemma 4.12. The idea now is to check segments oflength R to see if we can find a disconnected Whitehead graph. There are a lot ofthese. We streamline the process by only checking those long segments for whichevery sub-segment gives a disconnected Whitehead graph.Let X = {∗} .We proceed by induction. Suppose X i is defined.Start with X i +1 = X i . Consider pairs of points v and v (cid:48) such that d ( v, X i ) = d ( v (cid:48) , X i ) = 1, such that d ( v, ∗ ) = d ( v (cid:48) , ∗ ) = i + 1, and such that ∗ ∈ [ v, v (cid:48) ]. IfWh( X i ∩ [ v, v (cid:48) ]) \ [ v, v (cid:48) ] is not connected, add v and v (cid:48) to X i +1 .Continue until stage k = 1 + (cid:100) R (cid:101) . Apply Corollary 4.14 and Proposition 4.11:there are pairs v and v (cid:48) in X k \ X k − with ∗ ∈ [ v, v (cid:48) ] such that Wh( X k − ∩ [ v, v (cid:48) ]) \ [ v, v (cid:48) ] has more than one component that is not just a free edge if and only if thereare cut pairs in the decomposition space. (cid:3) Corollary 4.16.
If a Whitehead graph for a line pattern has the property thatdeleting any pair of vertices leaves at most one free edge and at most one otherconnected component, then the decomposition space has no cut pairs.
Unfortunately, this corollary does not apply if a Whitehead graph has more thanone edge between a pair of vertices. Indeed, consider the pattern in F = (cid:104) a, b (cid:105) generated by the word a ba ¯ b . The Whitehead graph in Figure 15 is reduced andcontains the complete graph on the four vertices, but q ( { a ∞ , a −∞ } ) is a cut pair,as is evident from Figure 16.4.5. Cut Sets When There are No Cut Pairs.
Let S be a finite set in thedecomposition space, and let H be the convex hull of q − ( S ). Lemma 4.7 tells usthat S is a cut set if and only if Wh( H ) is disconnected. We will pass to a finitesubset of H whose Whitehead graph contains the same connectivity information.Define the core C of q − ( S ), to be the smallest closed, connected set such that H \ C is a collection of disjoint infinite geodesic rays R j : [1 , ∞ ] → T . We use R j (0)to denote the vertex of the core that is adjacent to R j (1). Figure 15.
Wh( ∗ ) { a ba ¯ b } Figure 16.
Wh([ ∗ , a ]) { a ba ¯ b } \ [ a − , a ]Let ξ be a point in ∂ T . If q ( ξ ) is either a cut point or a bad point that is amember of a cut pair, it is not hard to see that there is a geodesic ray R with R ( ∞ ) = ξ such that Wh( R ([1 , ∞ )) \ R (0) is not connected.Conversely, if there exists a geodesic ray R : [0 , ∞ ] → T with R ( ∞ ) = ξ such that Wh( R ([1 , ∞ )) \ R (0) is not connected, then, arguing as in the proofof Lemma 4.12, q ( ξ ) is either a cut point or is a bad point that belongs to a cutpair.If there are no cut points or cut pairs, then Wh( R ([1 , ∞ )) \ R (0) is connectedfor any ray R . Proposition 4.17.
Suppose ξ is a point in ∂ T such that q ( ξ ) is a bad point that isnot a member of a cut pair. Then q ( ξ ) is not a member of any minimal finite cutset. In particular, if D has no cut pairs then no bad point belongs to any minimalfinite cut set.Proof. The assumption that q ( ξ ) is not a member of a cut pair implies that for anyray R : [0 , ∞ ] → T with R ( ∞ ) = ξ , the Whitehead graph Wh( R ([1 , ∞ ])) \ R (0) isconnected.Let S be a finite cut set in D with q ( ξ ) ∈ S . Let H and C be the hull and core of q − ( S ), respectively. Consider the ray R that is the component of H \ C containing ξ . Components of D \ S are in bijection with components of Wh( H ), which, in turn,are in bijection with components of Wh( H \ R ([1 , ∞ ])), since Wh( R ([1 , ∞ ])) \ R (0)is connected. This is just the hull of q − ( S \ { q ( ξ ) } ).Thus, S \ { q ( ξ ) } is still a cut set, so S was not minimal. (cid:3) For a finite collection of lines ˜ S = { l , . . . , l k } ⊂ L , the core is a finite tree. Theconvex hull minus the core is a collection of 2 k disjoint rays: {R (cid:15)i : [1 , ∞ ] → T | lim t →∞ R (cid:15)i ( t ) = l (cid:15)i , (cid:15) ∈ { + , −} , i = 1 . . . k } Lemma 4.18.
Let S be a finite set of good points of D , none of which is a cutpoint. Components of D \ S are in bijection with components of Wh( C ) \ ˜ S Proof.
