Right-angled Artin groups and Out I:
We construct quasi-isometric embeddings from right-angled Artin groups into the outer automorphism group of a free group. These homomorphisms are in analogy with those constructed in [CLM12], where the target group is the mapping class group of a surface. Toward this goal, we develop tools in the free group setting that mirror those for surface groups as well as discuss various analogs of subsurface projection; these may be of independent interest.
For a finite simplicial graph with vertex set , the right-angled Artin group is the group presented with generators and relators when and are joined by an edge in . Such groups, though simple to define, have been at the center of recent developments in geometric group theory and low-dimensional topology because of both the richness of their subgroups (and their residual properties) and the frequency with which they occur as subgroups of other geometrically significant groups. For example, Wang constructed injective homomorphisms from certain right-angled Artin groups into SL, for [Wan07]. In [Kap11], Kapovich proved that for any finite simplicial graph and any symplectic manifold , embeds into the group of Hamiltonian symplectomorphisms of . Koberda showed that in the mapping class group of a surface raising any collections of mapping classes that are supported on connected subsurfaces to a suitably high power generates a right-angled Artin group [Kob10]. In [CLM12], the authors constructed quasi-isometric embeddings of right-angled Artin groups into mapping class groups using partial pseudo-Anosov mapping classes supported on either disjoint or overlapping subsurfaces, depending on whether the mapping classes represented commuting or non-commuting vertex generators. Specifically, they prove the following:
Theorem 1.1 (Theorem 1.1 of [Clm12]).
Suppose that are fully supported on disjoint or overlapping non-annular subsurfaces. Then after raising to sufficiently high powers, the elements generate a quasi-isometrically embedded right-angled Artin subgroup of . Furthermore, the orbit map to Teichmüller space is a quasi-isometric embedding.
Corollary 1.2 (Corollary 1.2 of [Clm12]).
Any right-angled Artin group admits a homomorphism to some mapping class group which is a quasi-isometric embedding, and for which the orbit map to Teichmüller space is a quasi-isometric embedding.
In this paper, we develop the theory necessary to quasi-isometrically embed right-angled Artin groups into Out, in analogy with [CLM12] for the mapping class group. Here, we show the following (see Section 4 for definitions and a more general statement):
Suppose that are fully supported on an admissible collection of free factors. Then after raising to sufficiently high powers, the elements generate a quasi-isometrically embedded right-angled Artin subgroup of . Furthermore, the orbit map to Outer space is a quasi-isometric embedding.
Recall that Outer space , introduced in [CV86], is the space of -marked metric graphs, and the distance being considered is the asymmetric Lipschitz metric of [FM11] (see Section 12). The admissible collection condition on the set of free factors in our main theorem is meant to mimic the surface case where the subsurfaces considered are either disjoint or overlap. It should be noted that similar to the mapping class group case our conditions for generating a right-angled Artin group involve translation lengths of the outer automorphims on the free factor complexes of their supporting free factors. Further, if is the “coincidence graph” for the involved free factors, then the right-angled Artin group generated is . This is made precise in Section 4. We also obtain,
Any right-angled Artin group admits a homomorphism to , for some , which is a quasi-isometric embedding, and for which the orbit map to Outer space is a quasi-isometric embedding.
We remark that although much of the inspiration for this paper is drawn from [CLM12], there are several significant points of departure. First, as opposed to subsurface projections in the mapping class group situation, when working with there are different possible projections that one could employ. In [BF12], the authors define the projection of the free factor to the free splitting complex of the free factor when the two factors are in “general position.” These projections, though powerful in other settings, are not delicate enough for our application. In particular, the presence of commuting outer automorphisms in our construction precludes free factors from satisfying the conditions for finite diameter Bestvina-Feighn projections, as would be required. In [SS12], a different sort of projection is considered; the authors project a sphere system in , the doubled handlebody of genus , to the disk and sphere complex of certain submanifolds. Though interesting in their own right, these projections do not always give free splittings of free factors and so are not used in this paper. These difficulties are discussed in detail in Section 5.3. To resolve these issues we develop our own projections which are tailored for the applications in this paper. In the process, we demonstrate the relationship between the projections of [BF12] and [SS12], answering a question appearing in both papers.
Second, unlike in the mapping class group case, there are no immediate analogs of the Masur-Minsky formulas, giving coarse estimates of distance in , that apply to our construction. Instead, in Section 10 we use the partial ordering on the syllables of a word in the standard generators of to control the order in which distance can be made when projecting a geodesic in to the free factor complex of a free factor. This suffices for providing the lower bounds on -distance in our main theorem.
