aa r X i v : . [ m a t h . G R ] J un On the coarse-geometric detection of subgroups
Diane M. Vavrichek
Abstract
We generalize [Vav] to give sufficient conditions, primarily on coarsegeometry, to ensure that a subset of a Cayley graph is a finite Hausdorffdistance from a subgroup. Using this result, we prove a partial converseto the Flat Torus Theorem for CAT(0) groups. Also using this result,we give sufficient conditions for subgroups and splittings to be invariantunder quasi-isometries.
The Flat Torus Theorem is a well-known result in Geometric Group The-ory, which deduces a geometric property of a metric space from an alge-braic property of a group that acts nicely on that space. It says that if G is a group acting geometrically on a CAT(0) space X , and G contains afree abelian subgroup of rank n , then X must contain an n –flat (on whichthe subgroup acts cocompactly). It is a well-known question going backto Gromov [Gro93, Section 6.B ] whether the converse to this theoremholds, i.e. whether the existence of an n –flat in X implies the existenceof a free abelian subgroup of rank n in G . In the following we prove thisconverse, given some assumptions on the flat. In addition, we show that,up to finite Hausdorff distances, the abelian subgroup we obtain acts co-compactly on the given flat, and give examples where this conclusion isfalse without our assumptions on the flat.Specifically, if G is a finitely generated group acting geometrically ona CAT(0) space X , and F is an isometrically embedded copy of Euclideanspace E n in X that satisfies three conditions on complementary compo-nents (including that the complement of a uniform neighborhood of F has at least three components that are unbounded away from F ), thenwe show that G contains a subgroup isomorphic to Z n , with any orbit afinite Hausdorff distance from F (Theorem 5.5). This conclusion is falseif any one of our three hypotheses are removed.In developing the tools needed to prove this theorem, we show that cer-tain, mainly coarse-geometric conditions on a subset of the Cayley graphof a group imply that that subset is a finite Hausdorff distance from a sub-group (Theorem 4.3). We go on to show that related hypotheses implythat certain types of subgroups and splittings are invariant under quasi-isometries (Theorem 7.9 and Corollary 8.17 respectively). e shall proceed by giving careful statements of these results. Recallthat a group action is said to be geometric if it is a properly discontinuousand cocompact action by isometries. A geodesic space is said to be CAT(0)if geodesic triangles are “thinner” than their comparison triangles in theEuclidean plane (see, for instance, [BH99]). The Flat Torus Theoremstates that if G is a group acting geometrically on a CAT(0) space X , and H ∼ = Z n is a subgroup of G , then X contains an isometrically embeddedcopy of E n , on which H acts with a torus quotient. (See [BH99, ChapterII.7].)In order to state our partial converse to this result, we must define afew terms. Let X be a metric space and let Y, Z ⊆ X . We say that aneighborhood of Z is uniform if it is an r –neighborhood of Z , for some r ≥
0. For any r ≥
0, we say that a component of the complement of N r ( Y ) is shallow if it is contained in a uniform neighborhood of N r ( Y ),and we say it is deep otherwise. Y is said to satisfy the deep condition if,for every r ≥
0, there is some m ≥ m -neighborhood ofeach deep component of ( X − N r ( Y )) contains N r ( Y ). We say that Y sat-isfies the shallow condition if, for every r ≥
0, there is some m ≥ m -neighborhood of N r ( Y ) contains all shallow components ofthe complement of N r ( Y ). We say that Y satisfies the 3–separating con-dition if it has at least three deep complementary components. We use d Haus to denote Hausdorff distance.
Theorem 5.5.
Let X be a CAT(0) space, and let G be a finitely generatedgroup acting geometrically on X . Suppose that X contains an isometri-cally embedded copy F of E n that has a uniform neighborhood that satisfiesthe deep, shallow and 3–separating conditions.Then G contains a subgroup H ∼ = Z n , such that for any x ∈ X , d Haus ( Hx , F ) < ∞ . Remark 1.1.
The conclusion in Theorem 5.5 is false without the deep,shallow and 3–separating hypotheses. For example, consider the free groupon m generators, F m , together with its action on its standard Cayleygraph, C ( F m ) , which is a regular m -valent tree. Then any geodesic linein C ( F m ) is an isometrically embedded copy of R ∼ = E , that satisfies theshallow and 3–separating conditions, but not the deep condition. However,this line corresponds to a subgroup if and only if it is periodic with respectto its edge labels in the standard generating set.The 3–separating hypothesis is also necessary in general. Consider, forinstance, Z , together with the standard action on R . Any line in R isan isometrically embedded copy of E , and satisfies the deep and shallowconditions. However, a line corresponds to a subgroup of Z if and only ifit has rational slope. (Note that both this and the previous example can begeneralized to examples with isometrically embedded copies of E n for any n , by taking products of the groups with Z n − and the spaces with R n − .)Finally, Lemma 7.1 shows that the conclusion of the theorem is neces-sarily false if the shallow condition is not satisfied. artial converses to the Flat Torus Theorem have appeared in [Hru05],[CH09] and [CM09]. In Theorem 3.7 of [Hru05], Hruska shows the con-verse to the Flat Torus Theorem, assuming that X has an “isolated flats”property. We note that Theorem 5.5 does not require that the flats of X are isolated, or even that the copy of E n is a maximal flat subspace of X .The main result of [CH09] also overlaps with Theorem 5.5. In thispaper, Caprace and Haglund show, in particular, that if W is a Coxetergroup and the Davis complex of W contains an isometrically embeddedcopy of E n then W contains a subgroup that is isomorphic to Z n .Also, Caprace and Monod prove in Theorem 3.8 of [CM09] that if X is a proper CAT(0) symmetric space with cocompact isometry group thatacts minimally on X , and X ∼ = R n × X ′ , then any lattice in Isom ( X )contains a Z n subgroup that acts cocompactly on the Euclidean factor.We shall denote by C ( G ) the Cayley graph of a finitely generated group G . We say that a subset Y ⊆ C ( G ) satisfies the noncrossing conditionif there is some k > g ∈ G , gY is contained in the k –neighborhood of a deep component of the complement of Y .The following theorem about “subgroup detection” is the main ingre-dient used in the proof of Theorem 5.5. Theorem 4.3.
Let G be a finitely generated group, let Y be a subgraphof C ( G ) that satisfies the deep, shallow, 3–separating and noncrossingconditions. Then G contains a subgroup H such that d Haus ( Y, H ) < ∞ . We note that in general, H will be badly distorted in G . In particular,we will not typically get that H is quasi-isometrically embedded in G . InSection 6, we will see that we can still detect certain algebraic propertiesof H from geometric properties of Y .The proof of Theorem 4.3 makes crucial use of work of Papasoglu thatappears in [Pap07].In the case that Y is a “uniformly distorted” copy of R , it was shownin [Pap05] and [Vav] that Y satisfies the deep and noncrossing condi-tions as long as G is finitely presented and one-ended. Proposition 3.5 of[Vav] gives the conclusion of Theorem 4.3 in the special case that theseassumptions hold.We conjecture that the noncrossing condition often holds for con-nected, 3–separating subspaces; see Conjecture 5.6. However, Remark1.1 essentially shows that the conclusion of Theorem 4.3 is false withoutthe deep, shallow and 3–separating assumptions.We show in Sections 2 and 7 that, with the exception of the non-crossing condition, the hypotheses of Theorem 4.3 are invariant underquasi-isometries, and correspond to certain algebraic conditions for sub-groups. In addition, we discuss a theory of “coarse isometries” in Sec-tion 3 that allows us to conclude that finitely generated subgroups thatcorrespond under a quasi-isometry, up to finite Hausdorff distance, arequasi-isometric themselves. This allows us to conclude that, up to the atisfaction of the noncrossing condition, certain subgroups are preservedunder quasi-isometries, in the following strong sense. The notion of thenumber of coends of a subgroup is originally due to Kropholler and Roller,and is discussed in Section 7. Theorem 7.9.
Let G and G ′ be finitely generated groups and let f : C ( G ) → C ( G ′ ) be a quasi-isometry. Suppose that H is a subgroup of G that has atleast three coends in G , and, for any infinite index subgroup K of H , K has only one coend in G .If sufficiently large uniform neighborhoods of f ( H ) in C ( G ′ ) satisfythe noncrossing condition, then G ′ contains a subgroup H ′ such that d Haus ( H ′ , f ( H )) < ∞ . In addition, H is finitely generated if and only if H ′ is finitely generated,and in this case H is quasi-isometric to H ′ . This theorem generalizes the following result from [Vav].
Theorem 1.2. [Vav]
Let f : C ( G ) → C ( G ′ ) be a quasi-isometry betweenone-ended, finitely presented groups, and suppose that G contains a two-ended subgroup H that has at least three coends in G . Then there is asubgroup H ′ ∼ = Z of G ′ such that d Haus ( H ′ , f ( H )) < ∞ . In the setting above, the noncrossing condition is satisfied if G (hence G ′ ) is finitely presented and one-ended, and the hypothesis about sub-groups of H is equivalent to G being one-ended.In general, Theorem 7.9 is false without the hypotheses on coends.To see this, consider the examples given in Remark 1.1. If γ and γ ′ denote geodesics in the tree C ( F ) that are periodic and aperiodic re-spectively, with respect to edge labelings, then note that there is anisometry f : C ( F m ) → C ( F m ) that interchanges γ and γ ′ . The line γ is a finite Hausdorff distance from an infinite cyclic subgroup H of F m ,and f ( γ ) = γ ′ is an infinite Hausdorff distance from any subgroup of F m .However, F m is not one-ended, hence the hypothesis in Theorem 7.9 aboutinfinite index subgroups of H is not satisfied.Also consider C ( Z ) embedded in R in the standard way, and let f : C ( Z ) → C ( Z ) be a quasi-isometry given by rotation by an irrationalangle, composed with nearest point projection back into C ( Z ). If H isany infinite cyclic subgroup of Z , then H is a finite Hausdorff distancefrom a line in R with rational slope, hence f ( H ) is a finite Hausdorff dis-tance from a line with irrational slope. Thus f ( H ) is an infinite Hausdorffdistance from any subgroup of Z .Both of these counterexamples can be generalized to ones, for instance,where H is isomorphic to Z n for any n .Using the work of Dunwoody and Swenson [DS00], we show in Theo-rem 8.16 that we can choose the subgroup H in Theorem 4.3 to be suchthat G splits over H , as an amalgamated free product or HNN extension.This splitting will “have three coends” (meaning that H has three coends n G ), which allows us to draw some algebraic conclusions about the split-ting in Theorem 8.7. Finally, combining Theorem 8.16 with Theorem 7.9,we get that these types of splittings are invariant under quasi-isometries,in the following setting. Corollary 8.17.
Let G and G ′ be finitely generated groups and let f : C ( G ) → C ( G ′ ) be a quasi-isometry. Suppose that H is a finitely gen-erated subgroup of G such that for any infinite index subgroup K of H , K has one coend in G . Suppose also that G admits a splitting over H thathas three coends.If sufficiently large uniform neighborhoods of f ( H ) in C ( G ′ ) satisfythe noncrossing condition, then G ′ contains a finitely generated subgroup H ′ such that H ′ is quasi-isometric to H , d Haus ( H ′ , f ( H )) < ∞ , and G ′ admits a splitting over H ′ that has three coends. The outline of this paper is as follows. In Sections 2 and 3, we willgive most of our definitions and some basic facts. The deep, shallow, 3–separating and noncrossing conditions will be discussed in Section 2, anduniformly distorting maps and coarse isometries will be defined in Section3. In Section 4, we will prove that any subset of a Cayley graph thatsatisfies the four conditions mentioned is a finite Hausdorff distance froma subgroup. In Section 5, we will give our partial converse to the FlatTorus Theorem.In Section 6, we will discuss some coarse isometry invariants, and seehow they show that the coarse geometry of the subset from Section 4 hasimplications about the algebra of the nearby subgroup.In Section 7, we will see how our subgroup detection theorem can beapplied to show the quasi-isometry invariance of certain subgroups, and wewill also show the invariance of the associated commensurizer subgroups,given an assumption that involves the noncrossing condition. Finally, inSection 8, we will see that we get group splittings in the settings in whichwe are working.
Acknowledgments.
The author is grateful Peter Scott, for many valu-able discussions and comments. The author also thanks Mario Bonk,Matt Brin, and Panos Papasoglu for several helpful conversations. In ad-dition, the author gratefully acknowledges that this research was partiallysupported by the University of Strasbourg, and NSF grant DMS-0602191.
In this section, we will state our conventions, define some necessary ter-minology, and prove a few basic facts related to some of our nonstandardterms.We shall assume throughout this paper that balls are open and neigh-borhoods closed, i.e. for any metric space X , B r ( p ) = { x ∈ X : d ( p, x ) < r } , N r ( Y ) = { x ∈ X : d ( x, Y ) ≤ r } or all p ∈ X, Y ⊆ X and r ≥
0. The r –neighborhood of Y refers to N r ( Y ). Definition 2.1. If X is a metric space and Y ⊆ X , then we say that aneighborhood N of Y is a uniform neighborhood if N = N r ( Y ) for some r ≥ . If X and Y are metric spaces, Λ ≥ C ≥
0, then a map f : X → Y is a (Λ , C ) quasi-isometric embedding if, for all x, x ′ ∈ X ,1Λ d ( x, x ′ ) − C ≤ d ( f ( x ) , f ( x ′ )) ≤ Λ d ( x, x ′ ) + C. Definition 2.2. If f : X → Y is a map between metric spaces and C ≥ ,then we say that f is C –onto if the C –neighborhood of Im ( f ) in Y is equalto Y . We say that f is coarsely onto if f is C –onto for some C . We say that f is a (Λ , C ) quasi-isometry if f is a (Λ , C ) quasi-isometricembedding and is C –onto. In this case, we call Λ and C the parametersof f . We will say that f is a quasi-isometric embedding (quasi-isometryrespectively) if f is a (Λ , C ) quasi-isometric embedding ((Λ , C ) quasi-isometry respectively) for some Λ ≥ C ≥ f : X → Y has finite distance from a function f : X → Y if d ( f , f ) := sup x ∈ X d Y ( f ( x ) , f ( x )) < ∞ . If f : X → Y is a quasi-isometry, and f ′ : Y → X is also a quasi-isometrysuch that both compositions f ′ f and ff ′ are a finite distance from theidentity maps id X and id Y respectively, then we say that f ′ is a quasiinverse to f . It is a fact that any quasi-isometry f has a quasi inverse f ′ ,such that the parameters of f ′ depend only on the parameters of f .We will take all graphs to be metric graphs, with each edge of lengthone. Convention 2.3.
