Nonexistence of boundary maps for some
hierarchically hyperbolic spaces
Abstract
We provide negative answers to questions posed by Durham, Hagen, and Sisto on the existence of boundary maps for some hierarchically hyperbolic spaces, namely maps from rightangled Artin groups to mapping class groups. We also prove results on existence of boundary maps for free subgroups of mapping class groups.
1 Introduction
Let be a finite graph with vertex set . The rightangled Artin group determined by , denoted by , is the group with the following presentation:
Let be a connected, oriented surface of genus with punctures, and let denote the mapping class group of . Clay, Leininger, and Mangahas [5] and Koberda [7] construct “nice” embeddings of rightangled Artin groups to mapping class groups. In [1] and [2], a geometric structure called a hierarchically hyperbolic space (HHS) was introduced. Important examples of spaces that are HHS’s include mapping class groups of surfaces and rightangled Artin groups. In [6] Durham, Hagen, and Sisto construct a boundary for hierarchically hyperbolic spaces (see Section 2). In that paper, the authors ask the following question, motivated by a desire to develop a notion of geometrically finite subgroups of mapping class groups.
Question 1.1.
We prove that in general the answer to this question is no by providing, for each type of embedding, an explicit example where the embedding does not extend continuously.
Theorem 1.2.
There exists a surface , a rightangled Artin group , a Clay, Leininger, and Mangahas embedding , and a Koberda embedding such that, regardless of the HHS structure on , neither nor extends continuously to a map .
We also prove the following result which gives a complete characterization of Koberda embeddings of free groups, which send all generators to powers of Dehn twists, that have continuous extensions.
Theorem 1.3.
Let be a collection of pairwise intersecting curves in and the graph with and no edges. For sufficiently large , the homomorphism
is injective by the work of Koberda [7]. Moreover, extends continuously to a map if and only if pairwise fill , where is equipped with any HHS structure.
In fact, we prove something stronger than Theorem 1.3. We prove a nonexistence result (Theorem 5.3) for a class of Koberda embeddings of rightangled Artin groups that are not necessary free groups. We also prove an existence result (Theorem 6.1) for a class of embeddings of free groups that includes the Koberda embeddings described in Theorem 1.3 as well as a class of Clay, Leininger, Mangahas embeddings.
In Section 2 we will recall relevant definitions and theorems and introduce notation. Section 3 will establish a handful of lemmas that will be used for proving Theorem 1.2. Section 4 is devoted to proving Theorem 1.2 for a Clay, Leininger, Mangahas embedding, and in Section 5 we prove Theorem 1.2 for a Koberda embedding. Using similar techniques, we then prove that a more general class of Koberda embeddings of rightangled Artin groups do not extend continuously (Theorem 5.3), which will imply one direction of Theorem 1.3. In Section 6 we will prove Theorem 6.1, which will imply the other direction of Theorem 1.3.
Remark: Koberda [7] proved that both types of embeddings we discuss are injective. We call the embeddings that send generators of our rightangled Artin group to mapping classes that are pseudoAnosov on subsurfaces Clay, Leininger, Mangahas embeddings primarily to distinguish the two types, but also to emphasize that these types of embeddings have nice geometric properties (see Theorem 2.5).
Acknowledgments: The author was supported by the Department of Defense (DoD) through the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program and by a Research Assistantship through NSF Grant number DMS1510034. The author would like to thank her PhD advisor Chris Leininger for his numerous insights which inspired a great deal of this paper and also for his patience and consistent support. The author would also like to thank Mark Hagen and Matt Durham for helpful conversations.
2 Background
In this section, we recall some needed definitions and theorems.
Notation: Let be functions. Given constants and , we write to mean for all , and will just write when the constants are understood.
2.1 Curves and subsurfaces
Throughout this paper, we let denote a connected, oriented surface of genus with punctures. Define the complexity of to be . We will always assume . Additionally, we fix a complete hyperbolic metric on . That is, we assume that is of the form , where and acts properly discontinuously and freely on .