Let ˜ S = { l , . . . , l k } .For each i and (cid:15) , since q ∗ ( l i ) is not a cut point, Wh( R (cid:15)i ([1 , ∞ ])) \ R (cid:15)i (0) isconnected.Wh( H ) is obtained from Wh( C ) \{ ¨ l , . . . , ¨ l k } by splicing on each Wh( R (cid:15)i ([1 , ∞ ])) \R (cid:15)i (0). ine Patterns in Free Groups 25 This means to each deleted vertex of Wh( C ) \ { ¨ l , . . . , ¨ l k } we have spliced on aconnected graph, so we might have just as well not deleted those vertices. (cid:3) In fact, we can use the argument of Lemma 4.18 to reduce the convex hull evenfurther. If C is not just a vertex, then it has some valence one vertices, that we call leaves . The edge connecting a leaf to the rest of the core is called the stem .Suppose that for some leaf v of C every line of the ˜ S that goes through v goesthrough the stem of v . From Wh( v ), delete the interiors of edges corresponding tothe l i and the vertex corresponding to the stem. The resulting graph is connected,so connected components of Wh( C ) \ ˜ S are in bijection with connected componentsof Wh( C \ { v } ) \ ˜ S .Thus, we may prune some leaves off of the core without changing the connectivityof the Whitehead graphs.If ˜ S is not an edge cut set then we may prune the core down to a well definednonempty tree p C , the pruned core , such that every leaf contains a line of ˜ S thatdoes not go through the stem.If ˜ S is an edge cut set then the core can be pruned down to a pair of adjacentvertices, both of which look like prunable leaves. In this case define p C to be thesetwo vertices. Proposition 4.19.
An edge cut set that does not contain a cut point is minimal.Proof.
Let ˜ S = { l , . . . , l k } be the set of lines of L going through an edge e of T ,so that S = q ∗ ( ˜ S ) is an edge cut set. Let p C be the pruned core. There are twoconnected components of Wh( p C ) \ ˜ S ; they lie on opposite sides of e . By Lemma 4.18these correspond to two connected components of D \ S .Each of the l i has one endpoint in each component, so if any l i is omitted fromthe set the two components will have a point in common. (cid:3) Corollary 4.20. If D has no cut pairs, the good points and bad points are topo-logically distinguished.Proof. Bad points are the points that do not belong to any minimal finite cut set.Good points are the points that do. (cid:3)
Proposition 4.21.
Let S be a minimal finite cut set, none of whose elements aremembers of a cut pair. There are exactly two connected components of D \ S .Proof. By Proposition 4.17, S consists of good points. Let ˜ S = { l , . . . , l k } = q − ∗ ( S ). Components of D \ S are in bijection with components of Wh( C ) \ ˜ S . Thisis a finite graph, so D \ S has only finitely many components.Let A , . . . , A m be a list of the components of D \ S .If q ∗ ( l i ) is not a limit point of A j in D then A j is still a connected component in D \ ( S \ q ∗ ( l i )). This contradicts minimality of S , so each point of S is a limit pointin D of every A j . This implies that for each i and j , at least one of the points l + i and l − i is a limit point of q − ( A j ).Now H \ C is a collection of disjoint rays R (cid:15)i . The graph Wh( R (cid:15)i ([1 , ∞ ])) \ R (cid:15)i (0)is connected, so no l + i or l − i is a limit point of more than one q − ( A j ).Thus, there are exactly two components A and A of D \ S , and each line l i has one endpoint in q − ( A ) and the other in q − ( A ). (cid:3) Corollary 4.22.
Let S be a minimal finite cut set that is not an edge cut set, noneof whose elements are members of a cut pair. For every vertex v ∈ p C , the portionof Wh( p C ) \ ˜ S at v contains an edge from each component of Wh( p C ) \ ˜ S .Proof. If v is a leaf such that the portion of Wh( p C ) \ ˜ S at v belongs to a singlecomponent of Wh( p C ) \ ˜ S then v should have been pruned off.If v is not a leaf, p C \ { v } has at least two components. If the Whitehead graphover one of those components sees only one component of Wh( p C ) \ ˜ S then it wouldhave been possible to prune it off. Thus, every component of p C \ { v } must see twocomponents of Wh( p C ) \ ˜ S . There are only two components of Wh( p C ) \ ˜ S , so bothmust be able to connect to all components of p C \ { v } . In particular, they mustconnect through v . (cid:3) Indecomposable Cut Sets.
In this section we will assume that the decom-position space has no cut pairs.Our ultimate goal is to construct a cubing quasi-isometric to a bounded valencetree. For this purpose, we will need to choose a collection of cut sets in a such away that there is a bound on the number of cut sets in the collection that cross anyfixed cut set in the collection.Cut sets with disjoint pruned cores do not cross, so we could control crossings ifwe could control the diameters of the pruned cores of the cut sets in some collection.The following example shows that cut sets of a fixed size can have pruned coreswith arbitrarily large diameter. We subsequently introduce the property of inde-composability to rule out this kind of bad behavior.Let L be the line pattern in F = (cid:104) a, b (cid:105) generated by the words ab ¯ a ¯ b , a and b .The edge cut sets have size three. It can be shown that these are the only cut setsof size three and there are none smaller, see Section 6.2. It is also possible to findminimal cut sets of size four. Pick any two of the edge cut sets that share a line.The four lines of the symmetric difference are a minimal cut set. Figure 17 showsthe line pattern. The two dashed lines indicate edge cut sets of size three. The fourthickened lines make up the cut set of size four that is the symmetric difference.There is no bound on the size of the pruned core of such a cut set, nor on thenumber of such cut sets that cross each other. Figure 17.