We assume that every finitely generated group men-tioned in the following comes equipped with a fixed finite generating set. If G is a finitely generated group, we denote by C ( G ) the associatedCayley graph. Thus C ( G ) has vertex set equal to G , and g, g ′ ∈ G span anedge if and only if g ′ = gs , where s or s − is an element of the generatingset associated to G . We use d G to denote metric on C ( G ), which restrictsto the word length metric on G . Note that G acts on C ( G ) by isometrieson the left.We recall that any two finite generating sets for G will yield quasi-isometric Cayley graphs, so the geometry of G is uniquely determined upto quasi-isometry.Next, we will introduce many of the nonstandard terms that will beneeded later. (We note that this terminology differs from that of [Vav]:‘deep’, ‘shallow’ and ‘ n –separating’ replace ‘essential’, ‘inessential’ and ‘ n –parting’ respectively, and deep ( m ) is a stronger condition than ess ( m ). efinition 2.4. Let X be a metric space and let Y and Z be subsets of X . A component of ( X − Z ) is shallow if it is contained in some uniformneighborhood of Z . Otherwise, we say that the component is deep .If m : R ≥ → R ≥ is such that for each r ≥ , each shallow componentof ( X − N r ( Y )) is contained in the m ( r ) –neighborhood of N r ( Y ) , thenwe say that Y satisfies shallow ( m ) .If m : R ≥ → R ≥ is such that for each r ≥ and each point p ∈ N r ( Y ) , the ball B m ( r ) ( p ) meets each deep component of ( X − N r ( Y )) ,then we say that Y satisfies deep ( m ) . Note that m ( r ) ≥ r for all r .We say that Y satisfies the shallow ( deep respectively) condition if Y satisfies shallow ( m ) ( deep ( m ) respectively) for some m ( m respec-tively). Definition 2.5.
Let X and Y be as above, let n > , and we shall saythat Y is n –separating , or satisfies the n –separating condition , if ( X − Y ) has at least n deep components. Remark 2.6.
We will be primarily interested in subspaces Y when X isthe Cayley graph of a finitely generated group, say C ( G ) . Note that, as G acts on C ( G ) by isometries, if Y satisfies deep ( m ) , shallow ( m ) , or the n –separating condition, then so does any translate gY of Y . Definition 2.7.
A set of subsets of a metric space X , Y = { Y , Y , . . . } ,is said to satisfy noncrossing ( k ) if, for each i = j , Y i is contained in the k –neighborhood of some deep component of X − Y j .Suppose a group G acts on X , let Y ⊆ X and let k > . We saythat Y satisfies noncrossing ( k ) if { gY } g ∈ G satisfies noncrossing ( k ) inthe previous sense.We say that Y ( Y respectively) satisfies the noncrossing condition if Y ( Y respectively) satisfies noncrossing ( k ) for some k . We will see in Section 7 that there are interesting situations when sub-sets of Cayley graphs naturally satisfy many of these conditions.As we will show next, the deep, shallow and n –separating conditionsare invariant under quasi-isometries, in a suitable sense.The proof of the next lemma follows an analogous argument in [Vav]. Lemma 2.8.
Let f : X → X ′ be a quasi-isometry between geodesic spaces,with Y ⊆ X and Y ′ ⊆ X ′ such that d Haus ( Y ′ , f ( Y )) < ∞ .If Y satisfies the shallow condition then Y ′ also satisfies the shallowcondition.Proof. Suppose that Y satisfies shallow ( m ), let Y ′′ denote N r ( Y ′ ) forsome r ≥
0, and we shall show that there is some constant m ′ ( r ) suchthat any shallow component of the complement of Y ′′ is contained in the m ′ ( r )–neighborhood of Y ′′ .Let f ′ denote a quasi-inverse to f . We claim that there is some R >
C, C ′ are distinct components of ( X ′ − Y ′′ ),then no component of ( X − N R ( Y )) meets both f ′ ( C ) and f ′ ( C ′ ). To seethis, let Λ ′ , K ′ be such that f ′ is a (Λ ′ , K ′ ) quasi-isometry, let R ′ > Λ ′ K ′ and let R ≥ f ′ ( N R ′ ( Y ′′ )) ⊆ N R ( Y ). Note that any pointsin distinct components of the complement of Y ′′ that are also in the omplement of N R ′ ( Y ′′ ) are a distance of more than 2 R ′ , and hence morethan 2Λ ′ K ′ , from one another.Since f ′ is K ′ –onto and f ′ ( N R ′ ( Y ′′ )) ⊆ N R ( Y ), if some componentof ( X − N R ( Y )) met the image under f ′ of more than one component of( X ′ − Y ′′ ), then there must be distinct components C, C ′ of the comple-ment of Y ′′ and points p ∈ C, p ′ ∈ C ′ such that d ( f ′ ( p ) , f ′ ( p ′ )) ≤ K ′ . But(1 / Λ ′ ) d ( p, p ′ ) − K ′ ≤ d ( f ′ ( p ) , f ′ ( p ′ )), since f ′ is a (Λ ′ , K ′ ) quasi-isometry,and 2Λ ′ K ′ < d ( p, p ′ ), by the previous paragraph. Combining these equa-tions yields K ′ < d ( f ′ ( p ) , f ′ ( p ′ )), a contradiction. Thus each componentof the complement of N R ( Y ) is met by the image under f ′ of no morethan one component of the complement of Y ′′ .Since f ′ is coarsely onto, it follows that any deep component of thecomplement of N R ( Y ) is met only by the image under f ′ of a deep com-ponent of the complement of Y ′′ . Hence, for any shallow component S of the complement of Y ′′ , f ′ ( S ) is contained in the union of N R ( Y ) withshallow components of the complement of N R ( Y ), and hence is containedin the m ( R )–neighborhood of N R ( Y ). It follows that there is a constant m ′ ( r ), depending only on f , f ′ , R , m ( R ) and d Haus ( Y ′′ , f ( Y )), suchthat S is contained in the m ′ ( r )-neighborhood of Y ′′ .With slight alteration, and in light of Lemma 2.8, [Vav] shows that n –separation is a quasi-isometry invariant: Lemma 2.9.
Let f : X → X ′ be a quasi-isometry between geodesic spaces,with Y ⊆ X and Y ′ ⊆ X ′ such that Y and Y ′ both satisfy the shallowcondition and d Haus ( f ( Y ) , Y ′ ) < ∞ .Then there is some R ≥ such that for any n > , if Y is n –separatingthen N R ( Y ′ ) is n –separating. Our next result, which is about quasi-isometries and the deep condi-tion, also requires that the shallow condition is satisfied.
Lemma 2.10.
Let f : X → X ′ be a quasi-isometry between geodesicspaces, with Y ⊆ X and Y ′ ⊆ X ′ such that both Y and Y ′ satisfy theshallow condition and d Haus ( f ( Y ) , Y ′ ) < ∞ .If Y satisfies the deep condition, then Y ′ also satisfies the deep condi-tion.Proof. Suppose that Y satisfies deep ( m ), and let f ′ be a quasi-inverseto f . Suppose that f is a (Λ , K ) quasi-isometry and that f ′ is a (Λ ′ , K ′ )quasi-isometry. Fix r ≥
0, let Y ′′ = N r ( Y ′ ), and we will show that thereis some constant m ′ ( r ) such that B m ′ ( r ) ( p ′ ) meets all deep componentsof the complement of Y ′′ , for any p ′ ∈ Y ′′ .As we saw in the proof of Lemma 2.8, there is some R ≥ f ′ ( Y ′′ ) ⊆ N R ( Y ), and each component of the complement of N R ( Y ) meetsthe image under f ′ of at most one component of ( X ′ − Y ′′ ). Let D ′ be adeep component of ( X ′ − Y ′′ ). As Y satisfies the shallow condition, f ′ ( D ′ )must meet a deep component of ( X − N R ( Y )), say D . Since f ′ is K ′ –onto,by enlarging R by K ′ if necessary, we can assume that D ⊆ N K ′ ( f ′ ( D ′ )).Thus B m ( R )+ K ′ ( p ) meets f ′ ( D ′ ) for any p ∈ N R ( Y ). In particular,fix any p ′ ı nY ′′ and take p = f ′ ( p ′ ), and there is some d ′ ∈ D ′ suchthat d ( f ′ ( p ′ ) , f ′ ( d ′ )) < ( m ( R ) + K ′ ), and hence d ( ff ′ ( p ′ ) , ff ′ ( d ′ )) < ( m ( R ) + K ′ ) + K . The distance from ff ′ to the identity on X ′ isbounded by a function of Λ and K and hence d ( p ′ , D ′ ) ≤ d ( p ′ , d ′ ) ≤ d ( p ′ , ff ′ ( p ′ )) + d ( ff ′ ( p ′ ) , ff ′ ( d ′ )) + d ( ff ′ ( d ′ ) , d ′ ) ≤ (Λ( m ( R ) + K ′ ) + K ) + 2 d ( ff ′ , id X ′ ) . Thus our claim follows, for any m ′ ( r ) > Λ( m ( R )+ K ′ )+ K +2 d ( ff ′ , id X ′ ).It follows that Y ′ satisfies the deep condition. Next, we will introduce uniformly distorting maps and coarse isometries,and make some geometric observations about subgroups. Coarse isome-tries will provide a useful generalization of quasi-isometries, and we willsee how both of these types of functions arise naturally when consideringsubgroups in Cayley graphs. Moreover, we will see in Proposition 3.6 thatin many of the situations that we are concerned with, coarse isometriesare in fact quasi-isometries.
Definition 3.1.
Let ( X, d X ) and ( Y, d Y ) be metric spaces, and let φ and Φ be weakly increasing proper functions from R ≥ to R ≥ . Then we shallsay that a function f : X → Y is a ( φ, Φ)–uniformly distorting map if, forany x, x ′ ∈ X and r ∈ R ≥ ,1. if d X ( x, x ′ ) ≥ r then d Y ( f ( x ) , f ( x ′ )) ≥ φ ( r ) , and2. if d X ( x, x ′ ) ≤ r then d Y ( f ( x ) , f ( x ′ )) ≤ Φ( r ) .Thus φ ( d X ( x, x ′ )) ≤ d Y ( f ( x ) , f ( x ′ )) ≤ Φ( d X ( x, x ′ )) .We shall say that f is a uniformly distorting map if f is ( φ, Φ) –uniformly distorting for some φ and Φ . Note that we do not require a uniformly distorting map to be con-tinuous. Note also that the composition of uniformly distorting maps isuniformly distorting.
Definition 3.2. If f is both uniformly distorting and coarsely onto, thenwe say that f is a coarse isometry . Note that any quasi-isometry is a coarse isometry. In particular, if Y , Y ⊆ X are such that d Haus ( Y , Y ) < ∞ , then a nearest point projec-tion map of Y onto Y is a coarse isometry.In analogy to quasi inverses, we have the following. Lemma 3.3. [Vav] If f : X → Y is a coarse isometry between metricspaces, then there is a coarse isometry f ′ : Y → X such that f ′ f and ff ′ have finite distances from the identity functions id X and id Y respectively. Definition 3.4.
We shall call any function f ′ satisfying the conclusionof the above lemma a coarse inverse to f . Also we note the following observation. emma 3.5. Suppose that f : X → Y, g : Y → Z are coarse isometries.Then gf : X → Z is also a coarse isometry. In the next result, we make the key observation that often coarseisometries actually are quasi-isometries.
Proposition 3.6.
Let ( X, d X ) and ( Y, d Y ) be geodesic metric spaces.Then any coarse isometry between them is a quasi-isometry.Proof. Let f : X → Y be a ( φ, Φ)–uniformly distorting map that is acoarse isometry, and let f ′ be a coarse inverse to f . We will argue thatthere is some Λ ≥ , C ≥ φ and Φ such that, for all p , p ∈ X , d Y ( f ( p ) , f ( p )) ≤ Λ d X ( p , p ) + C . This will imply the proposition, for we can run this argument with f ′ replacing f to get constants Λ , C such that for all q , q ∈ Y , d X ( f ′ ( q ) , f ′ ( q )) ≤ Λ d Y ( q , q ) + C . As d ( id X , f ′ f ) < ∞ , we have d X ( p , p ) ≤ d X ( f ′ f ( p ) , f ′ f ( p ))+2 d ( id X , f ′ f ),and combining this with the above equation, taking f ( p i ) for q i , gives d X ( p , p ) ≤ Λ d Y ( f ( p ) , f ( p )) + ( C + 2 d ( id X , f ′ f )) . Hence (1 / Λ ) d X ( p , p ) − (1 / Λ )( C + 2 d ( id X , f ′ f )) ≤ d Y ( f ( p ) , f ( p )) ≤ Λ d X ( p , p ) + C , so f is a quasi-isometry.It remains to show that we can find Λ ≥ , C ≥ p , p ∈ X , d Y ( f ( p ) , f ( p )) ≤ Λ d X ( p , p ) + C . Fix ǫ > ǫ ′ = Φ( ǫ ). Then for any p , p ∈ X , there is a sequenceof points x = p , x , x , . . . , x k along a geodesic from p to p such that d X ( x i , x i +1 ) = ǫ for all i < k , d X ( x k , p ) ≤ ǫ , and k − X i =0 d X ( x i , x i +1 ) = kǫ ≤ d X ( p , p ) . Hence k ≤ d X ( p , p ) /ǫ. Note that { f ( x i ) } is a sequence of points from f ( p ) to f ( p ), suchthat for each i < k , d Y ( f ( x i ) , f ( x i +1 )) ≤ ǫ ′ and d Y ( f ( x k ) , f ( p )) ≤ ǫ ′ .Hence d Y ( f ( p ) , f ( p )) ≤ k − X i =0 d Y ( f ( x i ) , f ( x i +1 )) + d Y ( f ( x k ) , f ( p )) ≤ ǫ ′ ( k + 1) ≤ ǫ ′ [ d X ( p , p ) /ǫ + 1] = ǫ ′ ǫ d X ( p , p ) + ǫ ′ . Thus our claim follows, for Λ = max { , ǫ ′ ǫ } and C = ǫ ′ . et H be a finitely generated subgroup of a finitely generated group G . Then we can consider H with respect to its own intrinsic geometry,( H, d H ), or with respect to the geometric structure induced by G , ( H, d G ).Our main interest in coarse isometries stems from the fact that these twospaces are coarsely isometric: Lemma 3.7. [Vav]
Let G be a finitely generated group with finitely gen-erated subgroup H . Then the inclusion map i H : ( H, d H ) → ( H, d G ) isuniformly distorting, hence is a coarse isometry.Moreover, the bound on expansion can be taken to be linear. That is, i H is ( φ, Φ) –uniformly distorting, for some φ and Φ , where we can take Φ( r ) ≤ Lr for all r and some constant L > . Finally, we will need to understand from coarse geometry when sub-groups of finitely generated groups are finitely generated themselves. Tothis end, we introduce the following.