For let be a biinfinite path in with ends limiting to distinct points and on . We say that and link if the geodesic connecting to intersects the geodesic connecting to in the interior of .
By a curve in , we will always mean the geodesic representative in the homotopy class of an essential, simple, closed curve in . By a multicurve in , we will always mean a collection of pairwise disjoint curves in . We write to denote the geometric intersection number of curves and . We say that a pair of curves and fill if for every curve in we have or .
A nonannular subsurface of is a component of after removing a (possibly empty) collection of pairwise disjoint curves on . Additionally, we require that satisfies ; in particular, we do not consider a pair of pants to be a subsurface. We define to be the collection of curves in that are disjoint from and also are contained in the closure of , treating as a subset of . When , the path metric completion of is a surface with boundary, and the image of this boundary under the map induced by the inclusion is .
An annular subsurface of is define as follows. Let be a curve in . Choose a component of the preimage of in , and let be a primitive isometry with axis . Define
where and are the fixed points of on . Observe that is a compact annulus and is a covering. We say that is the annular subsurface of with core curve . We define to be .
For any subsurface of , we will write , even though when is an annulus, is not a subset of .
Given and a curve or simple biinfinite geodesic in , we define to be the curve or simple biinfinite geodesic obtained as follows. Consider a component of the preimage of in . Choose a representative in the isotopy class of and lift it to a map . We define to be the image in of the geodesic in that connects the endpoints of on . Given , if is nonannular we let denote the nonannular subsurface in its isotopy class. If is an annulus with core curve , we let denote the annular subsurface of with core curve .
2.2 Curve complex
Let be a subsurface of . If satisfies , the curve complex of , denoted , is the simplicial complex whose vertices are curves contained in , and if , a set of vertices form a simplex if and only if they are pairwise disjoint. If , then we define the simplices of differently. In the case that is a once punctured torus, a set of vertices form a simplex if and only if they pairwise intersect exactly once. If is a four times punctured sphere, a set of vertices form a simplex if and only if they pairwise intersect exactly twice.
Now let be a compact annulus. Consider all embedded arcs in that connect one boundary component to the other. We define two arcs to be equivalent if one can be homotoped to the other, fixing the endpoints of the arcs throughout the homotopy. In this case, the curve complex of is the simplicial complex whose vertices are equivalence classes of arcs, and a set of vertices form a simplex if and only if for each pair of vertices there exist representative arcs of each whose restrictions to are disjoint. The following simple formula will be useful to us: given inequivalent arcs in ,
(1) 
where denotes the algebraic intersection number of and .
2.3 Markings and subsurface projection
A marking on is a maximal collection of pairwise disjoint curves in , denoted , together with another collection of associated curves called transversals: for each its associated transversal is a curve that intersects minimally (i.e. once or twice) and is disjoint from all other curves in .
Let be a subsurface of and a multicurve in . We will now define the projection of to , which we will denote by . Suppose is not an annulus and is a single curve. If is disjoint from , define . If is contained in , define . Otherwise, is a collection of essential arcs in with endpoints on . For each such arc , take the geodesic representatives of the boundary components of a small regular neighborhood of that are contained in . Define to be the collection of all such curves over all arcs in . If is a multicurve, define to be the union of the projections to of each curve in .
Now let be an annular subsurface with core curve and the associated covering. Let be a multicurve or a biinfinite, simple geodesic in . Consider the full preimage of in . Each component is an arc in which we view as having endpoints on the boundary of . In this case, we define to be the (equivalence classes of) arcs in this collection that have an endpoint on each boundary component of . When convenient, we will write instead of .
We now describe how to project a marking to . If is nonannular or is an annulus whose core curve is not contained in , we define . Otherwise, is an annulus with core curve , and we define to be , where is the transversal associated to .
Given any subsurface , we define
where and are markings, collections of curves, or (when is an annulus) biinfinite simple geodesics in . A useful fact about subsurface projection is the following. For all
In this paper, we utilize the following theorem, which involves subsurface projections.
Theorem 2.1 (Lemma 2.3 in [10]).