A problematic minimal cut set ine Patterns in Free Groups 27
We say that a minimal finite cut set S ⊂ D is decomposable if there are minimalcut sets Q and R such that:(1) Q and R are non-crossing,(2) | Q | < | S | and | R | < | S | ,(3) S = Q ∆ R = ( Q \ R ) ∪ ( R \ Q )A minimal finite cut set S is indecomposable if it is not decomposable.The smallest cut sets in D are indecomposable since there are no smaller cutsets to decompose them into. Lemma 4.23.
Suppose S is a finite minimal cut set and the pruned core p C of ˜ S has an interior vertex v such that Wh( v ) \ p C has exactly two components, one ofwhich is a free edge, and no lines of ˜ S go through v . Then S is decomposable. l vQ R X X Y Y Figure 18.
Schematic diagram of decomposable cut set
Proof.
Let l be the line of L corresponding to the free edge in Wh( v ) \ p C . Let ˜ Q be l and the lines of ˜ S on one side of p C \ { v } , and let ˜ R be l and the lines of ˜ S onthe other side of p C \ { v } .Then Q ∩ R = q ∗ ( l ), and S = Q ∆ R .Let A and A be the components of D \ S . The line l does not belong to ˜ S , sowe may assume that q ∗ ( l ) ∈ A . Let X and Y be the two components of p C \ { v } .We may assume Q is on the X side.Let X be the portion X corresponding to A , and define X , Y and Y anal-ogously, see Figure 18. The edge of Wh( p C ) \ ˜ S corresponding to l is the onlyconnection between X and Y , so ˜ Q separates X from X ∪ Y ∪ Y .Thus, Q is a cut set. Moreover, Q is a minimal cut set since every edge corre-sponding to a line in ˜ Q has one end in X and one end in X ∪ Y ∪ Y .By a similar argument, R is a minimal cut set. Q and R are non-crossing because the only line of R that has an endpoint in X is l = ˜ Q ∩ ˜ R .Finally, as there are no cut pairs, we have:3 ≤ | Q | = | Q \ R | + | Q ∩ R | = | Q \ R | + 1 = ⇒ | Q \ R | ≥ | S | = | Q \ R | + | R \ Q | > | R \ Q | = | R ∩ Q | + | R \ Q | = | R | Similarly, | Q | < | S | . (cid:3) Theorem 4.24 (Edge cut sets are indecomposable) . Suppose we have chosen afree basis of F such that Wh( ∗ ) is minimal complexity. Then edge cut sets areindecomposable.Proof. Let e be an edge of T . Let ˜ S be the lines of L that cross e . Let S = q ∗ ( ˜ S ). S is minimal by Proposition 4.19. Suppose S decomposes into Q and R . Wemust have Q ∩ R (cid:54) = ∅ , otherwise Q and R are proper subsets of S that are cut sets,contradicting minimality of S . Since Q and R do not cross and S \ R = Q \ R , S doesnot cross R . Thus, since they are minimal, R does not cross S . Therefore, R \ S = R ∩ Q is contained in one component of D \ S . This means that q − ∗ ( Q ∩ R ) = ˜ Q ∩ ˜ R is contained in one component of T \ e .Let ∗ be the vertex of T incident to e on the ˜ Q ∩ ˜ R side.It is possible that pruning the cores of ˜ Q or ˜ R would remove ∗ . Let the partiallypruned core of ˜ Q , pp C ˜ Q , be the result of pruning the core of ˜ Q as much as possiblewithout pruning off ∗ . Note pp C ˜ Q = pp C ˜ R , so we may just call it pp C . | R \ Q | + | Q ∩ R | = | R | < | S | = | Q \ R | + | R \ Q | So | Q ∩ R | < | Q \ R | . Similarly, | Q ∩ R | < | R \ Q | .There are two connected components of Wh( pp C ) \ ˜ Q , call them component 0and component 1. Since Q and S do not cross, everything on the side of e opposite˜ Q ∩ ˜ R belongs to a single component.First suppose pp C = ∗ . Suppose the edge e oriented away from ∗ has label x ∈ B ∪ ¯ B ; suppose the corresponding vertex in Wh( pp C ) \ ˜ Q is in component 1.Suppose the vertex corresponding to the edge labeled ¯ x is in component 0. Thenthe Whitehead automorphism that pushes the vertices in Wh( ∗ ) in component 1through x changes the valence at x from | S | = | Q \ R | + | R \ Q | to | Q ∩ R | + | R \ Q | .Since | Q ∩ R | < | Q \ R | this contradicts the assumption that the Whitehead graphhad minimal complexity.Conversely, if the vertex ¯ x is in component 1 push Z = { x } ∪ { vertices of component 0 } through x and get a contradiction.Now suppose pp C is not just ∗ . Then there is some leaf v (cid:54) = ∗ . Suppose the stemof v (oriented away from the leaf) has label x ∈ B ∪ ¯ B , and suppose the vertex inWh( pp C ) \ ˜ Q corresponding to ¯ x is in component 1.Figure 19 shows a schematic diagram of Wh( pp C ).The labeling in the diagram is as follows: • X = the portion of component 0 on the v side of the stem. • X = the portion of component 1 on the v side of the stem. • Y = the portion of component 0 between the stem of v and e . • Y = the portion of component 1 between the stem of v and e . • Z = everything on the side of e opposite ˜ Q ∩ ˜ R . • lowercase letters represent the number of lines with endpoints in the spec-ified regions with: – a , b , c and h counting the lines of ˜ Q ∩ ˜ R – d and i counting the lines of ˜ Q \ ˜ R – e and j counting the lines of ˜ R \ ˜ Q – f and g counting the lines not in ˜ Q ∪ ˜ R crossing the stem. ine Patterns in Free Groups 29 * va bc de fg h ij X X Y Y Z QR S
Figure 19.