Definition 3.8.
We shall say that a subset Z ⊆ C ( G ) is coarsely 0–connected if there is some r ≥ such that N r ( Z ) is connected. [Vav] implies the next fact. Proposition 3.9.
Let G be a finitely generated group and let H be asubgroup of G . Then H is finitely generated if and only if H is coarsely0–connected, as a subset of C ( G ) . Suppose that there is some subset Y ⊆ C ( G ) such that d Haus ( Y, H ) < ∞ , and note that H is 0–connected if Y is connected. Hence Proposition3.9 implies the following. Corollary 3.10.
Let G be a finitely generated group with H a subgroupof G and Y a connected subset of C ( G ) . If d Haus ( Y, H ) < ∞ then H isfinitely generated. In this section, we will show that subsets of Cayley graphs that satisfythe deep, shallow, 3–separating and noncrossing conditions are, up to afinite Hausdorff distance, subgroups of the ambient group. First we willneed the following two lemmas.
Lemma 4.1.
Suppose that
Y, Y ′ ⊆ C ( G ) are 2–separating and satisfy deep ( m ) and the shallow condition, and that Y ′ ⊆ N r ( Y ) . Then there issome constant that we will denote by r ( r ) , but which depends on r and m , such that r ( r ) > r and Y ⊆ N r ( r ) ( Y ′ ) .Proof. We will show the lemma for r ( r ) = [2 m ( r + m ( r )) + m ( r )].Suppose that there are there are two deep components of the com-plement of Y ′ , D and D , such that Y meets each D i in a point p i that is not contained in the m ( r )-neighborhood of Y ′ . Thus, for each i , B m ( r ) ( p i ) ⊆ D i . Since Y ′ ⊆ N r ( Y ), the components of the complementof N r ( Y ) are contained in the components of the complement of Y ′ . Inparticular, any deep component of the complement of N r ( Y ) is disjoint rom at least one of D or D . But then this component must be dis-joint from B m ( r ) ( p ) or from B m ( r ) ( p ), contradicting that Y satisfies deep ( m ).Hence, since Y ′ is 2–separating, there is a deep component D ′ of thecomplement of Y ′ such that ( Y ∩ D ′ ) ⊆ N m ( r ) ( Y ′ ). Since D ′ is notcontained in any uniform neighborhood of Y ′ , it follows that it is alsonot contained in any uniform neighborhood of Y . Since N m ( r ) ( Y ′ ) ⊆ N r + m ( r ) ( Y ), and Y satisfies the shallow condition, there must be a deepcomponent D of the complement of N r + m ( r ) ( Y ) that is contained in D ′ . Let fr ( D ) denote the frontier of D , and note that since D is deep, d Haus ( fr ( D ) , Y ) ≤ m ( r + m ( r )).For any point p ∈ D ′ , consider a shortest path from p to Y . If thispath meets Y ′ , then d ( p, Y ′ ) ≤ d ( p, Y ). Otherwise, the path is entirelycontained in D ′ , so its endpoint is in ( Y ∩ D ′ ). Recall that ( Y ∩ D ′ ) ⊆ N m ( r ) ( Y ′ ), and it follows that d ( p, Y ′ ) ≤ ( d ( p, Y ) + m ( r )). Hence ineither case, d ( p, Y ′ ) ≤ ( d ( p, Y )+ m ( r )). Since fr ( D ) ⊆ N m ( r + m ( r )) ( Y ),and also fr ( D ) ⊆ D ′ , it follows that fr ( D ) is contained in the [ m ( r + m ( r )) + m ( r )]-neighborhood of Y ′ . Since Y ⊂ N m ( r + m ( r )) ( fr ( D )), itfollows that Y is contained in the [2 m ( r + m ( r )) + m ( r )]-neighborhoodof Y ′ , as desired.The next result is a slight generalization of a lemma from [Vav]. Lemma 4.2.
Let G be a finitely generated group and let Y be a collectionof 3–separating subsets of C ( G ) that satisfies noncrossing ( k ) , and assumethat each Y ∈ Y satisfies deep ( m ) and shallow ( m ) . Moreover, supposethat there is some ball B s ( v ) in C ( G ) that meets each Y ∈ Y .Then there is a constant x (which is independent of s ) such that if,for all distinct Y, Y ′ ∈ Y , d Haus ( Y, Y ′ ) > x , then Y is finite.Proof. By Lemma 4.1, for every r ≥ r ( r ) > r suchthat for all Y, Y ′ ∈ Y , Y ′ ⊆ N r ( Y ) implies d Haus ( Y, Y ′ ) ≤ r ( r ). Let x > m ( k ) and let x = r ( x ). Then suppose both that d Haus ( Y, Y ′ ) >x for all Y, Y ′ ∈ Y , and that Y is infinite. Hence for any Y, Y ′ ∈ Y , Y * N x ( Y ′ ).Choose any Y ∈ Y . As Y satisfies deep ( m ) and C ( G ) is locallyfinite, the complement of Y has only finitely many deep components.As Y satisfies noncrossing ( k ), there must be some deep component C whose k –neighborhood contains infinitely many elements of Y .Let Y = { Y ∈ [ Y − { Y } ] : Y ⊆ N k ( C ) } . Choose Y from Y ,and let C ′ be the deep component of the complement of Y whose k –neighborhood contains Y . As Y is infinite, there is some deep componentof the complement of Y whose k –neighborhood contains infinitely manyelements of Y . Let C denote this component, and let Y = { Y ∈ [ Y − { Y , Y } ] : Y ⊆ N k ( C ) } . Choose Y from Y , and continue on inthis manner.This process produces an infinite sequence of elements of Y , { Y i } ,and subsets of C ( G ), { C i } and { C ′ i } , such that, for each i , C i is a deepcomponent of the complement of Y i such that Y j ⊆ N k ( C i ) for all j > i ,and C ′ i is a deep component of the complement of Y i with Y j ⊆ N k ( C i )for all j < i (with perhaps C i = C ′ i ). Each Y i is 3–separating, so we may et D i to be a deep component of the complement of Y i that is not equalto C i or C ′ i .We will see next that the D i ’s are essentially disjoint. We have that( D i − N k ( Y i )) is a collection of deep and shallow components of the com-plement of N k ( Y i ). Since D i is a deep component of the complement of Y i and Y i satisfies shallow ( m ), it follows that ( D i − N k ( Y i )) must containa deep component of the complement of N k ( Y i ), say E i .Now fix i and j to be distinct. Since Y i is not contained in the x –neighborhood of Y j , there must be some point p ∈ Y i such that B x ( p )does not intersect Y j . As x > m ( k ) ≥ k , B x ( p ) is contained in C j or C ′ j . Also B x ( p ) meets each deep component of the complement of N k ( Y i ), and hence ( B x ( p ) ∪ E i ) is connected.As Y j is contained in the k –neighborhood of C i or C ′ i , we have that itis disjoint from ( D i − N k ( Y i )), hence Y j does not meet E i , or the union( B x ( p ) ∪ E i ). It follows that this union is contained in C j or C ′ j , so isdisjoint from D j , and hence from E j ⊆ D j . Thus, the E i ’s are disjoint.Now we recall that all Y ∈ Y meet the ball B s ( v ), and hence B s + m ( k ) meets each E i . But there are infinitely many E i ’s, which we now know tobe disjoint, while C ( G ) is locally finite, so we have reached a contradiction.The next theorem is one of our main results. It makes use of anargument given in [Pap07]. Theorem 4.3.
Let G be a finitely generated group, let Y be a subgraphof C ( G ) that satisfies the deep, shallow, 3–separating and noncrossingconditions. Then G contains a subgroup H such that d Haus ( Y, H ) < ∞ . Proof.
Suppose that Y is as in the statement of the theorem. We canassume that Y contains e ∈ G and is an infinite subgraph of C ( G ). Let Y satisfy deep ( m ) and noncrossing ( k ).Let Y = { gY : gY meets the closed ball B k ( e ) } . Note that Y satisfiesthe hypotheses of Lemma 4.2; let x be the constant from that lemma.Consider two elements of Y equivalent if they are of finite Hausdorffdistance from each other, and let { Y i } be the collection of equivalenceclasses of Y .First, we will show the following claim:(*) Y is made up of only finitely many equivalence classes Y i , and, foreach i and each gY ∈ Y i we have thatsup g ′ Y ∈ Y i d Haus ( gY, g ′ Y )is finite.For suppose that either there are infinitely many equivalence classes Y i , or that there is some i and some gY ∈ Y i such thatsup g ′ Y ∈ Y i d Haus ( gY, g ′ Y ) = ∞ . n either case, there must be an infinite sequence { g i Y } ⊆ Y such that d Haus ( g i Y, g j Y ) > x for all i = j . However, this violates the conclusionof Lemma 4.2. Thus (*) must hold.For each i , fix a representative g i Y ∈ Y i , and let µ > max i sup gY ∈ Y i d Haus ( g i Y, gY ) . Thus, for each i and any gY, g ′ Y ∈ Y i , d Haus ( g i Y, gY ) < µ and d Haus ( gY, g ′ Y ) < µ .An argument similar to the following was used by Papasoglu in theproof of Lemma 2.3 of [Pap07]. Let { C , . . . , C n } denote the deep compo-nents of the complement of Y , and let { D , . . . , D m } denote the deep com-ponents of the complement of N µ ( Y ). If g is a vertex in N k ( Y ), then g − Y meets B k ( e ) so g − Y ∈ Y i for some i , and hence d Haus ( g − Y, g i Y ) < µ .Thus note that, for any j , there exist k , . . . , k l such that g − C j contains g i D k ` . . . ` g i D k l , and is disjoint from g i D ˆ k for all ˆ k / ∈ { k , . . . , k l } .Recalling that all translates of Y satisfy the deep and shallow conditions,it follows that d Haus ( g − C j , [ g i D k ` . . . ` g i D k l ]) < ∞ .Thus, given a vertex g ∈ N k ( Y ), we can define a function f g : { C , . . . , C n } → P{ D , . . . , D m } , where P{ D , . . . , D m } denotes the power set on { D , . . . , D m } , such that f g ( C j ) = { D k , . . . , D k l } if g − Y ∈ Y i and g − C j is a finite Hausdorffdistance from [ g i D k ` . . . ` g i D k l ]. Then, if g, g ′ ∈ N k ( Y ), we shallwrite g ∼ g ′ if g − Y, ( g ′ ) − Y ∈ Y i for some i , and f g = f g ′ . Note thatthere are only finitely many equivalence classes. Note also that if g ∼ g ′ ,then 2 µ > d Haus ( g − Y, ( g ′ ) − Y ) = d Haus ( g ′ g − Y, Y ), and, for each j , d Haus ( g − C j , ( g ′ ) − C j ) = d Haus ( g ′ g − C j , C j ) is finite.Let R > B R ( e ) contains a member of each equivalenceclass from the equivalence relation on the vertices of N k ( Y ). Then foreach vertex g ∈ N k ( Y ), let τ g ∈ B R ( e ) denote a vertex of N k ( Y ) thatis equivalent to g . Let H be the subgroup of G that is generated by theelements gτ − g for all vertices g ∈ N k ( Y ). Thus, for each h ∈ H and forall j , both d Haus ( Y, hY ) and d Haus ( C j , hC j ) are finite.Note that the vertices of Y are contained in H ( B R ( e )) = N R ( H ), andhence Y ⊆ N R +1 ( H ). On the other hand, we claim that H is containedin a uniform neighborhood of Y . To see this, fix h ∈ H and suppose that hY is contained in some C i . If also Y ⊆ hC i ′ for some i ′ then, for all j = i ′ , the region hC j meets hY ⊆ C i and does not meet Y ⊆ hC i ′ , henceis contained in C i . Since Y , and hence hY , is 3–separating, we have that C i contains more than one component hC j . But d Haus ( C j , hC j ) < ∞ forall j , so this cannot be the case.Otherwise, we have that hY meets Y , hY meets more than one com-ponent C i , or Y meets more than one component hC i . In the lattertwo cases, the noncrossing condition implies that there is some vertex z ∈ ( hY ∩ N k ( Y )). Thus in any case, we have the existence of some such z . Hence e ∈ z − hY and z − Y meets B k ( e ), so z − hY, z − Y ∈ Y . Wenoted earlier that d Haus ( Y, hY ) < ∞ , so since d Haus ( z − Y, z − hY ) = Haus ( Y, hY ), we must have that z − Y and z − hY are both in some Y i and hence that d Haus ( z − Y, z − hY ) = d Haus ( Y, hY ) < µ . As h ∈ hY ,it follows that h ∈ N µ ( Y ) and thus H ⊆ N µ ( Y ).Thus d Haus ( H, Y ) is bounded by max { R + 1 , µ } .We conclude with a couple of observations related to what we saw inSection 3. Remark 4.4.
