BEYOND RELATIVE HYPERBOLICITY
LECTURES BY DENIS OSIN
This document is based on lecture notes from a mini-course given by Denis Osin at the
Centre de recherches mathématiques, Université de Montréal, October 2010 and was
compiled by Mark Hagen and Benjamin Smith, partly from notes shared by Svetla
Vassileva, who is hereby thanked.
Contents
1. Outline
2. Background on relatively hyperbolic groups
2.1. Farb’s notion of relative hyperbolicity
2.2. Bowditch’s notion of relative hyperbolicity
2.3. Osin’s notion of relative hyperbolicity
3. Hyperbolically embedded subgroups
3.1. Lecture 1 - October 6
3.2. Lecture 2 - October 7
4. Examples
References
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1. Outline
Preparatory to the material in Prof. Osin’s talks, in Section 2, we give background
on relatively hyperbolic groups, comparing the definitions given by Farb, Bowditch and
Osin. The main material begins in Section 3, where we define the notion of a hyperbolically embedded subgroup and show that the peripheral subgroups of a relatively hyperbolic group are hyperbolically embedded. The rest of the notes are devoted to showing
that the property of having hyperbolically embedded subgroups generalizes relative hyperbolicity. Important examples include mapping class groups of surfaces, discussed in
Section ??; other examples, using Dehn filling, are given in Section 4.
2. Background on relatively hyperbolic groups
The notion of a relatively hyperbolic group appears in Gromov’s essay [Gr87], and
there are several extant definitions of a relatively hyperbolic group which, up to some
juggling of hypotheses, are all equivalent. The basic idea being abstracted in all of the
definitions is the following.
Date: November 11, 2010.
1
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We begin with the familiar notion of a hyperbolic group, also due to Gromov.
Definition 2.1 (δ-hyperbolic metric space, word-hyperbolic group). The geodesic metric
space (X, d) is δ-hyperbolic if there exists δ ≥ 0 such that, if αβγ is a geodesic triangle
in X, for all a ∈ α, there exists b ∈ β ∪ γ with d(a, b) ≤ δ, and similarly for points in γ
and δ.
Now let G be a finitely generated group and d a word metric on G with respect to
some finite generating set (one can view d as the usual path metric on the Cayley graph
of G with respect to that generating set). Then G is word-hyperbolic if there exists δ such
that (G, d) is a δ-hyperbolic metric space. It is not difficult to show that the property
of being word-hyperbolic is independent of the generating set; the constant δ changes
when the generators are changed, but the existence of such a δ does not.
The above slim triangles characterization of hyperbolic metric spaces, generally attributed to Rips, is shown on the left side of Figure 1. It is easily seen that simplicial
trees are 0-hyperbolic, for example, and it is a nice exercise to find a constant δ such
that H2 is δ-hyperbolic1.
Figure 1. A slim triangle and a relatively slim triangle.
Now suppose that X contains a specified collection {Pi } of peripheral subspaces. In
the context of groups, X is the Cayley graph of the group G, and {Pi } = {gi Hj } is
the set of distinct left cosets of a (finite) collection {Hj } of subgroups of G. Intuitively,
we would like X to be δ-hyperbolic up to the declaration that we can travel “for free”
inside of the peripheral subspaces. In other words, each geodesic triangle αβγ in X is
“relatively slim” in the sense that if a ∈ α, then there is a path σ joining a to some
point of β ∪ γ such that the part of σ that is not contained in some Pi has length at
most δ. This notion is captured in the naive form of Farb’s definition given in the next
section. Slightly more must be added to the definition, however, to obtain a definition
that coheres with the whole notion of relative hyperbolicity in [Fa94]. This is bounded
coset penetration, a somewhat technical idea we shall introduce in Section 2.1.
A more combinatorial version of Farb’s definition was given by Bowditch [Bo97]. This
definition, discussed in Section 2.2, replaces the use of a Cayley graph of G, with “conedoff” peripheral cosets, by any graph admitting a G-action whose vertex-stabilizers are the
peripheral subgroups, subject to a few other conditions. The advantage of this definition
is that the bounded coset penetration condition translates into a combinatorial condition
– fineness – that is easy to verify.
1I think the answer is log(√(2) + 1).
BEYOND RELATIVE HYPERBOLICITY
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Osin’s notion of relative hyperbolicity [Os04a] is given in terms of areas, rather than
lengths. More specifically, it generalizes the following characterization of hyperbolic
groups.
Definition 2.2 (Dehn function). Let G ∼
= hS|Ri be a finitely presented group with
e be the universal cover of X. It is a fact that for each
presentation complex X. Let X
e (thought
(combinatorial) path σ : S 1 → X, there exists a (combinatorial) disc D in X
2
e whose boundary path is σ. The area Area(D) is the number of
of as a map D → X)
2-cells of D.
Of course, each such combinatorial path σ can be viewed as a word in the generating
set S that represents 1G , and the van Kampen diagram D has 2-cells corresponding to a
representation of σ as the product of conjugates of relators.