Finally, we note that there is another method to construct quasi-isometrically embedded right-angled Artin subgroups of . One could start with a once punctured genus surface and using the methods of [CLM12] build a quasi-isometric embedding from into . In [HH11], the authors show that the injective homomorphism , induced by the action of on , is itself a quasi-isometric embedding. Composing two such maps then gives a quasi-isometric embedding from into . These homomorphisms, however, have the property that they factor through mapping class groups and, hence, fix the conjugacy class in corresponding to the puncture. In our approach, homomorphisms into do not factor through mapping class groups and outer automorphism are supported on free factors, rather than cyclic factors. Further, our construction produces homomorphisms into that have quasi-isometric orbit maps into Outer space.
1.1 Plan of paper
The paper is organized as follows: Section covers background that is needed throughout the paper. Section defines the subfactor projections that we use, gives their basic properties, and relates them to the projections of [BF12]. Section defines the homomorphisms of right-angled Artin groups into that are of interest, gives a precise statement of our main theorem, and provides a few examples.
Central to the proof of our main theorem is a version of Behrstock’s inequality which controls the projection of a tree into the free factor complex of two free factors that overlap. To prove this, we work with a topological model of the projections developed in Section . This approach has the additional advantage that the methods developed in this section can be used to relate various notions of projection that exist in the literature. In Section , we prove the version of Behrstock’s inequality that is needed. This is followed by Sections and which give related notions of order for both free factors and syllables of , respectively. This is in preparation for Section which closely follows the arguments of [CLM12] and gives conditions when normal form words in give large projections to the free factor complex of (certain) free factors.
Having arranged large projection distances, the last step is to argue that for “non-disjoint” free factors these distances independently contribute to distance in ; this is done in Section . This section can be thought of as making up for the lack of lower bounds coming from Masur-Minsky type formulas. The proof that our homomorphisms are quasi-isometric embeddings into is then concluded in Section . Since showing that these homomorphisms give quasi-isometric orbit maps into Outer space involves different terminology and constants, this is done in the appendix.
The author is grateful to Patrick Reynolds, Chris White, and Nick Zufelt for helpful conversation. Thanks are also due to Chris Leininger and Chris Westenberger, who made constructive comments on an earlier version of this paper. Most importantly, the author is indebted to his advisor Alan Reid as well as Hossein Namazi for each’s encouragement, advice, and continuous feedback throughout this project.
Let and be metric spaces. A map is a - quasi-isometric embedding if for all
In addition, if every point in is within distance from the image , then is a quasi-isometry and and are said to be quasi-isometric. In this paper, the metric spaces of interest arise from finite dimensional simplicial complexes where a fixed complex is considered with the metric induced by giving each simplex the structure of a Euclidean simplex with side length one. Recall that if is a finite dimensional simplicial complex, then this piecewise Euclidean metric on is quasi-isometric to , the -skeleton of , with its standard graph metric [BH09]. Since we are interested in the coarse geometry of such complexes, i.e. their metric structure up to quasi-isometry, this justifies our convention of when working with a complex to consider only the graph metric on , rather than the entire complex. Here, and below, a graph is a -dimensional CW complex and a simply connected graph is a tree.
Fix and let denote the free group of rank and its group of outer automorphisms. When clear from context, the subscript will be dropped from the notation. In this section we recall the definition and basic facts concerning some relevant -complexes. First, a splitting of is a minimal, simplicial actions on a non-trivial simplicial tree, the action being determined by a homomorphism into the simplicial automorphisms of . An action on a tree is minimal if there is no proper invariant subtree. By a free splitting, we mean a splitting with trivial edge stabilizers and refer to a -edge splittings as a free splittings with natural edge orbits. From Bass-Serre theory, -edge splittings correspond to graph of groups decompositions of with edges, each edge with trivial edge group. Two actions and are conjugate if there is a -equivariant homeomorphism and the conjugacy class of an action is denoted by . We will usually drop the action symbol from the notation and refer to the splitting by . Finally, an equivariant surjection between -trees is a collapse map if all point preimages are connected. In this case, is said to be a refinement of .