In the proof of Theorem 4.3, we have given the subgroup H via an infinite generating set. However, if Y connected, then it followsfrom Corollary 3.10 that H must actually be finitely generated.In the case that H is finitely generated, we can consider H with respectto its own intrinsic geometry, ( H, d H ) . In general we might have that H isbadly distorted in G , so we cannot expect that ( H, d H ) is quasi-isometricto Y . However, it follows from Lemma 3.7 that the two spaces are coarselyisometric. We will see in Section 6 that certain coarse geometric informationabout Y implies algebraic information about H . The noncrossing condition is known to be satisfied by “quasi-lines” (i.e.uniform neighborhoods of images of R under uniformly distorting maps)in the setting that we are working in, if they are contained in Cayleygraphs of finitely presented, one-ended groups. Indeed, Proposition 2.1of [Pap05] shows essentially that any quasi-line contained in the Cayleygraph of a finitely presented, one-ended group that satisfies the shallowand 3–separating conditions also satisfies the noncrossing condition. (Seealso [Vav].) (In this setting, the quasi-line will automatically satisfy thedeep condition. See [Vav].)The situation for more general subsets appears to be trickier. In Propo-sition 5.1, we will prove that the noncrossing condition is satisfied in acertain CAT(0) setting.Recall that the Flat Torus Theorem implies that if a group G actsgeometrically on a CAT(0) space X , and H ∼ = Z n is a subgroup of G ,then X contains an isometrically embedded copy of E n , that H acts onwith torus quotient.The full converse to the Flat Torus Theorem is false — that is, if G acts geometrically on a CAT(0) space X , and F is a Euclidean flat in X ,then F will not necessarily be a finite Hausdorff distance from an orbitof a Z n subgroup of G , as we saw in Remark 1.1. However, by combiningProposition 5.1 below with Theorem 4.3, we will get Theorem 5.5, whichis a partial converse to the Flat Torus Theorem. Proposition 5.1.
Let X be a CAT(0) space and let F , F ′ ⊆ X be iso-metrically embedded copies of Euclidean space E n . Let R ≥ and let F = N R ( F ) and F ′ = N R ( F ′ ) . Suppose that both F and F ′ satisfy deep ( m ) and shallow ( m ) , and that F is 3–separating. hen there is some constant k ′ = k ′ ( m , m , R ) such that F is con-tained in the k ′ –neighborhood of a deep component of the complement of F ′ .Proof. We will prove the proposition for k ′ = ( m (0) + m (0) + R ).We will show that there is some component C of ( X − F ′ ) such that F ⊆ ( F ′ ∪ C ). If C is deep, then F ′ ⊆ N m (0) ( C ), so F ⊆ N m (0) ( C ),and hence F ⊆ N m (0)+ R ( C ). If C is shallow, then C ⊆ N m (0) ( F ′ ), so F ⊆ N m (0) ( F ′ ). If C ′ denotes any deep component of ( X − F ′ ) then,as F ′ ⊆ N m (0) ( C ′ ), we have that F ⊆ N m (0)+ m (0) ( C ′ ) and hence F ⊆ N m (0)+ m (0)+ R ( C ′ ). Thus in either case it will follow that theproposition holds.We have that X is CAT(0), hence is uniquely geodesic; for any pair ofpoints p, q ∈ X , we shall denote the geodesic segment connecting p and q by [ p, q ]. Note that F and F ′ are convex.Moreover, since X is CAT(0), its distance function is convex, i.e.,for any two geodesics c, c ′ : [0 , → X parameterized proportional to arclength, and for any t ∈ [0 , d ( c ( t ) , c ′ ( t )) ≤ (1 − t ) d ( c (0) , c ′ (0)) + td ( c (1) , c ′ (1)) . (See Proposition II.2.2 of [BH99].) It follows that F and F ′ are convex,and hence so are ( F ′ ∩ F ) and ( F ∩ F ′ ).Suppose for a contradiction that F is not contained in ( F ′ ∪ C ), forany component C of ( X − F ′ ). Then F meets two distinct componentsof ( X − F ′ ), say C and C . Thus we have that ( F ′ ∩ F ) separates F and is convex in F . Note that it follows that ( F ′ ∩ F ) is a uniformneighborhood of a hyperplane in F .Let v i ∈ ( F ∩ C i ), for i = 1 ,
2. Then there is some ǫ > B ǫ ( v i ) ⊆ C i . Let q ∈ ( F ′ ∩ F ), and let l i denote the geodesic ray in F that begins at q and contains v i . Since F ′ is convex, note that the subrayof l i that begins at v i is contained in C i . Let r i = d ( v i , q ).Fix i ∈ { , } , and let w ∈ l i be such that d ( w, q ) = r ′ > r i , let ǫ ′ >
0, and suppose that there is some p ∈ ( B ǫ ′ ( w ) ∩ F ′ ). Then theCAT(0) inequality implies that [ p, q ] meets the ( r i ǫ ′ /r ′ )–ball about v i .The convexity of F ′ implies that [ p, q ] ⊆ F ′ , hence ǫ < ( r i ǫ ′ /r ′ ). Itfollows that there is some w i ∈ F such that the m ( R )–ball about w i iscontained in C i .On the other hand, we have that ( F ∩ F ′ ) is convex in F ′ , so inparticular F ′ − ( F ∩ F ′ ) consists of no more than two components. Recallthat F is 3–separating, thus there is a deep component X of ( X − F ) thatdoes not meet F ′ . Since F satisfies the shallow condition, there is a deepcomponent X of ( X − N R ( F )) such that X ⊆ X . Since ( X ∩ F ′ ) = ∅ and F ′ = N R ( F ′ ), it follows that ( X ∩ F ′ ) = ∅ .But the m ( R )–ball about each w i must meet X , so X is a connectedregion in X that meets C and C , but not F ′ . Since C and C aredistinct components of ( X − F ′ ), this is impossible. Thus there is somecomponent C of ( X − F ′ ) such that F ⊆ ( F ′ ∪ C ).Recall the following well-known theorem of Gromov. heorem 5.2. If a finitely generated group is quasi-isometric to Z n thenit contains Z n as a subgroup of finite index. Recall also that two subgroups H and H of a group G are said to becommensurable if [ H : H ∩ H ] and [ H : H ∩ H ] are both finite. Remark 5.3.
For any finitely generated group G with commensurablesubgroups H and H , it is straightforward to show that the Hausdorffdistance between H and H in C ( G ) is finite. Thus we have the following.
Corollary 5.4. If G is a finitely generated group and H is a finitelygenerated subgroup of G that is quasi-isometric to Z n (with respect to itsintrinsic metric), then G contains a subgroup H ∼ = Z n such that d Haus ( H, H ) < ∞ , where we take d Haus to denote Hausdorff distance in C ( G ) . Now we can give our partial converse to the Flat Torus Theorem.
Theorem 5.5.
Let X be a CAT(0) space, and let G be a finitely gener-ated group acting geometrically on X . Suppose that X contains an iso-metrically embedded copy, F , of E n that has a uniform neighborhood thatsatisfies the deep, shallow and 3–separating conditions.Then G contains a subgroup H ∼ = Z n , such that for any x ∈ X , d Haus ( Hx , F ) < ∞ .Proof. Fix x ∈ X , let φ : C ( G ) → X be a quasi-isometry that takes each g ∈ G to gx , and let φ ′ be a quasi-inverse to φ . Let F be a connected uni-form neighborhood of F that satisfies the deep, shallow and 3–separatingconditions. By Lemmas 2.8, 2.9 and 2.10, there is some connected uniformneighborhood Y of φ ′ ( F ) that satisfies the deep, shallow and 3–separatingconditions. F is quasi-isometric to R n , so Y is quasi-isometric to Z n .Let r > φ ( Y ) ⊆ N r ( F ). As we saw in Lemma 2.8,we can enlarge r if necessary so that each component of the complementof N r ( F ) meets the image under φ of no more than one component of( C ( G ) − Y ). Proposition 5.1 implies that N r ( F ) satisfies the noncrossingcondition. It follows that there is some k = k ( r ) > g ∈ G , gφ ( Y ) is contained in the k –neighborhood of a deep componentof ( X − N r ( F )).As d Haus ( gφ ( Y ) , φ ( gY )) is bounded by a function of the parametersof φ (and is independent of g ), there is some constant k , depending onlyon k ( r ) and the parameters of φ , such that for all g ∈ G , gY is con-tained in the k –neighborhood of a deep component of the complement of Y . Hence Y satisfies noncrossing ( k ). It follows from Theorem 4.3 andCorollary 3.10 that G contains some finitely generated subgroup H suchthat d Haus ( Y, H ) < ∞ , and hence d Haus ( F, Hx ) < ∞ .Note that by Proposition 3.6, H is quasi-isometric to F , and by Corol-lary 5.4, we can assume that H ∼ = Z n .We end this section by mentioning that we expect the noncrossingcondition to hold in far more general settings. Specifically, we expect thefollowing. onjecture 5.6. Let G be a finitely generated group, and let Y be a3–separating connected subset of C ( G ) . If for all subsets Y ′ ⊆ Y with d Haus ( Y ′ , Y ) = ∞ , Y ′ is not 2–separating, and G is of type F n for suf-ficiently large n ∈ ( N ∪ {∞} ) depending on the geometry of Y , then Y satisfies the noncrossing condition. The reader should note that results in this direction could be veryinteresting when combined with the results from this paper. When com-bined with Theorem 4.3, such results could yield a subgroup detectiontheorem that depends only on coarse geometry. If combined with Theo-rem 7.9, such a result could give the quasi-isometry invariance of certaintypes of subgroups.
We showed in Theorem 4.3 that certain properties of a subset Y of aCayley graph C ( G ) imply that there is some subgroup H of G such that d Haus ( Y, H ) < ∞ . In Remark 4.4, we noted that H is finitely generated if Y is connected, and in this case Y is coarsely isometric to ( H, d H ). In thissection, we will consider a couple of basic invariants of coarse isometries,in order to see that the coarse geometry of Y determines aspects of thealgebraic structure of H . The first such invariant we will consider is coarse n –connectedness.If ( X, d ) is a discrete metric space and ǫ ≥
0, then we use
Rips ǫ ( X ) todenote the ǫ –Rips complex of X . Thus Rips ǫ ( X ) is the simplicial complexwith vertex set equal to X , and such that any finite subset X of X spansa simplex if and only if, for all x , x ∈ X , d ( x , x ) ≤ ǫ .The following is Definition 2.10 of [Kap]. Definition 6.1.
A discrete metric space X is said to be coarsely n –connected if, for each r ≥ , there is some R ≥ r such that the natu-ral simplicial map Rips r ( X ) → Rips R ( X ) induces the trivial map on i th homotopy groups, for every ≤ i ≤ n . Note that in the case that X is a discrete subset of a Cayley graph, ormore generally of a geodesic space, this definition of coarse 0–connectednessagrees with that given in Definition 3.8.The next theorem appears in [Vav], and is based on Corollary 2.15 of[Kap]. Theorem 6.2. [Vav]
Coarse n –connectedness is a coarse isometry in-variant. The proof of Theorem 2.21 of [Kap] shows that each coarsely n –connected group is of type F n +1 , which gives the next result. Corollary 6.3.
Let
G, Y and H be as in Theorem 4.3, assume that H isfinitely generated, and let n > . Then Y is coarsely n –connected if andonly if H is of type F n +1 . Next, we will use Gromov’s theorem about groups with polynomialgrowth to see that, assuming H is finitely generated, H has a nilpotentsubgroup of finite index if Y “coarsely” has slow growth in C ( G ). Forthe remainder of this section, if ( X, d ) is a metric space, x ∈ X and >
0, then let B n ( x, ( X, d )) denote the n –ball about x in ( X, d ), and let B n ( x, ( X, d )) denote the closure of that ball.Recall that, if (
X, d ) is a discrete metric space and x ∈ X , thenthe growth function of X with respect to the basepoint x is defined by β ( x, ( X, d ))( n ) := B n ( x, ( X, d )). If H is a group with generating set S , giving rise to the metric we denote by d H , then ( H, d H ) has a growthfunction that is independent of basepoints, and which we denote by β H ( n ).Following [dlH00], we say that a function β : R + → R + is weakly dom-inated by another function β ′ : R + → R + , denoted β w ≺ β ′ , if there areconstants Λ ≥ C ≥ β ( n ) ≤ Λ β ′ (Λ n + C ) + C for all n >
0. Note that w ≺ is a transitive relation. We will say that adiscrete metric space ( X, d ) has polynomial growth if there is some a ≥ β ( x, ( X, d ))( n ) is weakly dominated by the function n n a . Lemma 6.4.
Let H be a finitely generated subgroup of a finitely generatedgroup G , and let Y ⊆ C ( G ) be such that d Haus ( Y, H ) < ∞ . If π : C ( G ) → G denotes a nearest point projection map, then β H w ≺ β ( x, ( π ( Y ) , d G )) forany x ∈ π ( Y ) .Proof. Let
L > i H : ( H, d H ) → ( H, d G )is ( φ, Φ)–uniformly distorting, with Φ( r ) ≤ Lr for all r , as in Lemma 3.7.Let ρ : H → Y be a nearest point projection map, so ρ and the projectionmap π move any given point a distance of no more than d Haus ( Y, H ) and respectively, and let ζ = πρi H , so ζ : ( H, d H ) → π ( Y ) ⊆ G .For any h, h ′ ∈ H , note that d H ( h, h ′ ) ≤ r ⇒ d G ( h, h ′ ) ≤ Lr ⇒ d ( ζ ( h ) , ζ ( h ′ )) ≤ Lr + 2 d Haus ( Y, H ) + 1 . It follows that, for any h ∈ H , ζ ( B r ( h, ( H, d H ))) ⊆ B Lr +2 d Haus ( Y,H )+1 ( ζ ( h ) , ( π ( Y ) , d G )) . On the other hand, note that for any h, h ′ ∈ H , d G ( ζ ( h ) , ζ ( h ′ )) = 0 ⇒ d G ( h, h ′ ) ≤ d Haus ( Y, H ) + 1 , and hence d H ( h, h ′ ) < r for any r such that φ ( r ) > d Haus ( Y, H ) + 1.Thus ζ maps no more than β H ( r ) elements of H to any given point of π ( Y ). It follows that, for any r ≥
0, and any point x ∈ π ( Y ), β H ( r ) ≤ β H ( r ) β ( x, ( π ( Y ) , d G ))( Lr + 2 d Haus ( Y, H ) + 1) . Hence β H w ≺ β ( x, ( π ( Y ) , d G )) for any x ∈ π ( Y ).In particular, it follows that if ( π ( Y ) , d G ) has polynomial growth, thenso does H . We recall Gromov’s famous theorem about groups with poly-nomial growth: heorem 6.5. [Gro81] Let H be a finitely generated group. Then ( H, d H ) has polynomial growth if and only if H has a nilpotent subgroup of finiteindex. Thus we have the following.
Corollary 6.6.