Then let ∆(σ) = inf{Area(D) : ∂D = σ}. This is the filling area of σ. The Dehn
function f : N → R associated to the presentation is
n
o
e |σ| ≤ n .
f (n) = max Area(σ) | σ : S 1 → X,
In other words, for any fixed word length, the Dehn function is the number of conjugates
of relators needed to express, most frugally, the least efficient identical word of at most
that length.
It is not difficult to show that, given two presentations of G with Dehn functions f1 , f2 ,
we have constants A, A0 , B, B 0 , C, C 0 , D, D0 such that for all n, Cf1 (An + B) + D ≤ f2 (n)
and C 0 f2 (A0 n+B 0 )+D0 ≤ f1 (n). In general, when such constants exist, we write f1 ∼ f2 ;
this is an equivalence relation. Thus, up to this equivalence relation, the Dehn function
of G is an invariant of the group itself, and not simply of the given finite presentation.
Theorem 2.3 (Linear isoperimetric inequality [Gr87]). The finitely presented group G
is word-hyperbolic if and only if its Dehn function f is equivalent to the identity n 7→ n
or to some constant map.
In Section 2.3, we shall see the notion of a relative presentation of a group G relative
to a family of peripheral subgroups, and a generalization of hyperbolicity using the
characterization given in Theorem 2.3. Relative presentations provide the basis for the
notion of hyperbolically embedded subgroups discussed in the lectures.
2.1. Farb’s notion of relative hyperbolicity. In this section S is a finite set of
generators for the group G and G is the Cayley graph of G with respect to S. The
combinatorial metric on G is denoted d. We consider a collection {Hi }i∈I of peripheral
subgroups, with I finite and each Hi infinite.
2.1.1. Coning off peripheral cosets. Our first goal is to produce a δ-hyperbolic space from
G that pins down the intuition of Figure 1. The desired space is the coned-off Cayley
graph.
Definition 2.4. For each i ∈ I, let {gji Hi }j be the set of left cosets. The coned-off
Cayley graph Gb of the pair (G, {Hi }) with respect to S is the graph whose vertex set
consists of “original vertices” of G, along with “cone points” corresponding to the cosets
BEYOND RELATIVE HYPERBOLICITY
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gji Hi , and whose edge set consists of “original edges” of G, along with a “cone edge”
joining the cone-point gji Hi to each original vertex gji h corresponding to an element
b
h ∈ Hi . Let dˆ be the path metric on G.
Having had experience with short-circuits, apparently, Farb termed dˆ the “electric
metric”. The reason for this is that dˆ makes each peripheral coset bounded; indeed, if
a, b ∈ gji Hi , then there is a path of length 2 in Gb that joins them: travel from a to the
cone-point along a cone-edge, then travel from the cone-point to b.
Example 2.5. Consider the presentation Z2 ∼
= ha, b | [a, b]i and the peripheral subgroup
n
hai. There is a cone-point b hai for each n ∈ Z; part of the coned-off Cayley graph is
shown in Figure 2. The cosets bn−1 and bn are at distance 1, via an original edge, so
Figure 2. Part of the coned-off Cayley graph of Z2 relative to a cyclic subgroup.
that the corresponding cone-points are at distance 3. Each coset has diameter 2, and
any two points are separated by a unique sequence of cosets. From this, it is easy to see
that the coned-off Cayley graph is quasi-isometric (see Remark 2.8) to a line.
Example 2.6. Consider the free group F2 ∼
= ha, bi and the peripheral subgroup hai. Part
of the coned-off Cayley graph is shown in Figure 3. Note that there is a unique cone-point
for each element of F2 − habi, plus a cone-point for habi. Indeed, if f = an1 bm1 . . . ank bnk
and g = ap1 bq1 . . . apk bqk are reduced words, then g −1 f ∈ habi if and only if f, g ∈ habi, as
can be verified by a simple computation. We construct a tree T as follows. The vertices
of T are the left cosets of habi, and we declare f habi and ghabi to be adjacent if f habi
and ghabi contain elements represented by reduced words differing by a single generator.
The graph T is connected since the Cayley graph of F2 is connected. Moreover, T
cannot contain any simple cycles of positive length. Indeed, if γ is a simple cycle in T ,
then there is a coset f habi = ghabi with f, g distinct reduced words not in habi. This
is impossible and therefore T is a tree. It is easily checked that the map Fb2 → T that
sends each cone-point and corresponding coset to the corresponding vertex of T is a
quasi-isometry.
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Figure 3. Coning off some left cosets of habi ≤ ha, bi.
Definition 2.7 (Weakly hyperbolic group 1). G is weakly hyperbolic relative to {Hi } if
ˆ of G relative to {Hi } is δ-hyperbolic
b d)
some (and thus any) coned-off Cayley graph (G,
for some δ.
Remark 2.8. The map f : X → Y of metric spaces is a (λ, µ)-quasi-isometric embedding
if for all x, y ∈ X, we have
λ−1 dX (x, y) − µ ≤ dY (f (x), f (y)) ≤ λdX (x, y) + µ.