The free splitting complex of the free group is the simplicial complex defined as follows (see [HM11] for details): The vertex set is the set of conjugacy classes of -edge splittings of , and vertices determine a -simplex of if there is a -edge splitting and collapse maps , for each . That is, a collection of vertices span a simplex in if they have a common refinement. We will mostly work with the barycentric subdivision of the free splitting complex, denoted by , whose vertices are conjugacy classes of free splittings of and two vertices are joined by an edge if, up to conjugacy, one refines the other. Higher dimensional simplicies are determined similarly.
For , the free factor complex of is the simplicial complex defined as follows (see [HV98] or [BF11] for details): The vertices are conjugacy classes of free factors of and conjugacy classes span a -simplex if there are representative free factors in these conjugacy classes with . When , the definition is modified so that is the standard Farey graph. In this case, vertices of are conjugacy classes of rank free factors and two vertices are jointed by an edge if there are representatives in these conjugacy classes that form a basis for .
acts simplicially on these complexes. For , if is represented by an automorphism , we define . It is clear that this is independent of choice of and extends to a simplical action on all of . For the action is defined as follows: with and as above and with the action on given by the homomorphism , then is the conjugacy class of -tree determined by That is, the underlying tree is unchanged and the action is precomposed with the inverse of a representative automorphism for . Again, checking that this is a well-defined action that extends to all of (or ) is an easy exercise. These definitions have the convenient property that if is a conjugacy class of free splitting with vertex stabilizers , then has vertex stabilizers , for any .
There is a natural, coarsely defined map . For , we set equal to the set of free factors that arise as a vertex group of a -edge collapse of . That is, if and only if there is a tree , refines , and is a vertex group of . We are content to have this map defined only on the vertices of and to observe property that for any , diam. Letting denote distance in and setting diam, it is easily verified that is -Lipschitz. Note that here, and throughout the paper, the brackets that denote conjugacy classes of trees and free factors will often be suppressed when it should cause no confusion to do so.
Recent efforts to understand the free splitting and free factor complex have focused on their metric properties along with their similarity to the complex of curves of a surface. In particular, both complexes are now known to be Gromov-hyperbolic. Hyperbolicity of the free factor complex was first shown by [BF11] then [KR12]; hyperbolicity of the free splitting complex was first shown by [HM11] and then [HH12]. We record this as a single theorem.
Remark that the action is far from proper; all vertices have infinite point stabilizers. There is, however, an invariant subcomplex of that is locally finite, and the inherited action is proper. This is the spine of Outer space and we refer the reader to [CV86] or [HM10] for details beyond what is discussed here. Also, see [Hat95] or [AS11] for an alternative perspective.
The spine of Outer space is the subcomplex of span by vertices that correspond to proper splittings of . Recall that a splitting is proper if no element of acts elliptically, i.e. fixes a vertex, in . Hence, is proper if is a graph with fundamental group isomorphic to . Observe that since preserve the vertices of corresponding to proper splittings there is an induced simplicial action .
It is well-known that is a locally finite, connected complex and that the action is proper and cocompact (see [CV86]). Hence, for any tree , the orbit map defines a quasi-isometry from to by the S̆varc-Milnor lemma [BH09]. As remarked above, the metric considered is the standard graph metric on , the -skeleton of the spine of Outer space. This metric on will serve as our geometric model for .
2.3 The sphere complex
We recall the -equivalent identification between the free splitting complex and the sphere complex. See [AS11] for details. Take to be the connected sum of copies of , or equivalently, the double of the handlebody of genus . Let be with open -balls removed. Note that is isomorphic to and once and for all fix such an isomorphism. A sphere in is essential if it is not boundary parallel and does not bound a -ball, and a collection of disjoint, essential, pairwise non-isotopic spheres in is called a sphere system. By [Lau74], spheres and are homotopic in if and only if they are isotopic.
The sphere complex is the simplicial complex whose vertices are isotopy classes of essential spheres and vertices span a -simplex if there are representatives in these isotopy classes that are disjoint in . It is a theorem of [Lau74] that setting (Diff) there is an exact sequence
where is a finite group generated by “Dehn twists” about essential spheres. Since elements of act trivially on , we have a well-defined action . The following proposition of Arramayona and Souto identifies and . See Section 5.1 for how one constructs splittings from essential spheres.
Proposition 2.2 ([As11]).
For , and are -equivariantly isomorphic.
2.4 Translation length in
Fully irreducible outer automorphisms are those that have no positive power that fixes a conjugacy class of free factor, i.e. for any , implies that . Recall that the (stable) translation length of an outer automorphism on is defined as
where . It is not difficult to verify that is well-defined, independent of choice of , and satisfies the property for . Further note that if and only if for all , . The following proposition characterizes those outer automorphisms with positive translation length on .