Let
G, Y and H be as in Theorem 4.3, assume that H isfinitely generated, and let π : C ( G ) → G be a nearest point projection map.If ( π ( Y ) , d G ) has polynomial growth, then H has a nilpotent subgroup offinite index. Using Theorem 4.3 and a few results below, in this section we will givesufficient conditions for certain subgroups to be invariant under quasi-isometries. We will begin by showing that certain algebraic hypotheseson a group G and a subgroup H imply that the deep, shallow and 3–separating conditions are satisfied by a uniform neighborhood of H in C ( G ). We saw in Section 2 that these conditions are essentially preservedunder quasi-isometries, so if f : C ( G ) → C ( G ′ ) is a quasi-isometry, H is asubgroup of G that satisfies these algebraic hypotheses, and assuming thatthe noncrossing condition is suitably satisfied, the existence of a subgroup H ′ of G ′ such that d Haus ( H ′ , f ( H )) < ∞ will follow from Theorem 4.3.We will conclude the section by seeing that work from [Vav] impliesthat the commensurizers of such subgroups are also invariant under quasi-isometries, given an assumption involving the noncrossing condition.First, we show that any subgroup in a Cayley graph satisfies the shal-low condition. Lemma 7.1.
Let G be a finitely generated group, let H be a subgroupof G and let R ≥ . Then there is some constant m , depending on R ,such that all shallow components of C ( G ) − N R ( H ) are contained in the m –neighborhood of N R ( H ) .In particular, it follows that H satisfies the shallow condition.Proof. Let S be a shallow component of ( C ( G ) − N R ( H )), so S projectsto a finite component of ( H \ C ( G ) − H \ N R ( H )). Recall that H \ C ( G )is locally finite, so, as H \ N R ( H ) is finite, there are only finitely manycomponents of ( H \ C ( G ) − H \ N R ( H )). In particular there is some m ≥ H \ C ( G ) − H \ N R ( H )) are containedin the m –neighborhood of H \ N R ( H ). If follows that S is contained inthe m –neighborhood of N R ( H ).Next, we will see that the n -separating condition is detectable froman algebraic property of H < G . To give a careful statement, we mustmake some definitions. For two subsets
X, Y ⊆ G , let X + Y denote thesymmetric difference of X and Y . Definition 7.2.
Following [SS00], if G is a finitely generated group, H asubgroup of G and X a subset of G , then we say that X is H –finite if X iscontained in finitely many cosets Hg of H , or equivalently if X ⊆ N r ( H ) for some r ≥ . If X is not H –finite, then we say that X is H –infinite . et P ( G ) denote the power set of all subsets of G , let F H ( G ) de-note the set of all H –finite subsets of G , and consider the quotient set P ( G ) / F H ( G ), where X, Y ∈ P ( G ) are considered equivalent if X + Y is H –finite. This set forms a vector space over F , the field with twoelements, under the operation of symmetric difference. In addition, theset admits an action of G on the right. The fixed set under this action,( P ( G ) / F H ( G )) G , consists of equivalence classes with representatives X such that X + Xg is H –finite for all g ∈ G , and it forms a subspace of P ( G ) / F H ( G ). The following definition is due to Kropholler and Roller[KR89]. Definition 7.3.
Let G be a finitely generated group and let H be a subsetof G . Then define ˜ e ( G, H ) = dim F ( P ( G ) / F H ( G )) G . Following Bowditch [Bow02], we shall call ˜ e ( G, H ) the number of co-ends of H in G . (Kropholler and Roller called ˜ e ( G, H ) the number ofrelative ends of H in G .)Note that ˜ e ( G, H ) = 0 if and only if G is H –finite, i.e. [ G : H ] < ∞ . Remark 7.4.
It is a useful and well-known fact that a subset X of G represents an element of ( P ( G ) / F H ( G )) G if and only if H \ δX is a fi-nite set of edges in the quotient graph H \ C ( G ) , where we use δ to de-note the coboundary operator in C ( G ) . Thus X represents an elementof ( P ( G ) / F H ( G )) G if and only if there is some r ≥ such that δX iscontained in the r –neighborhood of H in C ( G ) . The next lemma shows that coends have a natural geometric interpre-tation, which is closely related to the n –separating condition. In light ofLemma 7.1, [Vav] provides an argument for it. Lemma 7.5.
Let G be a finitely generated group, let H be a subgroup of G and let n > . Then ˜ e ( G, H ) ≥ n if and only if there is some R > such that N R ( H ) is n –separating in C ( G ) .Moreover, ˜ e ( G, H ) = ∞ if and only if, for each n > , there is some R = R ( n ) such that N R ( H ) is n –separating. Next we show that uniform neighborhoods of H satisfy the deep con-dition, as long as “smaller” subgroups coarsely do not separate G . Lemma 7.6.
Let G be a finitely generated group with subgroup H , andsuppose that, for all subgroups K of infinite index in H , ˜ e ( G, K ) = 1 .Then H satisfies the deep condition. Our hypothesis about subgroups of H is generally a necessary one. Inthe special case that H ∼ = Z , we are imposing the condition that e ( G ) = 1;for a counterexample in the absence of this assumption, consider the freegroup F n , n >
1, with respect to a standard generating set, and let H bethe subgroup generated by one of the standard generators. Then certainly H will not satisfy the deep condition.Generalizing this counterexample, let G be an amalgamated free prod-uct A ∗ C B , where A ∼ = B ∼ = Z n and C ∼ = Z n − . Let H = A , and then H does not satisfy the deep condition, with the problem stemming from thefact that C ( G ) is coarsely separated by “smaller” regions, in particularthe subgroup C . roof of Lemma 7.6. Fix r ≥
0, and it shall suffice to show that there issome constant m ( r ) such that, for any p ∈ N r ( H ), B m ( r ) ( p ) meets allthe deep components of the complement of N r ( H ).If ( C ( G ) − N r ( H )) has no deep components, i.e. H is of finite indexin G , then the deep condition is vacuously satisfied. Suppose then thatthere is only one deep complementary component of N r ( H ), say D . Then hD = D for all h ∈ H , since in general H acts by permuting the deepcomplementary components of N r ( H ). Let M > d ( e, D )
1. But this contradictsour hypothesis about subgroups of H , unless stab H ( D ) is of finite indexin H . Hence this index is finite, so stab H ( D ) is a finite Hausdorff distancefrom H , and thus fr ( D ) is a finite Hausdorff distance from H as well.Next, we claim that there is a bound on the Hausdorff distances ofthe frontiers of these deep components to H . For suppose instead that( C ( G ) − N r ( H )) has deep components D , D , . . . such that d Haus ( H, fr ( D i )) →∞ . By passing to a subsequence, we shall assume that the values of d Haus ( H, fr ( D i )) are all distinct.Note that, for any i and any h ∈ H , d Haus ( H, fr ( hD i )) = d Haus ( hH, h · fr ( D i )) = d Haus ( H, fr ( D i )) and hence, for any h, h ′ ∈ H and i = j , D i = h ′ D j . Note also that hD i and h ′ D j are deep components of( C ( G ) − N r ( H )), and thus must be disjoint.Recall that fr ( D i ) ⊆ N r ( H ), and let h i ∈ H be such that fr ( D i )meets B r ( h i ) for each i , and consider { h − i D i } . This set is a collection ofdisjoint connected regions of C ( G ), all of which meet B r ( e ). As C ( G ) islocally finite, we have reached a contradiction.Thus there must be a uniform bound on d Haus ( H, D ) for all deepcomponents D of ( C ( G ) − N r ( H )). Let M denote this bound. Then anypoint in N r ( H ) is of a distance no more than ( M + r ) from any deepcomponent D , so our claim follows for m ( r ) = ( M + r ). Thus H satisfiesthe deep condition.We will also need the next lemma. Lemma 7.7.
Let f : C ( G ) → C ( G ′ ) be a quasi-isometry between Cayleygraphs of finitely generated groups, with H a finitely generated subgroupof G and H ′ a finitely generated subgroup of G ′ such that d Haus ( H ′ , f ( H )) < ∞ . Then ( H, d H ) and ( H ′ , d H ′ ) are coarsely isometric.Proof. Let π : f ( H ) → H ′ denote nearest point projection, and let ι H : ( H, d H ) → ( H, d G ) , ι H ′ : ( H ′ , d H ′ ) → ( H ′ , d G ′ )denote the identity maps. By Lemma 3.7, ι H and ι H ′ are coarse isometries;let ι ′ H ′ be a coarse inverse to ι H ′ . Note that π is a coarse isometry, and f | H is a quasi-isometry onto its image.It follows that ( ι ′ H ′ ) πfι H : ( H, d H ) → ( H ′ , d H ′ ) is a coarse isometry.Recall the characterization of subgroups being finitely generated, givenin terms of coarse 0–connectedness by Proposition 3.9. It follows fromTheorem 6.2 and Lemma 7.7 that, if f : C ( G ) → C ( G ′ ) is a quasi-isometry,and H and H ′ are subgroups of G and G ′ respectively such that d Haus ( H ′ , f ( H )) < ∞ , then H is finitely generated if and only if H ′ is finitely generated. Im-mediate from this observation and Proposition 3.6 is the following: Proposition 7.8. If f : C ( G ) → C ( G ′ ) is a quasi-isometry between Cay-ley graphs of finitely generated groups, and if H and H ′ are subgroups of G and G ′ respectively such that d Haus ( H ′ , f ( H )) < ∞ , then H is finitely generated if and only if H ′ is finitely generated, and inthis case H and H ′ are quasi-isometric. We can now prove the main result of this section:
Theorem 7.9.
Let G and G ′ be finitely generated groups and let f : C ( G ) → C ( G ′ ) be a quasi-isometry. Suppose that H is a subgroup of G such that ˜ e ( G, H ) ≥ , and, for any infinite index subgroup K of H , ˜ e ( G, K ) = 1 . f sufficiently large uniform neighborhoods of f ( H ) in C ( G ′ ) satisfythe noncrossing condition, then G ′ contains a subgroup H ′ such that d Haus ( H ′ , f ( H )) < ∞ . In addition, H is finitely generated if and only if H ′ is finitely generated,and in this case H is quasi-isometric to H ′ .Proof. Let
G, G ′ , H and f be as stated. By Lemmas 7.1 and 7.6, H sat-isfies the shallow and deep conditions, hence so does any uniform neigh-borhood of H in C ( G ). We have ˜ e ( G, H ) ≥
3, so by Lemma 7.5, there issome
R > N R ( H ) is 3–separating. It follows from Lemmas2.8, 2.9 and 2.10 that there is some uniform neighborhood Y of f ( H ) thatsatisfies the deep, shallow and 3–separating conditions.We have assumed that sufficiently large uniform neighborhoods of f ( H ) satisfy the noncrossing condition, hence, by replacing Y with abigger uniform neighborhood of f ( H ) if necessary, we have that Y sat-isfies the noncrossing condition. Note that replacing Y in this mannerwill not change that Y satisfies the deep, shallow, and 3–separating con-ditions. Thus, by Theorem 4.3, G ′ contains a subgroup H ′ such that d Haus ( Y, H ′ ) < ∞ , hence d Haus ( H ′ , f ( H )) < ∞ .Finally, by Proposition 7.8, H is finitely generated if and only if H ′ is also finitely generated, and in this case the two subgroups are quasi-isometric.We now turn our attention to commensurizer subgroups. Definition 7.10. If H and H are subgroups of a group G , then they aresaid to be commensurable if [ H : H ∩ H ] < ∞ and [ H : H ∩ H ] < ∞ .If H is a subgroup of G , then the commensurizer of H in G is defined to bethe subgroup Comm G ( H ) = { g ∈ G : H and g − Hg are commensurable } . The commensurizer has the following geometric characterization.
Lemma 7.11. [Vav] If G is a finitely generated group with subgroup H ,then Comm G ( H ) = { g ∈ G : d Haus ( H, gH ) < ∞} . The proof of next proposition follows that of an analogous result in[Vav].
Proposition 7.12.
Let G be a finitely generated group and let Y be a col-lection of pairwise finite Hausdorff distance 3–separating subsets of C ( G ) that satisfy deep ( m ) and shallow ( m ) . Suppose that { gY } g ∈ G,Y ∈ Y sat-isfies the noncrossing condition. Finally, fix any Y ∈ Y and by Theorem4.3 there is a subgroup H of G such that d Haus ( Y, H ) < ∞ .Then there is some R ≥ such that Y ⊆ N R ( Comm G ( H )) . This result, together with Theorem 7.9 and a further assumption aboutthe noncrossing condition being satisfied, imply the quasi-isometry invari-ance of the commensurizers of the subgroups under discussion:
Corollary 7.13.
Let
G, G ′ , H, H ′ and f be as in Theorem 7.9. Let f ′ bea quasi-inverse to f , and suppose there is some R > such that, for all R > R , both { gN R ( f ′ ( c ′ H ′ )) } g ∈ G,c ′ ∈ Comm G ′ ( H ′ ) nd { g ′ N R ( f ( cH )) } g ′ ∈ G ′ ,c ∈ Comm G ( H ) satisfy the noncrossing condition.Then d Haus ( Comm G ′ ( H ′ ) , f ( Comm G ( H ))) < ∞ . Proof.
As we saw in the proof of Theorem 7.9, there is some
R > R suchthat N R ( f ( H )) satisfies the deep, shallow and 3–separating conditions.We can further assume that R , m and m are such that for all g ∈ G , N R ( f ( gH )) is 3–separating and satisfies deep ( m ) and shallow ( m ).For all c ∈ Comm G ( H ), d Haus ( cH, H ) < ∞ and hence the elementsof { N R ( f ( cH )) } c ∈ Comm G ( H ) are of pairwise finite Hausdorff distance. ByProposition 7.12, there must be some R > [ c ∈ Comm G ( H ) N R ( f ( cH )) = N R ( f ( Comm G ( H ))) ⊆ N R ( Comm G ′ ( H ′ )) . Similarly there are R ′ , R ′ > [ c ′ ∈ Comm G ′ ( H ′ ) N R ′ ( f ′ ( c ′ H ′ )) = N R ′ ( f ′ ( Comm G ′ ( H ′ ))) ⊆ N R ′ ( Comm G ( H )) . Hence d Haus ( Comm G ′ ( H ′ ) , f ( Comm G ( H ))) < ∞ . Corollary 7.13 generalizes the main result of [Vav], which proves thecorollary in the case that H ∼ = Z , and G, G ′ are finitely presented. (Thenoncrossing condition is always satisfied in that setting, by Proposition2.1 of [Pap05].)We note that the commensurizer subgroups in Corollary 7.13 will ingeneral not be finitely generated. The corollary, together with Proposition7.8, implies the following. Corollary 7.14.