Schematic diagram for Wh( pp C )Since v (cid:54) = ∗ is a leaf of pp C we must have a > X and X nonempty.Minimality of Q implies that Wh( pp C ) \ ˜ Q has exactly two connected components.In the diagram they are X ∪ Y and X ∪ Y ∪ Z . X ∪ Y must belong to a connected component, so Y = ∅ if and only if f = 0. Y = ∅ also implies c = h = i = 0. Now d + i = | Q \ R | , so this would imply d > R is also a minimal cut set, so Wh( pp C ) \ ˜ R must have exactly two components.In the diagram they are X ∪ Y ∪ Z and X ∪ Y . Since X ∪ Y is connected, Y = ∅ if and only if g = 0.Thus, we have:(1) Y = ∅ ⇐⇒ f = 0 = ⇒ d > , c = h = i = 0(2) Y = ∅ ⇐⇒ g = 0 = ⇒ e > , b = h = j = 0The Whitehead automorphism that pushes Z = { x } ∪ { vertices of Wh( v ) in component 0 of Wh( pp C ) \ ˜ Q } through the stem changes the valence of vertex x from b + c + d + e + f + g to a + c + e + g . By our minimal complexity assumption, we must therefore have a ≥ b + d + f .Now | Q \ R | > | Q ∩ R | ≥ a + b + c ≥ b + c + d + f , which means that | Q \ R | > d ,so i > Q and R to a new decomposing pair Q (cid:48) and R (cid:48) for S withstrictly smaller partially pruned core.Let ˜ Q (cid:48) \ ˜ R (cid:48) be the lines of ˜ Q \ ˜ R that do not pass through v .Let ˜ R (cid:48) \ ˜ Q (cid:48) be the rest of ˜ S .Let ˜ Q (cid:48) ∩ ˜ R (cid:48) be ˜ Q ∩ ˜ R minus the lines contributing to a and b plus the linescontributing to f . | S | − | R (cid:48) | = | Q (cid:48) \ R (cid:48) | − | Q (cid:48) ∩ R (cid:48) | = | Q \ R | − d − ( | Q ∩ R | − a − b + f )= | Q \ R | − | Q ∩ R | + a + b − d − f ≥ | Q \ R | − | Q ∩ R | + ( b + d + f ) + b − d − f = | S | − | R | + 2 b ≥ | S | − | R | >
0A similar computation shows | S | − | Q (cid:48) | ≥ | S | − | Q | + 2( b + d ) > Q (cid:48) and R (cid:48) are non-crossing minimal cut sets. * va bc de fg h ij X X Y Y Z Q (cid:48) R (cid:48) S Figure 20.
Schematic diagram for modified Wh( pp C )The components of Wh( pp C ) \ ˜ Q (cid:48) are Y and X ∪ X ∪ Y ∪ Z . To see that thelatter is connected, note that a > e + j > Y = ∅ or g > Q (cid:48) is a minimal cut set since Wh( pp C ) \ ˜ Q (cid:48) has exactly two connectedcomponents and every line of ˜ Q (cid:48) goes from one component to the other.By similar considerations, R (cid:48) is a minimal cut set since Wh( pp C ) \ ˜ R (cid:48) has com-ponents Y ∪ Z and X ∪ X ∪ Y .That Q (cid:48) and R (cid:48) are non-crossing follows from the observation: Y ∩ ( X ∪ X ∪ Y ) = ∅ We have not added anything to ˜ Q (cid:48) ∩ ˜ R (cid:48) except possibly some lines going through v , so the new partially pruned core is contained in the old one minus the vertex v .If ¯ x is in component 0, repeat the argument with the roles of Q and R reversedand reach a similar conclusion.Thus, by repeating this process, we can reduce the partially pruned core until wefind some decomposing pair Q and R so that the partially pruned core is just ∗ . Wehave already seen that that leads to a contradiction, so S is indecomposable. (cid:3) Theorem 4.25 (Indecomposables are bounded) . The diameter of the pruned coreof an indecomposable cut set S is bounded in terms of L and | S | .Proof. If p C is a point or two points we are done. Otherwise it is a tree with leaves.Each leaf contains a line from ˜ S that does not go through its stem, so there are atmost | S | leaves.Suppose X is a segment of p C that does not have any lines of ˜ S going through it.By Corollary 4.22, at every vertex of p C there are edges of Wh( p C ) \ ˜ S from bothcomponents. Since S is indecomposable, by Lemma 4.23 it is not the case that oneof these components is a free edge. Now apply Corollary 4.14 and conclude thatthere is a bound R on the length X .Similarly, if X is a segment of p C that meets exactly one of the l i then it haslength bounded by R .It follows that the diameter of p C is at most 2 R ( | S | − (cid:3) Rigidity
The Problem with Cut Pairs. If D has cut pairs then it has either a cutpoint or a bad cut pair, by Corollary 4.13. In either case, there is a cut set suchthat the preimage in ∂ T is two points { g ∞ , g −∞ } . The convex hull H of these two ine Patterns in Free Groups 31 points is a line, and Wh( H ) has multiple components, A , . . . , A k . For each i , let X i be the union of components of T \ H corresponding to A i .The action of g may permute these components, but g k ! fixes them.Let φ : T → T be the quasi-isometry: φ ( x ) = (cid:40) g k ! x if x ∈ X x otherwiseThis “shearing” quasi-isometry moves the X component along H , fixing the restof the tree.It is not hard to see that φ n is not bounded distance from an isometry for n (cid:54) = 0.Since F acts by isometries it follows that F φ a and F φ b are not the same coset of F in QI ( F, L ) when a (cid:54) = b , so F is infinite index subgroup.It is possible to show directly that φ could not be conjugate into an isometrygroup. Alternatively, notice that we can stack shearing quasi-isometries to producea sequence of quasi-isometries with unbounded multiplicative quasi-isometry con-stants, see Figure 21. Take an element h of F such that h H is contained in the X component with h X ⊂ X .The desired sequence of quasi-isometries is (Φ i ), where:Φ i = φ i ◦ ( hφ i ¯ h ) ◦ ( h φ i ¯ h ) ◦ · · · That is, for any x ∈ T there exists some j such that x ∈ h j X \ h j +1 X . For m > jh m φ i ¯ h m ( x ) = x , so:Φ i ( x ) = φ i ◦ ( hφ i ¯ h ) ◦ ( h φ i ¯ h ) ◦ · · · ◦ ( h j φ i ¯ h j )( x ) h H h H h HH Φ −→ Figure 21.