Proposition 2.3 ([Bf11]).
Let , is fully irreducible if and only if .
It appears to be an open question whether there is a uniform lower bound on translation length for fully irreducible outer automorphisms for a fixed rank free group, as is the case for pseudo-Anosov mapping classes acting on the curve complex [MM99]. It is worthwhile to note that when , Proposition 2.3 reduces to the statement that if an outer automorphism is infinite order and does not fix a conjugacy class of a primitive element in , then it acts hyperbolically on , which as noted above is the Farey graph. This well-known statement is what is used in most of our applications.
3 Projections to free factor complexes
For a finitely generated subgroup , let and denote the free splitting complex and free factor complex of , respectively. The subgroup is self-normalizing if , where is the normalizer of in . Less formally, a subgroup is self-normalizing if the only elements that conjugate back to itself are those elements in . When is self-normalizing the complexes and depend only on the conjugacy class of in . More precisely, if for , then induces an isomorphism between and (and between and ) via conjugation. For any other with we see that normalizes and so . In this case, and it is easily verified that and induce identical isomorphisms between and . Hence, when is self-normalizing we obtain a canonical identification between the free splitting complex of and the free splitting complex of each of its conjugates. The same holds for the free factor complex of . This allows us to unambiguously refer to the free splitting complex or free factor complex for the conjugacy class . Finally, recall that a subgroup is malnormal if implies that . For example, free factors of are malnormal and malnormal subgroups are self-normalizing.
3.1 Projecting trees
Given a free splitting and a finitely generated subgroup denote by the minimal -subtree of . This is the unique minimal -invariant subtree of the restricted action and is either trivial, in which case fixes a unique vertex in , or is the union of axes of elements in that act hyperbolically on . When is not trivial, we define the projection of to the free splitting complex of as , where the brackets denote conjugacy of -trees. Note that this projection is a well-defined vertex of and depends only on the conjugacy class of . This is the case since any conjugacy between -trees will induce a conjugacy between their minimal -subtrees. Further define the projection to the free factor complex of to be the composition , where is the -Lipschitz map defined in Section 2.2. Hence, is the collection of free factors of that arise as a vertex group of a one-edge collapse of the splitting . When is also self-normalizing, e.g. a free factor, these projections are independent of the choice of within its conjugacy class. The following lemma verifies that such projections are coarsely Lipschitz.
Let be a free splitting and finitely generated with non-trivial. Let be a refinement of with equivalent collapse map . Then there is an induced map which is also a collapse map. Hence, is a refinement of .
Since is an invariant -tree, it contains . Also, the axis in of any hyperbolic , which exists because is nontrivial, is mapped by to either ’s axis in or a singe vertex stabilized by ; each of which is contained in . Since is the union of such axes, we see that . Hence the map described in the lemma is given by restriction. It remains to show that is a collapse map. This is the case since for any ,
is the intersection of two subtrees of and is therefore connected. ∎
For a free factor of we use the symbol to denote distance in and for -trees we use the shorthand
when both projections are defined. The following proposition follows immediately from the definitions in this section and Lemma 3.1.
Proposition 3.2 (Basic properties I).
Let be adjacent vertices in , , and a finitely generated and self-normalizing subgroup of containing , up to conjugacy. Then we have the following:
3.2 Projecting factors
Consider rank free factors and of the free group . Define and to be disjoint if they are distinct vertex groups of a common splitting of . Disjoint free factors are those that will support commuting outer automorphisms in our construction. Define and to meet if there exist representatives in their conjugacy classes whose intersection is nontrivial and proper in each factor. In this section, we show that this intersection provides a well-defined projection of to , the free factor complex of . Note that if and meet, then .
Fix free factors and in . Define the projection of into as
where conjugacy is taken in . Observe that and meet exactly when . We show that members of are vertex groups of a single (non-unique) free splitting of and so has diameter in . Since the projection is independent of the conjugacy class of , this provides the desired projection from to the free factor complex of .
Suppose the free factors and meet. Then diam.