For all subsurfaces of , given any marking or multicurve such that , we have that . If is an annulus, then .
Masur and Minsky [10] define the marking graph of , denoted , to be the graph whose vertices are markings and vertices are adjacent if one can be obtained from the other by an elementary move; see [10] for a complete definition. Giving the path metric and a word metric , there is an action of on by isometries for which every orbit map is a quasiisometry. The following theorem gives a relationship between distances in and subsurface projections.
Theorem 2.2 (Lemma 3.5 in [10]).
For any subsurface of and any markings and on , we have that .
We say that distinct subsurfaces and are disjoint if and . We say that is a proper subsurface of , denoted , if and . We say that and are overlapping, denoted , if and . In the case where and are not annuli, these relationships, respectively, are disjointness, proper containment, and intersection without containment as subsets of . We say and fill if for every curve in we have or .
The following theorems will be used to prove our results. The first theorem was proved in [4] and later a simpler proof with constructive constants appeared in [8].
Theorem 2.3 (Behrstock inequality: Theorem 4.3 in [4], Lemma 2.13 in [8]).
Let and be overlapping subsurfaces of and a marking on . Then
Theorem 2.4 (Bounded Geodesic Image Theorem: Theorem 3.1 in [10]).
There exists a constant depending only on such that the following is true. Let and be subsurfaces of with a proper subsurface of . Let be any geodesic segment in satisfying for all . Then
2.4 Partial order on subsurfaces
Let be markings on and . Let denote the collection of subsurfaces of such that . Behrstock, Kleiner, Minsky, and Mosher [3] define the following partial order on . Given such that , define if and only if one of the following equivalent conditions is satisfied:
That these conditions are equivalent is a consequence of Theorem 2.3; see Corollary 3.7 in [5].
2.5 Embedding RAAGs in Mod(S)
If is such that there exists a representative in the isotopy class of that pointwise fixes the complement of a nonannular subsurface , we say that is supported on . Given such an , we define the translation length of on to be
where is any marking on . If is a power of a Dehn twist about a curve , we say that is supported on the annular subsurface with core curve , and define to be the absolute value of the power. In either case, we say that fully supports if By the work of Masur and Minsky [9], when is nonannular, fully supports if and only if is pseudoAnosov on .
Clay, Leininger, and Mangahas [5] proved the following result, which allows us to find quasiisometrically embedded rightangled Artin subgroups inside .
Theorem 2.5 (Theorem 2.2 in [5]).
Let be a finite graph with , and let be a collection of nonannular subsurfaces of . Suppose is an edge in if and only if and are disjoint, and is not an edge in if and only if or . Then there exists a constant such that the following holds. Let be a set of mapping classes of such that is pseudoAnosov on and satisfies for all . Then the homomorphism
is a quasiisometric embedding, and in particular is injective.
Koberda [7] also has a result which produces rightangled Artin subgroups of . Below we give a special case of Koberda’s result that we will use.
Theorem 2.6 (Theorem 1.1 in [7]).
Let be a collection of distinct curves in . Let be the graph with and with an edge in if and only if . Then for sufficiently large , the homomorphism
is injective, where denotes a Dehn twist about .
2.6 Gromov boundary of hyperbolic spaces
A geodesic metric space is Gromov hyperbolic (or just hyperbolic) if there exists a such that given any geodesic triangle in , each side is contained in the neighborhood of the union of the other two sides. Given a Gromov hyperbolic space and points , the Gromov product of and with respect to is defined as
We say that a sequence in converges at infinity if for some (any) . We define two such sequences and to be equivalent if for some (any) . The Gromov boundary of is the collection of all such sequences up to this equivalence, and is denoted or just when it is clear from context that we are using the Gromov boundary.
One Gromov hyperbolic space that this paper is concerned with is the curve complex of , which was proved to be Gromov hyperbolic by Masur and Minsky [10]. We can now state a corollary of Theorem 2.4 that will be useful later.
Corollary 2.7.
Let and be subsurfaces of with a proper subsurface of . Suppose is a sequence of markings on such that for some . Then
Proof.