Shearing5.2.
Rigidity When There are No Cut Pairs.
Let L be a line pattern in F .Choose a free basis B for F so that Wh( ∗ ) = Wh B ( ∗ ) {L} has minimal complexity,and let T be the Cayley graph of F with respect to B . Assume that D = D L hasno cut pairs. We will construct a cubing X , a quasi-isometry φ : T → X , and anisometric action of QI ( F, L ) on X that agrees with φ QI ( F, L ) φ − ⊂ Isom( X ), upto bounded distance , completing the proof of the Main Theorem. The action of F on X will be cocompact, implying that QI ( F, L ) has a complex of groups structure. Constructing the Cubing.
Let b be the maximum valence of a vertex inWh( ∗ ). Let { S i } i ∈ I be the collection of indecomposable cut sets of size at most b in D . For each i ∈ I , S i is a finite minimal cut set, by definition, and, since there areno cut pairs, D \ S i has exactly two connected components, by Proposition 4.21.Let the two connected components of D \ S i be A i and A i . LetΣ = { A i } i ∈ I ∪ { A i } i ∈ I Recall from Section 2.1 that from this information we define a graph as follows:A vertex is a subset V of Σ such that:(1) For all i ∈ I exactly one of A i or A i is in V .(2) If C ∈ V and C (cid:48) ∈ Σ with C ⊂ C (cid:48) then C (cid:48) ∈ V .Two vertices are connected by an edge if they differ by only one set in Σ.This gives a graph; it remains to select a path connected component of this graphto be the 1-skeleton of the cubing.Define a bad triple ¯ x = { x , x , x } to be an unordered triple of distinct badpoints in D .There are no bad points in minimal cut sets, so for any bad triple and any S i ,¯ x ⊂ A i ∪ A i . We let ¯ x decide democratically whether it will associate with A i or A i : say ¯ x ∈ A (cid:15)i if at least two of the x j ’s are in A (cid:15)i .Define V ¯ x = { A (cid:15)i ∈ Σ | ¯ x ∈ A (cid:15)i } . This is a vertex of X . Define the 0-skeletonof the cubing to be the set X (0) of all vertices that are connected by a finite edgepath to V ¯ x for some bad triple ¯ x .The following lemma replaces Lemma 3.4 of [15]. Lemma 5.1.
For any bad triples ¯ x and ¯ y , there are only finitely many S i separatingthem.Proof. Let ¯ x = { x , x , x } and ¯ y = { y , y , y } be bad triples.The preimage q − (¯ x ) = { q − ( x i ) } i =1 , , consists of three distinct points in ∂ T .The convex hull of three points in the boundary of a tree is a tripod. The core,as previously defined, is the unique vertex that is the branch point of the tripod.Denote this point C ¯ x .It is not hard to see that a cut set S i separates ¯ x from ¯ y only if the pruned core p C of S i intersects the finite geodesic segment joining C ¯ x and C ¯ y in T .By Theorem 4.25, there is a uniform bound a on the diameter of the pruned coreof any S i . Since L is locally finite, this means there is a uniform bound c on thenumber of S i such that ∗ ∈ p C S i . If Y is any finite collection of vertices in T , thenumber of S i such that p C S i ∩ Y (cid:54) = ∅ is at most c |Y| .Thus, the number of S i separating ¯ x from ¯ y is at most c · (1 + d T ( C ¯ x , C ¯ y )). (cid:3) Add edges to the 0-skeleton as above to get the 1-skeleton X (1) of the cubing.With Lemma 5.1 replacing Lemma 3.4 of [15], the following theorem follows by thesame proof as in [15]: Theorem 5.2. [15, Theorem 3.3] X (1) is connected. The rest of the construction of the cubing follows as in Section 2.1.
Remark.
We are forced to use this alternate way of choosing the vertices of thecubing because every good point in D belongs to infinitely many of the cut sets.Also, Lemma 5.1 is false if one tries to use just bad points instead of bad triples.Two bad points are separated by infinitely many of the S i . ine Patterns in Free Groups 33 Remark.