First, observe that uniquely determines the class up to double coset in . Precisely, if and only if ; this follows from the fact that free factors are malnormal. Now choose any marked graph which contains a subgraph whose fundamental group represents up to conjugacy. Let be the cover of corresponding to free factor and let denote the core of . By covering space theory, the components of are in bijective correspondence with the double cosets and the fundamental group of the component corresponding to is . Since the core carries the fundamental group of , all nontrivial subgroups correspond to double cosets representing components of in the core . Hence, is a marked -graph that contains disjoint subgraphs whose fundamental groups (up to conjugacy in ) are the subgroups of . This completes the proof. ∎
If and stabilizes , then induces an outer automorphism of , denoted . In this case, let represent the translation length of on . By Proposition 2.3, if is fully irreducible in , then . The following proposition provides the addition properties of the projections that will be needed throughout the paper. Its proof is a straightforward exercise in working through the definition of this section.
Proposition 3.4 (Basic Properties II).
Let so that and meet and and are disjoint. Let stabilize the free factors and with in . Finally, let and be arbitrary. Then induces an isomorphism and we have the following:
f(A) and f(B) meet and .
For the applications in this paper, a slightly stronger condition than meeting is necessary on free factors and . In particular, we need their meeting representatives to generate the “correct” subgroup of . More precisely, say that two free factors and of overlap if there are representatives in their conjugacy classes, still denoted and , so that is proper in both and and the subgroup generated by these representatives is isomorphic to . Note that the first condition here is exactly that and meet.
Suppose the free factors overlap and select representatives in their conjugacy classes so that is nontrivial and proper in both and . Note that as in Lemma 3.3 the free factor is not necessarily unique up to conjugacy, but once the conjugacy class of is fixed the subgroup generated by these conjugacy class representatives is itself determined up to conjugacy in . Since and overlap, can be chosen so that and it is not difficult to verify that is finitely generated and self-normalizing. So, for example, if , then by Lemma 3.2. Projections of meetings factors, however, may slightly change. In particular, and are free factors of that overlap, but with conjugacy now considered in , is their unique intersection up to conjugacy. In general, we use the notation to denote the projection of into the free factor complex of when is considered as a free factor of the free group . Note that in this case and so although the choice of and, hence, is not uniquely determined by the overlapping free factors and , this ambiguity is not significant when considering projections.
3.3 The Bestvina-Feighn Projections
In [BF12], the authors show that there is a finite coloring of the vertices of the free factor complex and an so that if and are free factors either having the same color or with , then there is a well-defined projection whose diameter is less than or equal to . Moreover, these projections have properties that are analogous to subsurface projections. Their projection is defined by choosing any so that the marked graph contains an embedded subgraph whose fundamental group represents and taking It is shown that when and satisfy the stated conditions, this projection is coarsely independent of the choice of .
Free factors that meet, however, do not satisfy the conditions stated above and it is easy to construct examples where and meet but the projection does not have finite diameter in (as the choice of is varied). Despite this, Lemma 3.3 shows that if we further project to the free factor complex of we obtain a set with finite diameter. This shows that when the free factors and meet, the projection defined in this paper agrees coarsely with the projection . In Section 5.3, we relate the projections discussed here with those of [SS12].
4 The homomorphisms
In this section, we present the most general version of our theorem. Technical conditions are unavoidable since, unlike the surface case, free factors do not uniquely determine splittings and defining the support of an outer automorphism is more subtle. After presenting general conditions, we give a specific construction to which our theorem applies. The idea is to replace the surface in the mapping class group situation with a graph of groups decomposition of .
4.1 Admissible systems
Let be a collection of (conjugacy classes of) rank free factors of such that for either
and are disjoint, that is they are vertex groups of a common splitting, or
and overlap, so in particular .
Then we say that is an admissible collection of free factors of . Let be the coincidence graph for . This is the graph with a vertex for each and an edge connecting and whenever the free factors and are disjoint.
An outer automorphism is said to be supported on the factor if for each in the star of and for each in the link of . Informally, is required to stabilize and act trivially on each free factor in that is disjoint from as well as stabilize itself. We say that is fully supported on if in addition is fully irreducible. Finally, we call the pair an admissible system if the are fully supported on the collection of free factors and for each joined by an edge in , and commute in (this condition is made unnecessary in the construction of the next section).
Given an admissible system , we have the induced homomorphism
defined by mapping Our main theorem is the following:
Given an admissible collection of free factors for with coincidence graph there is a so that if outer automorphisms are chosen to make an admissible system with then the induced homomorphism is a quasi-isometric embedding.