Let H and H ′ be as in Corollary 7.13. Then Comm G ( H ) is finitely generated if and only if Comm G ′ ( H ′ ) is finitely generated. The settings of Theorems 4.3 and 7.9 imply the existence of splittings ofthe ambient groups, as we will see in this section. It will follow that,assuming the satisfaction of the noncrossing condition as before, a largeclass of splittings are invariant under quasi-isometries.For the arguments in this section, we will need to introduce the numberof ends of a group, the number of ends of a pair, almost invariant sets, andrelated notions. For this we follow Scott and Swarup (see, for instance,[SS00]).For any locally finite, connected simplicial complex X , define the num-ber of ends of X to be e ( X ) = sup { infinite components of ( X − K ) } , where the supremum is taken over all finite subcomplexes K of X . Thus e ( X ) may take any value in Z ≥ ∪ {∞} . or any finitely generated group G , the number of ends of G , e ( G ),is defined to be e ( C ( G )). The number of ends is invariant under quasi-isometries, and therefore this definition does not depend on our choice offinite generating sets for G .It is a fact due to Hopf [Hop44] that the number of ends of a finitelygenerated group can be only 0 , , ∞ . We have that e ( G ) = 0 if andonly if G is finite, e ( G ) = 2 if and only if G has a finite index Z subgroup,and by Stallings’ Theorem, G splits as an amalgamated free product orHNN extension over a finite subgroup if and only if e ( G ) = 2 or ∞ . See[SW79] for more details. Definition 8.1. If H is a subgroup of a finitely generated group G , then the number of ends of the pair ( G, H ) , denoted e ( G, H ) , is defined to bethe number of ends of the quotient graph H \ C ( G ) . We defined ˜ e ( G, H ) in the previous section, and showed that, thoughit is defined algebraically, its value can be detected from coarse separationproperties of neighborhoods of H in C ( G ) (Lemma 7.5). In fact ˜ e ( G, H ) isclosely related to e ( G, H ). For instance, in [KR89], Kropholler and Rollershowed that e ( G, H ) ≤ ˜ e ( G, H ), and that, if H ⊳ G , then ˜ e ( G, H ) = e ( G, H ) = e ( G/H ).Recall that a subset of G is said to be H –finite if it is contained infinitely many cosets Hg , and that this condition is equivalent to the subsetbeing contained in some uniform neighborhood of H . We say that a subsetis H –infinite if it is not H –finite.We say that subsets X and Y of G are H –almost equal if their sym-metric difference, X + Y , is H –finite. Let X ∗ denote ( G − X ). We willsay that X is nontrivial if neither X nor X ∗ is H –finite. We will say that X, Y ⊆ G are H –almost complementary if X ∗ and Y are H –almost equal,and we will say that X is H –almost contained in Y , denoted X H ⊆ Y , if X ∩ Y ∗ is H –finite. Definition 8.2. If G is a group, H is a subgroup of G and X is a subsetof G , then X is said to be H –almost invariant if X is invariant underthe left action of H and represents an element of ( P ( G ) / F H ( G )) G . Thatis, X is H –almost invariant if X = HX and X + Xg is H –finite for all g ∈ G . ( H –almost invariant sets are related to e ( G, H ) — in fact we couldhave defined e ( G, H ) as we did ˜ e ( G, H ), replacing ( P ( G ) / F H ( G )) G withequivalence classes of H –almost invariant sets.)We now turn our attention to group splittings. Recall that a group G is said to split over a subgroup H if G can be written as an amalgamatedfree product A ∗ H B , with A = H = B , or as an HNN extension A ∗ H .Equivalently, G splits over H whenever G acts on a simplicial tree T without edge inversions or any proper G –invariant subtrees, and with H the stabilizer of some edge of T . (See, for instance, [Ser77] or [SW79].)The numbers of ends of pairs are related to splittings by [SW79,Lemma 8.3], which shows that if G is a finitely generated group thatsplits over a subgroup H then e ( G, H ) ≥
2. (The converse to this state-ment is false in general. Understanding when the existence of a subgroup H of G such that e ( G, H ) ≥ G is n important question, on which much work has been done. See [Wal03]for a survey.)As e ( G, H ) ≤ ˜ e ( G, H ), we have the following.
Proposition 8.3. If G is a finitely generated group that splits over H ,then ˜ e ( G, H ) ≥ . Definition 8.4.
Suppose that G admits a splitting over a subgroup H .We say that this splitting has three coends if ˜ e ( G, H ) ≥ . The subgroup H associated to a splitting of G is well-defined up toconjugacy. So to see that this definition makes sense, note that any innerautomorphism of G is a quasi-isometry, so Lemmas 2.9 and 7.5 imply that˜ e ( G, H ) = ˜ e ( G, gHg − ) for any g ∈ G .Our work in the previous sections will imply the existence and quasi-isometry invariance of splittings with three coends, under the appropriatehypotheses. Thus it is relevant to investigate these types of splittings —we provide a characterization in Theorem 8.7 below. Before stating thisresult, we must define the notion of interlaced cosets. Definition 8.5.
Let H be an infinite index subgroup of a finitely generatedgroup G . Then we say that H has interlaced cosets in G if, for every r > and every pair of deep components D, D ′ of C ( G ) − N r ( H ) thereis a sequence g , . . . , g n ∈ G such that g i H meets components C i , C ′ i ofthe complement of N r ( H ) , C = D , C ′ n = D ′ and C ′ i = C i +1 whenever ≤ i < n . Note that in particular if ˜ e ( G, H ) = 1, i.e. no uniform neighborhood of H in G is 2-separating, then H has interlaced cosets. If ˜ e ( G, H ) > H satisfies the noncrossing condition,then H does not have interlacing cosets in G . Remark 8.6.
Suppose that H has interlaced cosets in G . As H satisfiesthe shallow condition, it follows that, for any r, r ′ > and given any pair D, D ′ of components of ( C ( G ) − N r ( H )) , there is a sequence g , . . . , g n and components C i , C ′ i as above, such that each g i meets C i and C ′ i outsideof the r ′ –neighborhood of N r ( H ) . Our characterization of when certain types of splittings have threecoends is the following.
Theorem 8.7.
Let G be a finitely generated group that splits over a sub-group H . Assume that the vertex groups of the splitting are finitely gen-erated, and that for all infinite index subgroups K of the inclusion(s) of H into G , ˜ e ( G, K ) = 1 .Suppose that the splitting is of the form G = A ∗ H B . Then it has threecoends if and only if none of the following hold: • [ A : H ] = [ B : H ] = 2 , • [ A : H ] = 2 , Comm B ( H ) = H and H has interlaced cosets in B , orwith the roles of A and B reversed, or • Comm A ( H ) = Comm B ( H ) = H and H has interlaced cosets in A and B . uppose instead that G = A ∗ H . Let t denote the stable letter, let i , i : H ֒ → A be the associated inclusions, and let H = i ( H ) and H = i ( H ) . Then the splitting has three coends if and only if none of thefollowing hold: • H = A = H , or • Comm A ( H ) = H , Comm A ( H ) = H , t / ∈ Comm A ( H ) (orequivalently t / ∈ Comm A ( H ) ) and both H and H have interlacedcosets in A . ([Hou74, Theorem 3.7] is a characterization of a splitting of G over H having three coends, under the assumption that H has infinite index inits normalizer.)For an example of a splitting that is one of the “interlacing cosettypes” above, and hence ˜ e ( G, H ) = 2, consider the following. Let Σ bea closed surface of genus at least two, let γ be a closed curve in Σ thatis homotopically nontrivial and has positive self-intersection number, andlet Σ ′ denote two copies of Σ, identified along γ . Let G = π (Σ ′ ), let A and B denote subgroups corresponding to the two copies of Σ in Σ ′ andlet H be an infinite cyclic subgroup that is induced by γ : S → Σ ′ . Then Comm A ( H ) = Comm B ( H ) = H , and H has interlaced cosets in A and B . A similar construction is given by replacing Σ above with a closedhyperbolic 3–manifold M , and letting γ denote any closed curve in M thatis homotopically nontrivial. Let G be the fundamental group of two copiesof M identified along γ , let A and B denote subgroups corresponding tothe two copies of M and let H ∼ = Z correspond to γ . Again Comm A ( H ) = Comm B ( H ) = H , and ˜ e ( A, H ) = ˜ e ( B, H ) = 1, so H has interlaced cosetsin A and B here as well.In the next proofs, we will use the following notation. If Y is a con-nected subset of C ( G ), X ⊆ Y and r ≥
0, then we will write N r ( X, Y ) todenote the r –neighborhood of X in Y , with respect to the induced pathmetric for Y . We will let N r ( X ) denote the r –neighborhood of X in C ( G ).For any subset Z of G , let Z denote the subgraph of C ( G ) consisting of Z together with all edges that have both vertices contained in Z . Proof of Theorem 8.7.
This proof will make use of Lemma 7.5 and Propo-sitions 8.8, 8.9 and 8.10 below.Assume first that G = A ∗ H B . It suffices to take the associatedgenerating set for G to be the union of the finite generating sets associatedto A and B . Thus for all g ∈ G and C ∈ { A, B } , gC = ( gC ) is simpliciallyisomorphic to C ( C ).Let T A , T B be sets of transversals for H in A and B respectively, so T A contains exactly one representative of each coset aH of H in A , with e representing H , and similarly for T B . It is shown in [SW79] that eachelement of G has a unique representation of the form a b a b · · · b n h, where n ≥ h ∈ H and each a i ∈ T A and b i ∈ T B , with a i = e onlyif i = 1 and b i = e only if i = n . In the following, when we write a b · · · b n or a b · · · a n − , it will be understood that these are subwords s the notation indicates of words in this normal form. Hence G is theunion of the cosets of the form a b a · · · b n A and a b · · · a n − B .The action of G on the associated Bass-Serre tree allows one to see that A ∩ B = H . Similarly one can see that, of all the cosets a b a · · · b n A and a b · · · a n − B , A meets precisely those equal to aB for some a ∈ T A , in aH , and B meets precisely the cosets bA , for b ∈ T B , in bH . In addition,each a b a · · · a n − b n A meets precisely the cosets a b a · · · a n − b n aB , a ∈ T A , in a b a · · · b n aH , and a b a · · · a n − B meets precisely all a b a · · · a n − bA such that b ∈ T B , in a b a · · · a n − bH .The Bass-Serre tree encodes all these intersections, in the followingsense. One edge of the tree has vertices say v A and v B , with stabilizers A and B respectively, and the cosets of A ( B respectively) translate v A ( v B respectively) to the different vertices in its orbit. Two vertices are adjacentprecisely when the corresponding cosets intersect. Since our generatingset for G is the union of the generating sets for A and B , note that twovertices in C ( G ) are connected by an edge only if they are in the samecoset of A or B .It follows from all of this that H separates A from B in C ( G ), andsimilarly for any g ∈ G , gH separates gA from gB . In particular, let X B = { a b a · · · b n h : a = e, n > } and note that X B is preciselythe vertex set of the component(s) of C ( G ) − H that meet B . Then X ∗ B = ( H ∪ { a b a · · · b n h : a = e } ) is the union of H with the verticesof the components that meet A . It follows that δX B ⊆ N ( H ), so X B represents an element of ( P G/ F H ( G )) G .We claim that X B is nontrivial. For it follows from [SW79] that each g ∈ G has a unique “reversed” normal form ha b · · · b n for h ∈ H and a b · · · b n as above. Hence the elements of the form a b a · · · b n are con-tained in distinct cosets Hg , and so both X B and its complement are H –infinite. Thus X B is nontrivial and so we recover that ˜ e ( G, H ) ≥ A and B . We will begin by notingthe standard result that ˜ e ( G, H ) = 2 if [ A : H ] = [ B : H ] = 2 (thus˜ e ( A, H ) = ˜ e ( B, H ) = 0). If 3 ≤ [ A : H ] < ∞ (so again ˜ e ( A, H ) = 0),then we will see that ˜ e ( G, H ) ≥