For a fixed vertex v ∈ T , there are uncountably many bad triples ¯ x with C ¯ x = v . However, these give only finitely many distinct vertices V ¯ x in X , becausethe V ¯ x can only differ in the finitely many coordinates i such that the pruned coreof S i contains v . Even this is an over count. If S e is an edge cut set associated to anedge e incident to v , then every bad triple with C ¯ x = v lies in the same componentof D \ S e . If our set of indecomposables is exactly the collection of edge cut setsthen the cubing is isomorphic to the tree T .Notice X is defined in terms of the topology of D , so we have: Lemma 5.3.
Any homeomorphism of D induces an isomorphism of X . Estimates on the Cubing.
Recall from the proof of Lemma 5.1, we have abound a on the diameter of the pruned core of any S i , and there is a c such that if Y is any finite collection of vertices in T , the number of S i such that p C S i ∩ Y (cid:54) = ∅ is at most c |Y| . S i and S j are non-crossing if their pruned cores are disjoint, so we have a uniformbound c (2 n ) a on the number of S j that cross a fixed S i .A k -cube in X corresponds to a collection of k pairwise crossing cut sets, so thecubing is finite-dimensional.Pick a vertex x ∈ X . Let e and e (cid:48) be edges incident to x . There are distincthyperplanes H e and H e (cid:48) associated to these edges. Since e and e (cid:48) are incident toa common vertex, there is no third hyperplane separating H e from H e (cid:48) . Therefore,the valence of a vertex in X is bounded by the maximum size of a subcollection { S i } i ∈ J of the indecomposable cut sets such that for any j and k in J , there is no i ∈ I such that S i separates S j and S k . If S j and S k have disjoint pruned coresthen there is an edge cut set separating them, so the maximum size of the set J isat most c (2 n ) a . Thus, X is uniformly locally finite.A hyperplane H corresponds to an equivalence class of edges in X . The 1-neighborhood of H is the set of vertices that are endpoints of these edges. If k is the number of hyperplanes crossing H , then the 1-neighborhood of H has atmost 2 k +1 vertices and diameter at most k + 1. Crossing hyperplanes correspondto crossing cut sets, so k is at most c (2 n ) a .5.2.3. The Rigidity Theorem.
Theorem 5.4. X is quasi-isometric to T .Proof. For each edge e ∈ T there is a corresponding edge cut set S e . By construc-tion, S e ∈ { S i } , so in the cubing X there is a corresponding hyperplane H e . Define φ ( e ) to be the set of vertices in the 1-neighborhood of H e . Recall from the previoussection that this is a set of boundedly many vertices with bounded diameter. d X ( φ ( e ) , φ ( e (cid:48) )) is the number of hyperplanes separating H e and H e (cid:48) . This is atleast the number of edges separating e from e (cid:48) in T , which is d T ( e, e (cid:48) ), and at mostthe number of { S i } such that p C S i meets the geodesic between e and e (cid:48) in T , whichis bounded by c · d T ( e, e (cid:48) ). This shows that φ is a quasi-isometric embedding.Suppose there is a vertex x ∈ X not in the image of φ . This x has some incidentedge, corresponding to some S ∈ { S i } . The hypothesis that x is not in the image of φ implies that S does not cross any edge cut set, which means that p C s is a singlevertex v ∈ T . There are boundedly many such S , and the distance from x to φ ( T )is less than this bound, so φ is coarsely onto, hence a quasi-isometry. (cid:3) The quasi-isometry φ gives a collection of quasi-lines φ ( L ) in X . In fact, wecan see this collection of quasi-lines directly from D . Each good point in D belongs to infinitely many indecomposable cut sets. For l ∈ L , the collection { S | S indecomposable , | S | ≤ b, q ∗ ( l ) ∈ S } corresponds to a collection of hyper-planes in X . The union of these hyperplanes is coarsely equivalent to φ ( l ). Theorem 5.5 (Rigidity Theorem) . For i = 1 , , let F i be a free group with linepattern L i . Let D i be the decomposition space corresponding to L i in F i .Suppose, for each i , D i has no cut pairs.Let φ i : F i → X i be the quasi-isometry to the cube complex constructed above.Then: φ QI{ ( F , L ) → ( F , L ) } φ − = Isom { ( X , φ ( L )) → ( X , φ ( L )) } Proof.
Elements of
QI{ ( F , L ) → ( F , L ) } give homeomorphisms D → D .Homeomorphism take indecomposable cut sets to indecomposable cut sets of thesame size and preserve crossing and intersection. Therefore, we get isometries X → X respecting the line patterns. (cid:3) The Rigidity Theorem answers Questions 1 and 2 for rigid patterns.The free group acts on itself by pattern preserving isometries via left multiplica-tion. Let ∗ be the identity vertex in T . For any indecomposable cut set S , there isan element g ∈ F such that ∗ ∈ g ( p C S ). There are only finitely many indecompos-able cut sets of bounded size with ∗ ∈ p C , so F acts cocompactly on X . Therefore, QI ( F, L ) ∼ = Isom( X, φ ( L )) acts cocompactly on X . This gives an explicit finitepresentation for Isom( X, φ ( L )) as a complex of groups. Moreover, as the F actionis already cocompact, we have: Corollary 5.6. If L is a rigid line pattern and if QI ( F, L ) acts on X with finitestabilizers then F is a finite index subgroup of QI ( F, L ) . Examples
Whitehead Graph is the Circle.