For each , choose . Because , we can choose large so that for all we have
(2) 
where the Gromov product is computed in . Consider . Let be a geodesic in with endpoints and . If there exists a vertex on with , then and form a multicurve in which implies that
But this contradicts Inequality (2), so we conclude that for all on . We can now apply Theorems 2.1 and 2.4 to see that for all
where is an in Theorem 2.4. Therefore,
∎
2.7 Hierarchically hyperbolic spaces
In [1] Behrstock, Hagen, and Sisto define the notion of a hierarchically hyperbolic space. Roughly, a hierarchically hyperbolic space is a quasigeodesic metric space , equipped with additional structure which we will call a hierarchically hyperbolic space (HHS) structure. An HHS structure consists of an index set and for each a Gromov hyperbolic space and a projection map . The elements of and the projection maps must satisfy a long list of properties. See [1] and [2].
The first example of a hierarchically hyperbolic space is , where here is the collection of all subsurfaces of , is the curve graph of for , and projection is given by composing an orbit map for the action of on with the subsurface projection map defined in Section 2.3. The work of Masur and Minsky [9],[10] and Behrstock [4] imply that is a hierarchically hyperbolic space. See [2] Section 11 for details. In fact, the notion of hierarchical hyperbolicity was motivated by a desire to generalize some of the machinery surrounding mapping class groups.
In [1] it is shown that a large class of cube complexes can be equipped with a hierarchically hyperbolic structure, including the universal covers of Salvetti complexes associated to rightangled Artin groups. The cube complex we are primarily concerned with is the Cayley graph of when has no edges (that is, is a free group). We equip with a hierarchically hyperbolic structure by equipping with such a structure and then associating with .
2.8 Boundary of hierarchically hyperbolic spaces
In [6] the authors construct a boundary for hierarchically hyperbolic spaces. Here we will describe convergence in this boundary for and for free groups. With the exception of Theorem 5.3, these will be the only examples we will need.
As a set, the HHS boundary of is defined as follows:
(3) 
In [6], the authors define a topology on . In this topology, Definition 2.10 of [6] tells us that a sequence of mapping classes in converges to a point in , where for all , , and for pairwise disjoint subsurfaces , if and only if the following statements hold: For a fixed marking on ,

for each ,

, and for each

for every (any) and every subsurface that is disjoint from for all .
Let be a graph with no edges, and let be the corresponding free group, equipped with an HHS structure. The HHS boundary of will be denoted by . We do not define here because Theorem 4.3 in [6] implies that the identity map extends to a homeomorphism . Thus, two sequences in converge to the same point in if and only if they converge to the same point in . (See Section 2 of [6] for the definition of .)
Another useful fact on convergence is that and are sequentially compact (see Theorem 3.4 of [6]).
To understand Question 1.1 and the statements of our theorems, one last definition is needed.
Definition 2.8.
Let be an injective homomorphism and let and be equipped with any fixed HHS structures. We say that extends continuously to a map if there exists a function such that (1) , (2) , and (3) is continuous at each point in .
Remark 2.9.
To establish that extends continuously, it is enough to show that for all , given any two sequences and in that converge to , we have that and converge to the same point in . This follows from a diagonal sequence argument (see the end of the proof of Theorem 5.6 in [6] for details).
3 Lemmas on subsurface projections
The following lemmas are the heart of our proof of Theorem 1.2.
Lemma 3.1.
Suppose and are disjoint subsurfaces of , and if is an annulus, then the core of is not contained in . If and are markings and a mapping class supported on , then .
Proof.
If is not an annulus, then so the claim clearly holds. Assume then that is an annular subsurface of with core , and let be the associated covering. Because and is not in , we can find a curve in , distinct from , that intersects and satisfies . If is not an annulus, define to be the component of that contains . If is an annulus with core , let be the component of containing after removing a small regular neighborhood of . The neighborhood should be taken small enough so that is contained in . Let be the component of the preimage of in that is a closed curve. Let be the component of the preimage of in that contains .