3. Next, we will show that ˜ e ( G, H ) ≥ e ( A, H ) > Comm A ( H ) = H , or if H does not have interlacedcosets in A (hence ˜ e ( A, H ) > e ( G, H ) = 2 if ˜ e ( A, H ) = 0 with [ A : H ] = 2, Comm B ( H ) = H and H has interlaced cosets in B , and finally Proposition 8.9 below gives˜ e ( G, H ) = 2 if
Comm A ( H ) = Comm B ( H ) = H and H has interlacedcosets in A and B . This will exhaust all possible cases when G = A ∗ H B .If [ A : H ] = [ B : H ] = 2 (so ˜ e ( A, H ) = ˜ e ( B, H ) = 0), then it is well-known that ˜ e ( G, H ) = 2. Indeed, it follows that H is normal in A and B , hence in G , and so G/H ∼ = Z ∗ Z , which is two-ended. It is shownin [KR89] that if H is normal in G , then ˜ e ( G, H ) = e ( G, H ) = e ( G/H ),thus ˜ e ( G, H ) = 2 in this case.For our next case, we consider ˜ e ( A, H ) = 0 with [ A : H ] ≥
3, and let e, a, a ′ ∈ T A be distinct. Recall that ˜ e ( A, H ) = 0 if and only if [ A : H ] < ∞ , or equivalently A ⊆ N r ( H ) for some r ≥
0. Fix some such r and itfollows that N r ( H ) contains aH and a ′ H . The unions of the components f ( C ( G ) − A ) that meet each of B, aB and a ′ B are X B , aX B and a ′ X B respectively hence they are not contained in any uniform neighborhood of H . As H satisfies the shallow condition, the complement of N r ( H ) has adeep component contained in each of these, and hence ˜ e ( G, H ) ≥
3. Wesimilarly get this conclusion if ˜ e ( B, H ) = 0 and [ B : H ] ≥ e ( A, H ) >
0, i.e. [ A : H ] = ∞ , and suppose that Comm A ( H ) = H . Let a ∈ Comm A ( H ) ∩ ( T A − { e } ), and let r > aH ⊆ N r ( H ). As we saw in the previous case, there are twodeep components of ( C ( G ) − N r ( H )) that are contained in X B and aX B respectively, and neither meets A . As [ A : H ] = ∞ , A contains infinitelymany cosets Ha ′ of H . Since H satisfies the shallow condition, it followsthat A meets another deep component of the complement of N r ( H ). So N r ( H ) is 3-separating and ˜ e ( G, H ) ≥ e ( B, H ) > Comm B ( H ) = H .If H does not have interlacing cosets in A , so ˜ e ( G, H ) >
1, then there issome r ≥ A − N r ( H, A )) has two deep components that are notconnected by a sequence of cosets of H , as in the definition of interlacingcosets. Thus the components of ( A − N r ( H, A )) can be partitioned intotwo sets, D and D , such that each contains at least one deep component,and no coset aH meets components in both D and D . Note that thereis an H –infinite component C i of ( C ( G ) − N r ( H )) that meets each D i ,and it follows from the definition of the D i ’s and the “tree” configurationof the cosets of A, B and H in G that C ∩ C = ∅ . Moreover, B meetsan H –infinite component of ( C ( G ) − H ) that is disjoint from A , and thiscomponent must contain a deep component of the complement of N r ( H ),that is disjoint from C and C . It follows that N r ( H ) is 3-separating, so˜ e ( A, H ) ≥ A : H ] = 2 (so ˜ e ( A, H ) = 0),
Comm B ( H ) = H and H has interlacing cosets in B , then ˜ e ( G, H ) = 2,and similarly if we exchange the roles of A and B .Finally, Proposition 8.9 shows that ˜ e ( G, H ) = 2 if
Comm A ( H ) = Comm B ( H ) = H and H has interlaced cosets in A and B . This com-pletes the proof of the theorem in the case that G = A ∗ H B .Now let us consider the HNN extension case, so G = A ∗ H = ( A ∗ h t i ) /N, where N denotes the normal closure of { i ( h ) − t − i ( h ) t : h ∈ H } in( A ∗ h t i ). Let H k = i k ( H ) for k = 1 ,
2, so H = t − H t and hence˜ e ( G, H ) = ˜ e ( G, H ). Thus the splitting G = A ∗ H has three coends if andonly if ˜ e ( G, H ) = ˜ e ( G, H ) ≥
3. The argument that follows will differfrom the amalgamated free product case in subtle ways.We shall take the generating set for G to be the union of t with afinite generating set for A . Thus C ( A ) ∼ = A ∼ = gA = ( gA ) for any g ∈ G .Let T k be a set of transversals for H k in A , so T k contains exactly onerepresentative of each coset aH k of H k in A , and we take e to represent H . Scott and Wall [SW79] show that any g ∈ G has a unique normal form a t ǫ a t ǫ · · · a n t ǫ n a n +1 , here n ∈ Z ≥ , a n +1 ∈ A , and for all k ≤ n , ǫ k = ± a k ∈ T if ǫ k = 1, a k ∈ T if ǫ k = −
1, and a k = e if k > ǫ k − = ǫ k . When we write a t ǫ · · · a n t ǫ n a n +1 or a t ǫ · · · a n t ǫ n in the following, we shall assume thatthe words are of this form. Thus G is the disjoint union of the cosets of A of the form a t ǫ · · · a n t ǫ n A .Let us consider which a t ǫ · · · a n t ǫ n are such that A is connected to a t ǫ · · · a n t ǫ n A by an edge. The edges meeting A are of the form [ a, as ],for a ∈ A and s ∈ A or s = t ± . If s ∈ A then [ a, as ] ⊆ A . If s = t − thenwrite a = a ′ h for a ′ ∈ T and h ∈ H , and then at − = a ′ ht − = a ′ t − h ′ ,where h ′ ∈ H . Note that a ′ t − h ′ is in normal form, so a ′ H ⊆ A isconnected to a ′ t − H ⊆ a ′ t − A by an edge in C ( G ), for any a ′ ∈ T . If s = t then write a = a ′ h for a ′ ∈ T and h ∈ H , and similarly at = a ′ th ′ for h ′ ∈ H , which is also the normal form. Thus a ′ H ⊆ A is alsoconnected to a ′ tH ⊆ a ′ tA by an edge, for any a ′ ∈ T . Hence A isconnected by single edges precisely to a ′ t − A , a ′ ∈ T , and a ′ tA , a ′ ∈ T .Similarly for any g ∈ G , gA is connected by single edges to only ga ′ t − H ⊆ ga ′ t − A through ga ′ H for any a ′ ∈ T and gatH ⊆ gatA through ga ′ H for any a ′ ∈ T .Note that the Bass-Serre tree associated to G = A ∗ H encodes thisadjacency information, similarly to the amalgamated free product case.In particular, for any g ∈ G , N ( gH ) contains gH and gH t = gtH ,hence separates gA and gtA from one another.Let X := { a t ǫ · · · a n t ǫ n a n +1 : a = e, ǫ = 1 } . It follows from ourdiscussion that δX ⊆ N ( H ), so X represents an element of ( P G/ F H ) G .Consider any g = a t ǫ · · · a n t ǫ n a n +1 , and note that if we use transver-sals T ′ k of H k \ A , then “pushing” elements of H , H to the left in this worduniquely determines a reversed normal form g = ha ′ t ǫ · · · a ′ n t ǫ n a ′ n +1 ,where a k ∈ T ′ if ǫ k − = 1 and a k ∈ T ′ if ǫ k − = −
1. Hence if theexponents ǫ k for two different elements of G do not match, then the ele-ments are in different cosets H ( g ′ ). In particular, it follows that X andits complement are H –infinite, and so represent nontrivial elements of( P G/ F H ) G . Thus we recover the conclusion of Proposition 8.3 in thiscase: ˜ e ( G, H ) ≥ H and H . First, we will give the standard result that˜ e ( G, H ) = 2 if A = H = H . Next we will show that ˜ e ( G, H ) ≥ e ( A, H ) = 0 and A = H . We shall see that if A = H , then ˜ e ( G, H ) =0, which puts us in one of the two previous cases. Then we have that˜ e ( G, H ) ≥ e ( A, H ) > Comm A ( H ) = H . The same resultfollows if H does not have interlaced cosets in A . If Comm A ( H ) = H , Comm A ( H ) = H , t / ∈ Comm G ( H ) and both H and H have interlacedcosets in A , then Proposition 8.10 below gives ˜ e ( G, H ) = 2. If instead Comm A ( H ) = H , Comm A ( H ) = H and H and H have interlacedcosets in A , but t ∈ Comm G ( H ) then it follows that ˜ e ( G, H ) ≥ A = H = H then ˜ e ( G, H ) = 2, since in this case H ⊳ G , with G/H ∼ = Z so by [KR89], ˜ e ( G, H ) = e ( G/H ) = e ( Z ) = 2.Suppose that ˜ e ( A, H ) = 0 and A = H . Then there is some nontrivial a ∈ T , and note that tA , atA and for instance t − A are all containedin distinct components of ( C ( G ) − A ). Each of these components is H – nfinite, for the component containing tA has vertex set X defined earlier,and the component containing atA has vertex set aX , hence containswords with normal forms containing arbitrarily large numbers of “ t ’s” andso is H –infinite by the observation made above. Finally, the componentcontaining t − A , call it Y , also contains all words that have normal form a t ǫ · · · a n t ǫ n a n +1 such that a = e and ǫ = −
1, so this set must also be H –infinite, by the same argument. Since ˜ e ( A, H ) = 0, there is some r > A ⊆ N r ( H ). Also H satisfies the shallow condition in C ( G ),and it follows that each of X , aX and Y contains a deep component of( C ( G ) − N r ( H )). Thus N r ( H ) is 3-separating and hence ˜ e ( G, H ) ≥ e ( G, H ) ≥ e ( A, H ) = 0 and A = H .Note that if A = H then H ⊆ H . As d Haus ( H = t − H t, t − H ) < ∞ , it follows that t − H is contained in some uniform neighborhood of H . By Lemma 7.6, H and its translates satisfy the deep condition, soLemma 4.1 implies then that d Haus ( H , t − H ) < ∞ . It follows that d Haus ( H , H ) < ∞ , so [ A : H ] < ∞ . This puts us in one of the previouscases, and similarly for A = H .If ˜ e ( A, H ) > Comm A ( H ) = H , then let a ∈ Comm A ( H ) ∩ ( T − { e } ), and, as in the amalgamated free product case, we considera neighborhood N r ( H ) that contains aH . Then N r ( H ) must separateregions of C ( G ) that meet X , aX and A . Since also [ A : H ] = ∞ and H satisfies the shallow condition, it follows that each of X , aX and A must meet a deep component of ( C ( G ) − N r ( H )), thus ˜ e ( G, H ) ≥ H is replaced by H .Suppose that [ A, H ] = ∞ and that H does not have interlacing cosetsin A , so in particular ˜ e ( A, H ) >
1. As in the amalgamated free productcase, we can take a neighborhood N r ( H , A ) whose complement in A contains deep components D , D that are not connected by a sequenceof cosets of H . It follows that ( C ( G ) − N r ( H , A )) contains distinct H –infinite components that meet D , D and tA , in fact X , respectively.Thus N r ( H , A ) is contained in a 3-separating uniform neighborhood of H in C ( G ), and it follows that ˜ e ( G, H ) ≥
3. Similarly we get ˜ e ( G, H ) ≥ A, H ] = ∞ and H does not have interlacing cosets in A It remains to consider the case that
Comm A ( H ) = H , Comm A ( H ) = H and both H and H have interlaced cosets in A . Recall that t / ∈ A ,so we can consider whether or not t ∈ Comm G ( H ). If t / ∈ Comm G ( H )then Proposition 8.10 below shows that ˜ e ( G, H ) = 2. If t ∈ Comm G ( H ),then d Haus ( H , H ) < ∞ . Let r > H ⊆ N r ( H ), and asin the case that ˜ e ( A, H ) = 0 and A = H , we have that ( C ( G ) − N r ( H ))contains a deep component that meets X , and a deep component thatmeets { a t ǫ · · · a n +1 : a = e, ǫ = − } , and neither of these meets A . Inaddition we have that [ A : H ] = ∞ , so ( C ( G ) − N r ( H )) contains a deepcomponent that has H –infinite intersection with A , hence ˜ e ( G, H ) ≥ roposition 8.8. Suppose that G = A ∗ H B , with A and B finitelygenerated, and that, for every infinite index subgroup K of H , ˜ e ( G, K ) =1 . If [ A : H ] = 2 , Comm B ( H ) = H and H has interlaced cosets in B ,then ˜ e ( G, H ) = 2 .Proof.