We will show in this section that whenthe Whitehead graph is a circle we get a quasi-isometrically flexible line pattern.
Theorem 6.1.
For a line pattern L in F , the following are equivalent: (1) Every Whitehead graph Wh B ( ∗ ) that has no cut vertex is a circle. (2) Some Whitehead graph Wh B ( ∗ ) is a circle. (3) D is a circle. (4) Every minimal cut set of D has size two.Proof. Clearly (1) = ⇒ (2), because Whitehead automorphisms will eliminate cutvertices.If some Whitehead graph Wh B ( ∗ ) is a circle then we can realize the free group F n as the fundamental group of a surface with boundary, and the generators of theline pattern L as the boundary labels. We can give this surface a hyperbolic metricso that the universal cover is just T fattened, and the boundary components arehorocycles that are in bijection with the lines of L . This gives us a homeomorphismbetween the decomposition space and S = ∂ H . Thus (2) = ⇒ (3).(3) = ⇒ (4) is a topological property of circles.Now, suppose every minimal cut set of D has size two. Then D is connectedwith no cut points. Choose a free basis B so that Wh B ( ∗ ) is connected without ine Patterns in Free Groups 35 cut points. The edges incident to a vertex of Wh B ( ∗ ) correspond to an edge cutset. This is a minimal cut set by Proposition 4.19, so by hypothesis has size two.Therefore, Wh B ( ∗ ) is a finite, connected graph with all valences equal to two, hence,a circle. Thus, (4) = ⇒ (1). (cid:3) Remark.
Otal proves [13, Theorem 2] that the decomposition space is a circle ifand only if the the collection of words can be represented as the boundary curvesa compact surface. The proof is essentially the same.
Theorem 6.2.
Let F and F (cid:48) be free groups, possibly of different rank. Let L and L (cid:48) be line patterns in F and F (cid:48) , respectively. Suppose D L is a circle. There is apattern preserving quasi-isometry from F to F (cid:48) if and only if D L (cid:48) is also a circle.Proof. The “only if” direction is clear, as a pattern preserving quasi-isometry in-duces a homeomorphism of decomposition spaces.Suppose both D L and D L (cid:48) are circles. By Theorem 6.1, there exist free bases B of F and B (cid:48) of F (cid:48) such that Wh B ( ∗ ) {L} and Wh B (cid:48) ( ∗ ) {L (cid:48) } are circles.As in the proof of Theorem 6.1 we can associate each pattern with the boundarycurves of the universal cover of a surface with boundary. It is a theorem of Behrstockand Neumann [1] that there are many boundary preserving quasi-isometries of suchsurfaces. (cid:3) For example, recall the example from the Introduction. Let F = F = (cid:104) a, b (cid:105) .Let L be the line pattern generated by the word ab ¯ a ¯ b .Let L be the line pattern generated by the words ab and a ¯ b .For each of these Wh( ∗ ) {L i } is a circle, so the two patterns are quasi-isometricallyequivalent.This example also shows that neither the number of generators of a line patternnor the widths of the generators are quasi-isometry invariants.6.2. Whitehead Graph is the Complete Graph.
Let K n be the completegraph on 2 n vertices, the graph consisting of 2 n vertices with exactly one edgejoining each pair of vertices.Suppose L is a line pattern in F = F n so that for some free basis B , Wh B ( ∗ ) {L} = K n .The decomposition space D has no cut pairs.Suppose S is a minimal finite cut set of D that is not an edge cut set. Wh( p C ) \ ˜ S has two components. The portion of Wh( p C ) \ ˜ S at a leaf contains vertices fromboth components.The portion of Wh( p C ) \ ˜ S at a leaf is a graph obtained from K n be deleting avertex, corresponding to the stem of the leaf, and interiors of some number of edgescoming from lines of ˜ S that go through the leaf but not through the stem. Theresult is a disconnected graph with at least one vertex in each of the components.Thus, we have partition of 2 n − m and 2 n − − m , for some1 ≤ m ≤ n −
2, and the number of edges between them is m (2 n − − m ) ≥ n − | S | ≥ n − > n −
1. The edge cut sets havesize 2 n −
1, so our construction of a cubing uses only the edge cut sets. Thus, thecubing is just the tree T .In this case it is easy to compute: QI ( F, L ) ∼ = Sym(2 n ) ∗ Sym(2 n − (Sym(2 n − × Sym(2))
Here, Sym(2 n ) is the symmetric group on 2 n objects, stabilizing a vertex of thetree and permuting the incident edges, and Sym(2 n − × Sym(2) is the stabilizerof an edge of T .This discussion proves the following theorem: Theorem 6.3.