It is worth noting that since right-angled Artin groups are torsion-free, homomorphisms from that are quasi-isometric embeddings are injective. Since it requires a different set of terminology as well as constants that need to be determined, we save the statement and proof that these homomorphisms induce quasi-isometric orbit maps into Outer space for the appendix.
4.2 Splitting contruction
Here we present a particular type of graph of groups decompositions of to which our theorem applies and use it to construct examples. Let be a free splitting of along with a family of collapse maps
to splittings , satisfying the following conditions:
Each splitting has a preferred vertex so that all edges of are incident to .
Setting we require that for one of the two following conditions hold: either and are disjoint, meaning that , or is a subgraph whose induced subgroup is nontrivial and proper in each of the subgroups induced by and . In the latter case, we say the subgraphs overlap.
We call the splitting satisfying these conditions a support graph and note that the above data is determined by the collection of subgraphs . For such a splitting of , we set , the vertex groups of the vertex in . It is clear from the above conditions that such a collection of free factors forms an admissible collection and that is precisely the coincidence graph of the in .
Next, consider the outer automorphisms that will generate the image of our homomorphism. For each , chose an which preserves the splitting , induces the identity automorphism on the underlying graph of , and restricts to the identity on the complement of in . In this case, we say that is supported on (or ) and if the restriction of to the free factor is fully irreducible, we say that is fully supported on (or ). With these choices, the pair is an admissible system. Indeed, the only condition to check is that if and represent disjoint free factors, then the outer automorphisms and commute. Observe that since and are disjoint subgraphs of we may collapse each to a vertex to obtain a common refinement of and which has vertices with associated groups and . Label these vertices of and corresponding to the subgraphs and of . From the fact that and are supported on and , respectively, it follows that they both stabilize the common refinement and are each supported on distinct vertices, namely and . This implies that and commute in . Hence, is an admissible system inducing a homomorphism
as before. With this setup, our main result can be restated as follows:
Suppose is a free splitting of that is a support graph with subgraphs for having coincidence graph . There is a so that if for each , is fully supported on with , then the induced homomorphism is a quasi-isometric embedding.
We remark that once a support graph is constructed with , there is no obstruction to finding fully supported on with large translation length on . Corollary 4.2 then implies that there exist homomorphisms which are quasi-isometric embeddings.
4.3 Constructions and applications
We use our main theorem to construct quasi-isometric homomorphisms into beginning with an arbitrary right-angled Artin group . We provide a bound on given a measurement of complexity of .
First, it is an easy matter to use the splitting construction to start with a graph and find a quasi-isometric embedding , with depending on . We first illustrate this with an example and then give a general procedure. Note that although using the splitting construction is simple, it will always require that is rather large compared to . As demonstrated in Example , more creative choices of admissible system can, however, be used to reduce .
Let be the pentagon graph with vertices labeled counter-clockwise as in Figure 1, and let be the same graph with vertices labeled cyclically . Take to be the graph of groups with underlying graph , the barycentric subdivision of , with trivial vertex group labels on the vertices of and infinite cyclic group labels on the subdivision vertices. Note that . Set equal to the subgraph of consisting of the vertex labeled , its two adjacent subdivision vertices, and the edges joining these vertices to . Observe that and have empty intersection if and only if and are joined by an edge in , and if and intersect then their intersection is a vertex with nontrivial vertex group. Hence, is a support graph with subgraphs whose coincidence graph is and so by Corollary 4.2 there is a constant such that choosing any collection of outer automorphisms fully supported on the collection with determines a homomorphism that is a quasi-isometric embedding. In Example , we improve this construction by modifying .
Now fix any simplicial graph with vertices labeled . We give a general procedure for producing a support graph with subgraphs whose coincidence graph is . By Corollary 4.2, this allows one to construct homomorphisms which are quasi-isometric embeddings for any right-angled Artin group. First, assume that the complement graph is connected. This is the subgraph of the complete graph on the vertices of with edge set given by the complement of the edge set of . Let be the barycentric subdivision of where we reserve labels for the vertices of that are vertices of and label the vertex of corresponding to the edge of by . Hence, in the vertex is valence two and is connected by an edge to both and . Set equal to the star of the vertex in , i.e. is the union of edges incident to together with their vertices. Now take to be the graph of groups with underlying graph and infinite cyclic vertex group labels for each vertex , . For vertices there are two cases for vertex groups: If has valence one in then we label it with infinite cyclic vertex group and otherwise we give it trivial vertex group.