Abusing notation, we let denote a representative in the isotopy class of that fixes pointwise. Let denote the lift of that fixes a point on , and thus fixes pointwise. Let be a component of the preimage of in that intersects . Then is contained in , implying that fixes pointwise.
Lemma 3.2.
Given a homomorphism and a marking on , there exists a constant such that the following holds. Let , where each . Then for all subsurfaces .
Proof.
Lemma 3.3.
Let be a homomorphism. Let be a sequence of elements in and a marking on . Suppose for some subsurface there exist constants and , that do not depend on , such that where denotes the word length of with respect to the standard generating set for . Further suppose that and that converges to some point in . Then all accumulation points of in are in and are of the form , where .
Proof.
After passing to a subsequence, we may assume that converges. By assumption, . Combine this with Theorem 2.2 to see that . Because is quasiisometric to via orbit maps, it follows that . Thus, it must be that .
Suppose for constants and . We will now argue that . Let be such that . If , we are done. So we assume . By definition of the topology on , we have that . If , then Corollary 2.7 implies that . But this cannot be since . Similarly, we cannot have for then Corollary 2.7 implies that , contradicting that . Now suppose that . Then by Theorem 2.3, after passing to a subsequence, we have that
If for all , then for all
contradicting that . Similarly, if for all , then is bounded independent of contradicting that . So it is not the case that . Therefore it must be that and are disjoint for all with .
Fix with . Lemma 3.2 together with the fact that implies that
(4) 
where is as in Lemma 3.2. Since , Equation (4) implies
Therefore by definition of the topology of , we have as desired.
∎
4 Clay, Leininger, Mangahas RAAGs
In this section, we prove the first part of Theorem 1.2. We begin with a description of a Clay, Leininger, Mangahas embedding .
Embedding construction: Let be the graph with vertex set and no edges. Let and be the surfaces indicated in Figure 1. For short, let denote . Let be a component of the preimage of in , and let be a geodesic in that is in the boundary of .
Let be a geodesic in that links with and maps to a simple biinfinite geodesic in . Further suppose that is an infinite ray and let be its endpoint on . For example, take to be the simple biinfinite geodesic in with one end spiraling around a curve essential in and the other end spiraling around a curve in as in Figure 1, and take to be an appropriate lift of . Choose so that is pseudoAnosov on . To simplify arguments, we abuse notation and let denote a representative in the isotopy class of that fixes all points outside . This ensures that fixes pointwise, where is the lift of fixing some point on . Thus, the extension of to fixes pointwise and the endpoints and of . Additionally, we choose to have the following properties:

links with , where is a primitive isometry with axis , and

where is as in Theorem 2.5.
We note that a pseudoAnosov on satisfying (1) can be obtained from any mapping class that is pseudoAnosov on by postcomposing with some number of Dehn twists (or inverse Dehn twists) about . Finally, a pseudoAnosov on satisfying (1) and (2) can be obtained from one satisfying (1) by passing to a sufficiently high power.
Let be any mapping class that is pseudoAnosov on and satisfies . Theorem 2.5 says that the homomorphism
is a quasiisometric embedding.
Equip with any HHS structure. In the remainder of this section we will prove the following theorem, which proves the first part of Theorem 1.2.
Theorem 4.1.
The sequences and converge to the same point in , but and do not converge to the same point in .
We will divide the proof of Theorem 4.1 into two propositions.
Proposition 4.2.
The sequences and converge to the same point in .
Proof.
Let be the Cayley graph of . By the discussion in Section 2.8, to show that and converge to the same point in , it is enough to show that they converge to the same point in . Now the Gromov product
Thus, in , as desired.
∎
Throughout the rest of this section will denote a fixed marking on . To continue, we require the following lemma.
Lemma 4.3.
There exist constants and such that for all we have . Consequently, after passing to a subsequence, converges to a point in .
Proof.
We begin by establishing the following claim.
Claim 1: Let . Then has endpoint and links with for all .
Proof of Claim 1: By our choice of and