Let T A = { e, a } and T B be transversals for H in A and B respec-tively. As above, it suffices to assume that the finite generating set for G is a union of finite generating sets for A and B , and take the generatingset for A to be a together with a finite generating set for H . Thus wehave C ( A ) ∼ = A ∼ = gA and C ( B ) ∼ = B ∼ = gB for all g ∈ G .Recall the subset X B of G from the proof of Theorem 8.7, whichrepresents a nontrivial element of ( P G/ F H ( G )) G , and suppose that X also represents an element of ( P G/ F H ( G )) G . We will show that X B H ⊆ X or ( X ∩ X B ) is H –finite, and a similar argument shows the analogousresult for X ∗ B . It follows that X is H -almost equal to X B , X ∗ B , G or ∅ .Hence { X B , G } generates ( P G/ F H ( G )) G over Z , so ˜ e ( G, H ) = 2.We have that the identity map (
B, d B ) ֒ → ( B, d G ) is a ( φ, Φ)-uniformlydistorting map, for some φ and Φ. Our choice of generating sets gives C ( B ) ∼ = B , so we can take Φ to be the identity. For any r ≥
0, we have N r ( H, B ) ⊆ ( N r ( H ) ∩ B ). On the other hand, let φ ′ : R ≥ → R ≥ be suchthat φ ( φ ′ ( r )) > r , and it follows that, for any r ≥
0, ( N r ( H ) ∩ B ) ⊆ N φ ′ ( r ) ( H, B ). Thus φ ′ ( r ) ≥ r for all r . We also have ( N r ( gH ) ∩ gB ) ⊆ N φ ′ ( r ) ( gH, gB ) and N r ( gH, gB ) ⊆ ( N r ( gH ) ∩ gB ) for any g ∈ G .As X is a representative of an element of ( P G/ F H ( G )) G , δX is H –finite, so there is some r ≥ δX ⊆ N r ( H ) and ( δX ∩ B ) ⊆ N φ ′ ( r ) ( H, B ). Thus each component of ( B − N φ ′ ( r ) ( H, B )) is either con-tained in X or is contained in X ∗ .We claim that the deep components of ( B − N φ ′ ( r ) ( H, B )) are all con-tained in X , or are all contained in X ∗ . Since H has interlacing cosets in B , it suffices to show that, for a fixed constant ρ , any pair of componentsthat meet the same coset bH outside of N ρ ( H ) must both be in X or bothbe in X ∗ .Suppose first that r <
2, and that bH meets components C, C ′ of( B − N φ ′ ( r ) ( H, B )) outside of N r +1 ( H ), in points g and g respectively.Then g i a ∈ bHa = baH , and g i a is connected to g i by an edge that is notin δX ⊆ N r ( H ) for each i .Recall that B and baB are disjoint and separated by bA = ( bH ` baH ),hence any path from H to baH (or to baB ) must pass through bH . Itfollows that any pair of points in H and baH respectively are of distanceat least two from one another. Hence N r ( H ) is disjoint from baH , and so δX is disjoint from baB . Therefore baB is contained in X or X ∗ . Sincethe edges [ g i , g i a ] connect C, C ′ to baB without meeting δX , it followsthat baB ⊆ X implies that C ∪ C ′ ⊂ X , and baB ⊆ X ∗ implies that C ∪ C ′ ⊂ X ∗ . Hence the deep components of ( B − N φ ′ ( r ) ( H, B )) are allcontained in X or are all in X ∗ in this case, as desired.Suppose instead that r ≥
2. Let ρ = ( r + m ( φ ′ ( r − bH meets components C and C ′ of ( B − N φ ′ ( r ) ( H, B ))outside of N ρ ( H ), in points g and g respectively. Thus, for each i , B m ( φ ′ ( r − ( g i ) is disjoint from δX ⊆ N r ( H ). We have that g i a ∈ baH s of distance 1 from g i , so B m ( φ ′ ( r − ( g i a ) also is disjoint from δX . Us-ing that H satisfies deep ( m ), it follows that there is a path from each g i to any deep component D of ( baB − N φ ′ ( r − ( baH )), that does not meet δX .Recall that any point of baB is of distance at least two from H . Hence( δX ∩ baB ) ⊆ ( N r − ( baH ) ∩ baB ) ⊆ N φ ′ ( r − ( baH, baB ). Thus D doesnot meet δX , and it follows that g can be connected to g by a path thatdoes not meet δX . Therefore C ⊆ X if and only if C ′ ⊆ X , so since H has interlacing cosets in B , the deep components of ( B − N φ ′ ( r ) ( H, B ))are all contained in X , or are all contained in X ∗ , as claimed. We shallcall this argument (*), for future reference.Thus, for every r >
0, the deep components of ( B − N φ ′ ( r ) ( H, B )) areall contained in X or are all contained in X ∗ ; assume that they are allcontained in X . We will use this to show that ( X ∗ ∩ X B ) is H –finite,i.e. that X B H ⊆ X . By considering X ∗ instead, it follows that if all deepcomponents are instead contained in X ∗ then ( X ∩ X B ) is H –finite, asdesired.All deep components of ( B − N φ ′ ( r ) ( H, B )) are contained in X , so( X ∗ ∩ B ) is contained in the union of N φ ′ ( r ) ( H, B ) with the shallow com-plementary components in B , i.e. for r = ( φ ′ ( r ) + m ( φ ′ ( r ))), we have( X ∗ ∩ B ) ⊆ N r ( H, B ) ⊆ N r ( H ).Let b ∈ T B − { e } and consider b H . Recall that Comm B ( H ) = H ,hence b / ∈ Comm B ( H ). Note that b H is 2–separating, and by Lemma7.6, both H and b H satisfy the deep condition. Thus it follows fromLemma 4.1 that b H is not contained in any uniform neighborhood of H .Hence for any ρ ′ ≥
0, there is some g ∈ b H such that B ρ ′ ( g ) ∩ H = ∅ .Note that if r <
2, then δX does not meet b aB , so a simplified versionof our earlier argument shows that b aB ⊆ X .Assume instead that r ≥
2. Part of the argument (*) shows that( δX ∩ b aB ) ⊆ N φ ′ ( r − ( b aH, b aB ), hence each component of ( b aB − N φ ′ ( r − ( b aH, b aB )) is entirely contained in X or X ∗ . Let ρ ′ > max { r , r + m ( φ ′ ( r − } and let g ∈ b H be such that B ρ ′ ( g ) ∩ H = ∅ . As g ∈ B , g / ∈ N r ( H ), we have that g ∈ X . As ρ ′ > ( r + 1), ga / ∈ δX ⊆ N r ( H ),so the edge [ g, ga ] is not contained in δX , and it follows that ga ∈ X .As ρ ′ > ( r + m ( φ ′ ( r − B m ( φ ′ ( r − ( ga ) ∩ N r ( H ) = ∅ , and itfollows that B m ( φ ′ ( r − ( ga ) ⊆ X . This ball meets all deep componentsof ( b aB − N φ ′ ( r − ( b aH, b aB )), so they all must be contained in X .Hence ( X ∗ ∩ b aB ) is contained in the union of N φ ′ ( r − ( b aH, b aB )together with the shallow components of its complement, i.e. we have( X ∗ ∩ b aB ) ⊆ N φ ′ ( r − m ( φ ′ ( r − ( b aH, b aB ).In fact we can say more. We saw above that if ρ > g ∈ b H and B ρ ( g ) ∩ H = ∅ , then ga ∈ ( b aH ∩ X ). It follows that, for any ga ∈ b aH , if B ρ +1 ( ga ) ∩ H = ∅ , then ga ∈ X . In other words, ( b aH − N ρ +1 ( H )) ⊆ X ,and thus ( X ∗ ∩ b aH ) ⊆ N ρ +1 ( H ).Recall that any component of ( b aB − N φ ′ ( r − ( b aH, b aB )) is entirelycontained in X or in X ∗ . Let S be a shallow component, and suppose that S ⊆ X ∗ . Let fr ( S, b aB ) denote the frontier of S in b aB , and let fr ( S )denote the frontier of S in C ( G ). Then fr ( S, b aB ) ⊆ N φ ′ ( r − ( b aH ),so any point p ∈ fr ( S ) can be connected to some p ′ ∈ b aH by a path of ength ≤ φ ′ ( r − p ′ ∈ X ∗ , then p ∈ N φ ′ ( r − ρ +1 ( H ). If p ′ ∈ X , thenthis path must meet δX , so p ∈ N r + φ ′ ( r − ( H ). Thus, if ρ ′′ > ( φ ′ ( r −
2) + ρ ′ + 1) > ( r + φ ′ ( r − fr ( S, b aB ) ⊆ N ρ ′′ ( H ). Let r = ( ρ ′′ + m ( φ ′ ( r − S ⊆ N r ( H ).As r does not depend on our choice of S , we have ( X ∗ ∩ b aB ) ⊆ N r ( H ), so in particular ( X ∗ ∩ b aB ) is H –finite. Also r does not dependon our choice of b ∈ T B − { e } , so it follows that X ∗ ∩ [ b ∈ T B −{ e } b aB ⊆ N r ( H ) , and therefore this intersection is H –finite.As for the cosets in X B of the form b ab aB , note that if we translateeverything in the above argument by ( b a ) and replace r with ( r − δX ∩ b ab aB ) ⊆ N r − ( b ab aH ), and we also get that( X ∗ ∩ b ab aB ) is contained in some uniform neighborhood of H that isindependent of our choice of b ∈ T B − { e } . It follows that X ∗ ∩ [ b ab aB is also H –finite, where the union is taken over all choices of b , b ∈ T B −{ e } .Continuing in this manner, we see that( δX ∩ b n ab n − a · · · b aB ) ⊆ N r − n ( b n a · · · b aH ) , so for all n > r/ δX does not meet any coset of the form b n a · · · b aB (norany coset b n +1 ab n a · · · b A ). An analogous argument to the discussion of r < X . Also weget that for every n ≤ r/ X ∗ ∩ [ b n a · · · b aB is H -finite, where the union is taken over all choices b , . . . , b n ∈ T B − { e } .Note that X B is the union of the cosets of the form b n a . . . b aB , andhence X ∗ ∩ X B is H -finite, as desired.Similar methods prove the following. Proposition 8.9.
Suppose that G = A ∗ H B for A , B finitely generated.If Comm A ( H ) = H = Comm B ( H ) and H has interlaced cosets in A and B , then ˜ e ( G, H ) = 2 . Proposition 8.10.
Suppose G = A ∗ H , with A finitely generated and no-tation from the proof of Theorem 8.7. If Comm A ( H ) = H , Comm A ( H ) = H , t / ∈ Comm G ( H ) , and both H and H have interlaced cosets in A ,then ˜ e ( G, H ) = 2 . Now we will show that, under suitable hypotheses, splittings with threecoends are detectable from the coarse geometry of a group, and hence areinvariant under quasi-isometries. Our results will rely on [DS00].Recall that two subgroups are said to be commensurable if their inter-section is of finite index in each. heorem 8.11. Let G and H be as in Theorem 4.3, and suppose that H is finitely generated. Then G admits a splitting over a subgroup commen-surable with H , which has three coends. In Lemma 7.6 we showed that if, for all infinite index subgroups K of H , ˜ e ( G, K ) = 1, then H satisfies the deep condition. Towards the proofof Theorem 8.11, we have the following, the proof of which shows theconverse to this. Lemma 8.12.
Let G and H be as in Theorem 4.3. Then, for all infiniteindex subgroups K of H , e ( G, K ) = 1 .Proof.
Let
G, H and Y be as in Theorem 4.3, so Y satisfies the deep,shallow, 3–separating and noncrossing conditions, and d Haus ( Y, H ) < ∞ .By Lemma 7.1, H satisfies the shallow condition. Note that there is aquasi-isometry of C ( G ) to itself that takes Y to H , and hence by Lemma2.10, H satisfies the deep condition — say deep ( m ′ ).Note that e ( G, K ) = 0 if and only if [ G : K ] < ∞ , so if K is an infiniteindex subgroup of H then e ( G, K ) ≥
1. As e ( G, K ) ≤ ˜ e ( G, K ) for anysubgroup K of G , the lemma will follow if we can show that ˜ e ( G, K ) ≤ K of H . Suppose for a contradictionthat H contains an infinite index subgroup K such that ˜ e ( G, K ) > R > N R ( K ) is2–separating.Next, we follow the argument from the proof of Lemma 4.1. Supposein addition that there are two components of the complement of N R ( K ),say C and C , such that H meets each C i in a point p i that is not con-tained in the m ′ ( R )–neighborhood of N R ( K ). So the m ′ ( R )–ball about p i contained in C i for each i .Also we have N R ( H ) ⊇ N R ( K ), so the components of the complementof N R ( H ) are contained in the components of the complement of N R ( K ).In particular, N R ( H ) must have a deep complementary component thatis disjoint from C or is disjoint from C , and hence does not meet the m ′ ( R )–ball about p or p . But this contradicts that H satisfies deep ( m ′ ).Thus H minus the m ′ ( R )–neighborhood of N R ( K ) must be containedin a single component of the complement of N R ( K ), say C . It followsthat H is contained in the m ′ ( R )–neighborhood of N R ( K ) ∪ C . As N R ( K ) is 2–separating, there is a deep complementary component C that meets H only in the m ′ ( R )–neighborhood of N R ( K ). Recall that H satisfies the shallow condition, so there must be a deep component D of thecomplement of the ( R + m ′ ( R ))–neighborhood of H that is contained in C . In particular, note that fr ( D ) is contained in a uniform neighborhoodof K .We note that K is an infinite index subgroup of H if and only if H is not contained in any uniform neighborhood of K . Thus H is notcontained in any uniform neighborhood of fr ( D ). But this contradictsthat H satisfies the deep condition. Hence, for any infinite index subgroup K of H , ˜ e ( G, K ) ≤
1, and therefore e ( G, K ) = 1.As an immediate corollary to Lemma 8.12, we have:
Corollary 8.13.
Let G and H be as in Theorem 4.3. Then for anysubgroup H of H and any infinite index subgroup K of H , e ( G, K ) = 1 . e will also need the following. Lemma 8.14.
Let G and H be as in Theorem 4.3. Then there is afinite index subgroup H of H such that there is a nontrivial H –almostinvariant set B with BH = B . For the proof of this lemma, we roughly follow the proof of Proposition3.1 in [DS00].
Proof.
Let G , H and Y be as in Theorem 4.3, and recall that d Haus ( Y, H ) < ∞ . As Y is 3–separating and satisfies the noncrossing condition, it fol-lows, in particular, that there are some r, k = k ( r ) ≥ N = N r ( H ) has at least two deep complementary components, and satisfies noncrossing ( k ). We saw in the proof of Lemma 8.12 that H , hence N ,satisfies the deep condition. In particular, it follows that N has onlyfinitely many deep complementary components, so there is a finite indexsubgroup H of H that stabilizes each deep component.Let U be one of the deep components of ( C ( G ) − N ), and let U ∗ =( C ( G ) − U ). Thus, for any g ∈ G , gN is contained in the k –neighborhoodof U or the k –neighborhood of U ∗ . Let B = { g ∈ G : gN ⊆ N k ( U ) } .Since H stabilizes both N and U , it follows that BH = B = H B .We claim that B is H –almost invariant. Since H B = B , it sufficesto show that δB is H –finite. Let E be an edge in δB , so E has endpoints b ∈ B and bs / ∈ B , where s is a generator of G . Then b ∈ bN and bN ⊆ N k ( U ), while bs ∈ bsN , with bsN ⊆ N k ( U ∗ ). Thus b ∈ N k ( U ) and bs ∈ N k ( U ∗ ), so we must have E ⊆ N k +1 ( N ).As d Haus ( N, H ) < ∞ and d Haus ( H, H ) < ∞ since [ H : H ] < ∞ , itfollows that δB is contained in a uniform neighborhood of H . Hence δB is H –finite, so B is H –almost invariant.It remains to prove that B is nontrivial. As U is a deep componentof the complement of N , there is a sequence of vertices u , u , . . . in U such that d ( u i , N ) → ∞ as i → ∞ , and hence { u i } is H -infinite. Wecan assume that u i / ∈ N k ( U ∗ ) for all i , thus u i N * N k ( U ∗ ) for each i , so { u i } ⊆ B , and hence B is H –infinite.Since N is 2–separating, U ∗ contains a deep component of the com-plement of N , and an analogous argument shows that the complement of B is not H –finite. Thus B is a nontrivial H –almost invariant set suchthat BH = B , as desired.Finally, Theorem 3.4 of [DS00] is the following: Theorem 8.15. [DS00]
Let G be a finitely generated group and let H be a finitely generated subgroup of G . Suppose that, for every infiniteindex subgroup K of H , e ( G, K ) = 1 . If G contains a nontrivial H –almost invariant set B such that BH = B , then G splits over a subgroupcommensurable with H . Theorem 8.11 is immediate from this, Corollary 8.13 and Lemma 8.14.By combining Theorem 8.11 with the observation made in Remark 5.3,we can see that we could have originally chosen H in Theorem 4.3 so that G splits over H : heorem 8.16. Let G be a finitely generated group and let Y be a con-nected subset of C ( G ) . If Y satisfies the deep, shallow, 3–separating andnoncrossing conditions, then G contains a finitely generated subgroup H such that d Haus ( Y, H ) < ∞ and such that G has a splitting over H which has three coends. Lastly, the following is an immediate corollary to Theorems 7.9 and8.16.
Corollary 8.17.
Let G and G ′ be finitely generated groups and let f : C ( G ) → C ( G ′ ) be a quasi-isometry. Suppose that H is a finitely generated subgroupof G such that for any infinite index subgroup K of H , ˜ e ( G, K ) = 1 . Sup-pose also that G admits a splitting over H that has three coends.If sufficiently large uniform neighborhoods of f ( H ) in C ( G ′ ) satisfythe noncrossing condition, then G ′ contains a finitely generated subgroup H ′ such that H ′ is quasi-isometric to H , d Haus ( H ′ , f ( H )) < ∞ , and G ′ admits a splitting over H ′ that has three coends. References [BH99] Martin R. Bridson and Andr´e Haefliger.
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