Suppose L is a line pattern in F = F n such that Wh B ( ∗ ) {L} = K n .Suppose that F (cid:48) = F m is another free group, possibly of different rank, with linepattern L (cid:48) .There is a pattern-preserving quasi-isometry F → F (cid:48) if and only if D L (cid:48) has thefollowing properties: (1) There are no cut sets of size less than n − . (2) The collection of cut sets of size n − yields a cubing that is a n -valenttree. (3) The induced line pattern in the cubing restricts to the complete graph K n in the star of any vertex. For example, the line pattern L in F = F with basis B = { a, b } generated by a , b , and ab ¯ a ¯ b has Whitehead graph Wh B ( ∗ ) {L} = K .Compare this to the line pattern L (cid:48) in F (cid:48) = F with basis B (cid:48) = { x, y, z } generatedby y , zx , z ¯ x ¯ y and xy ¯ z . The Whitehead graph Wh B (cid:48) ( ∗ ) {L (cid:48) } looks like two copiesof K spliced together, see Figure 22. xz ¯ x ¯ zy ¯ y Figure 22. Wh { x,y,z } ( ∗ ) { y, zx, z ¯ x ¯ y, xy ¯ z } It is not hard to show that the smallest cut sets are the obvious ones of size three.These yield a cubing that is a 4-valent tree, essentially blowing up each vertex of F into a pair of vertices.This pattern is quasi-isometric to the K pattern in F .6.3. A Rigid Example for which the Free Group is not Finite Index in theGroup of Pattern Preserving Quasi-isometries.
Consider the line pattern in F = (cid:104) a, b (cid:105) generated by the words a , b , and aba ¯ b ¯ ab ¯ a ¯ b . Let T be the Cayley graphof F with respect to { a, b } .It is easy to check that Whitehead graph in Figure 23 is reduced and the decom-position space has no cut pairs, so the pattern is rigid.The edge cut sets have size five. Deleting any vertex of the Whitehead graphleaves a graph that requires at least three more edges to be deleted to disconnectthe graph. Thus, any other cut sets have size at least six. As the edge cut sets arethe only cut sets of size less than or equal to five, the rigid cube complex is justthe tree T . ine Patterns in Free Groups 37 Figure 23.
Wh( ∗ ) { a, b, aba ¯ b ¯ ab ¯ a ¯ b } (loose)We will show that F is not a finite index subgroup of QI ( F, L ). Not only arethe vertex stabilizers in QI ( F, L ) not finite, they are not even finitely generated.Define an isometry φ of T piecewise as follows. First, note that the automorphism α of F that exchanges a with ¯ a preserves the pattern. It inverts a , fixes b , and takes aba ¯ b ¯ ab ¯ a ¯ b to a cyclic permutation of itself. To the branch of the tree consisting ofwords beginning with b , apply the automorphism α . To each branch of the treebeginning with a n b for some n , apply the automorphsim a n ◦ α ◦ ¯ a n . Fix the restof the tree.The isometry φ is built piecewise from pattern preserving automorphisms of F .It fixes the “bottom half” of T , fixes the b –line through a n for each n , and reflectseach branch beginning with a n b through the b –line through a n .There are lines of the pattern that pass through multiple pieces of the domainof φ , so we check that the φ is defined consistently for these lines. As illustratedin Figure 24, the only lines shared by the bottom half of the tree and the verticalbranches are the fixed b –lines (green lines in the figure are fixed). The reflectionsin adjacent vertical branches agree on the two lines they share (the two thickenedblue lines are exchanged). Therefore, φ pieces together to give a pattern preservingisometry.Thus, for any n , b n ◦ φ ◦ ¯ b n is a pattern preserving isometry that fixes every linein the n -neighborhood of the identity vertex, but is not the identity map. It followsthat the stabilizer of the identity vertex is not finitely generated.6.4. A Cube Complex That is Not a Tree.
Finally, we give an example of arigid line pattern for which our argument does not produce a cube complex that isa tree.Consider the line pattern in F = (cid:104) a, b, c (cid:105) generated by the four words ¯ abc , ¯ acb ,¯ ab and ¯ ac . The Whitehead graph (with loose ends), is shown in Figure 25.The reader may verify that this is a minimal Whitehead graph and there are nocut points or cut pairs in the decomposition space. In fact, the smallest cut setsare the edge cut sets of size four corresponding to the a –edges. These are the onlycut sets of size four.The other edge cut sets have size five, so we construct a cube complex usingindecomposable cut sets of size four and five. Figure 26 depicts the Whiteheadgraph (along with portions of the Whitehead graph over two neighboring vertices)with the 1–skeleton of the cube complex overlaid. ∗ ¯ a b ¯ ab Figure 24.
Wh([¯ ab, b ]) { a, b, aba ¯ b ¯ ab ¯ a ¯ b } (loose) ab c ¯ a ¯ b ¯ c Figure 25.
Wh( ∗ ) { ¯ abc, ¯ acb, ¯ ab , ¯ ac } Note that every cut set of size five is crossed by another cut set of size five.However, the edge cut sets are still topologically distinguished! They are the cutsets of size five that are crossed minimally (once) by another cut set of size five.The other cut sets of size five are crossed by either two or five other cut sets of sizefive.Had we said, “build the cube complex associated to the cut sets of size four andthose of size five that are crossed by exactly one other cut set of size five” we wouldhave recovered the tree as the cube complex. ine Patterns in Free Groups 39
Figure 26.
Whitehead graph with cube complexIn every example we know, it is possible, after computing the cube complex, topick out a topologically distinguished collection of cut sets whose associated cubecomplex is a tree. We do not know whether this is true in general.
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Christopher H. CashenDepartment of MathematicsUniversity of UtahSalt Lake City, UT 84112, USA
E-mail address : [email protected] URL : Nataˇsa MacuraDepartment of MathematicsTrinity UniversitySan Antonio, TX 78212, USA
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