With these vertex groups, becomes of graph of groups decomposition for which is a support graph for the collection of subgraphs with coincidence graph . Indeed, and have nonempty intersection in if and only if and are joined by and edge in . When this is the case, their intersection is a single vertex with infinite cyclic vertex group and this vertex group is proper in each of the groups induced by and . We can also calculate the rank of . By construction, the rank of is equal to the rank of the fundamental group of the underlying graph plus the number of nontrivial vertex groups on . Since there is a nontrivial vertex group for each edge of and each vertex of of valence one, the rank of equals
Translating this into a function of , we see that the rank of is
and we refer to this quantity as the complexity of , denoted .
When is not connected it decomposes into components and it is not difficult to show that . In this case, we set and the corresponding supported graph is constructed as follows: Let be the support graph constructed as above for the graph , or any support graph with coincidence graph . Take to be the support graph built by taking the wedge of intervals (at one endpoint of each) and attaching the other endpoint of the th interval to an arbitrary vertex of . The graph of groups structure on is induced by that of with a trivial group label at the wedge vertex. Then is a support graph with coincidence graph and complexity . As noted above, the existence of a support graph with coincidence graph implies the following (the statement about Outer space is established in the appendix):
For any simplicial graph , admits a homomorphism into , with , which is a quasi-isometric embedding, and for which the orbit map to Outer space is a quasi-isometric embedding.
The next example shows how Theorem 11.1 can be used to give quasi-isometric embeddings into for smaller than using support graphs, as in our construction above.
Again, let be the pentagon graph with vertices labeled counter-clockwise as in Figure 1. Take as in Figure 1. This is a graph of groups decomposition for ; the central vertex has trivial vertex group and the valence one vertices joined to the central vertex each have infinite cyclic vertex group, with generators labeled . can be thought of as a “folded” version of the support graph that appears in Example . For , let be the smallest connected subgraph containing the vertices labeled and , with indices taken mod . Note that together with the subgraphs is not a support graph; for example and intersect in a vertex with trivial vertex group. Despite this, for , does form an admissible collection of free factors with coincidence graph . Hence, by Theorem 11.1 there exists a so that if there are outer automorphisms making an admissible system with then the induced homomorphism is a quasi-isometric embedding. Choosing such a collection in this case is straightforward. Specifically, let and choose for so that
the restriction is fully irreducible with , and
the restriction .
With these choices, it is clear that each is fully supported on and that and commute if and only if and are joined by an edge of . This makes into an admissible system with and so the induced homomorphism
is a quasi-isometric embedding. In fact, as we shall see in the proof of the main theorem, the required translation length is simple to determine. Further, as each of free factors in the admissible system is rank , the free factor complex is the Farey graph where translation lengths can be computed.
As an application, it is well-known that contains quasi-isometrically embedded copies of , the fundamental group of the closed genus surface (see [CW04]). Restricting the homomorphism constructed above to such a subgroup we obtain quasi-isometric embeddings
As we will see in the appendix, these homomorphisms can be chosen to give quasi-isometric orbit maps into Outer space, .
Let and set for . It is not difficult to verify that for , and . See Stallings’s paper [Sta83] for how to efficiently compute such intersections. Hence, for , the collection is an admissible collection of pairwise overlapping free factors and so there are such that choosing any collection of outer automorphisms fully supported on the collection with determines a homomorphism that is a quasi-isometric embedding. In fact, we will see in the proof of Theorem 11.1 that we may take to be constant over all and obtain a uniform lower bound on -word length of in terms of word length of in , independent of .
5 Splittings and submanifolds
It is important to have a topological interpretation of our projections in order to prove the version of Behrstock’s inequality that appears in the next section. We first review some facts about embedded surfaces in -manifolds and the splittings they induce.
5.1 Surfaces and splittings
It is well-known that codimension submanifolds induce splittings of the ambient manifold group [Sha01]. We review some details here, focusing on the case when then inclusion map is not necessarily -injective.
For our application, begin with an orientable, connected -manifold possibly with boundary and a property embedded, orientable surface . We do not require that is connected or that each component of is -injective. Working, for example, in the smooth setting, choose a tubular neighborhood of in whose restriction is a tubular neighborhood of the boundary of in . Let denote the graph dual to in . This is the graph with a vertex for each component of and an edge for each component that joins vertices corresponding to the (not necessarily distinct) components on either side of . We may consider as embedded in and, after choosing an appropriate embedding, is easily seen to be a retract of