If, in addition, all of Y lies in the µ-neighborhood of f (X), then f is a (λ, µ)-quasiisometry. It is readily seen that, if X, Y are Cayley graphs of G with respect to different
finite generating sets, then they are quasi-isometric, and also that the property of being
hyperbolic is a quasi-isometry invariant (though the constant changes). This explains
the “and thus any” statement in Definition 2.7.
One useful application of Remark 2.8 is that it shows easily that Z2 and F2 are
weakly hyperbolic relative to cyclic subgroups. Indeed, Examples 2.5 and 2.6 show that
the coned-off Cayley graphs are quasi-isometric to trees, which are 0-hyperbolic, and
hyperbolicity is a quasi-isometry invariant. In practice, finding a quasi-isometry to a
tree or proving, in particular, that the coned-off space is bounded (i.e., quasi-isometric
to a point) are useful ways of seeing that a group is weakly hyperbolic.
2.1.2. Bounded coset penetration. Recall that a quasigeodesic in a geodesic space X is a
quasi-isometric embedding R → X. Much of the following discussion comes from [Fa98].
Given a path γ in the Cayley graph X of G, we form a path γ̂ in Gb as follows. Fix
a set Yi of words in our generators such that Yi generates Hi . For each maximal string
b write
z in the word γ of elements of Yi , replace the subpath z by a length-2 path in G,
γ = γ1 zγ2 , and replace z by a length-2 path in Gb joining γ1 to γ1 z via the cone-point
γ1 Hi . Note that when |I| > 1, there is some choice in how this is done, depending on
the order in which the Yi are considered. γ̂ is the result of performing all such possible
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replacements. See Figure 4. If γ̂ contains a cone-point gHi , we say that γ penetrates
Figure 4. Hatting a path.
b
the coset gHi . If γ̂ is a G-geodesic,
we call γ a relative geodesic. Similarly, if γ̂ is a
λ-quasigeodesic, then γ is a relative λ-quasigeodesic. The path γ backtracks if γ̂ enters
the same cone-point more than once.
Definition 2.9 (Bounded coset penetration). The pair (G, {Hi }) satisfies bounded coset
penetration if, for all λ ≥ 1, there is a constant c = c(λ) such that if u, v are relative λquasigeodesics such that the parts of û and v̂ in Γ lie in a 1-neighborhood of one another
in Γ, and u, v do not backtrack, then, with all distances measured in Γ,
(1) If u penetrates gHi and v does not, then u travels a distance of at most c in gHi .
(2) If u, v both penetrate gHi , then the vertex at which u enters gHi lies within c of
the vertex at which v enters, and similarly for the exit vertices.
Given a close pair of relative quasigeodesics, Definition 2.9.(1) says that any coset
penetrated a sufficiently large amount by one of the paths is penetrated by the other.
Definition 2.9.(2) says that, while u and v may travel far from one another on their
respective journeys through a coset that they both penetrate, they must enter and leave
in roughly the same place.
A consequence of bounded coset penetration whose proof is somewhat illuminating is
that it implies that the peripheral subgroups form an almost malnormal collection.
Remark 2.10 (Malnormality). The collection {Hi } of subgroups of G is almost malnormal if, for all g ∈ G, if gHi g −1 ∩ Hj is infinite, then i = j and g ∈ Hi . The collection
is malnormal if the same conclusion holds whenever gHi g −1 ∩ Hj 6= {1}.
If G satisfies bounded coset penetration relative to {Hi }, then the latter collection is
almost malnormal.
For simplicity, consider the case of a single peripheral subgroup H. Suppose that H
is not almost malnormal, i.e. suppose there exists g ∈ G such that H g ∩ H is infinite.
Note that gH lies in a a = |g|-neighborhood of H. Since H is not almost malnormal,
there exist infinitely many hj ∈ H such that dG (hj , ghj ) ≤ a. Consider the geodesic
rectangle with vertices 1, g, ghj , hj . The sides [g, ghj ] penetrate gH arbitrarily deeply as
j increases, by local finiteness of the Cayley graph. Similarly, the sides [1, hj ] penetrate
H arbitrarily deeply. Thus the distance between the sides [ghj , hj ] and [hj , 1] becomes
arbitrarily large as j grows, for otherwise (2) would be violated. But by the a-closeness
BEYOND RELATIVE HYPERBOLICITY
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of H and gH, the other pair of sides fellow-travel in G, but one deeply penetrates H and
the other does not, violating (1). Hence H must be almost malnormal.
In [Fa98], Farb defines a relatively hyperbolic group using Definition 2.7, but frequently
discusses groups that are “relatively hyperbolic and satisfy bounded coset penetration”;
that paper implicitly contains the following definition.
Definition 2.11 (Relatively hyperbolic group [Fa94]). If (G, {Hi }) satisfies bounded
coset penetration and the coned-off Cayley graph Gb is hyperbolic, then G is hyperbolic
relative to {Hi }.
2.2. Bowditch’s notion of relative hyperbolicity.
2.2.1. Generalizing coset graphs. Consider the coned-off Cayley graph Gb of Section 2.1.
b extending the action of G by left multiplication on the Cayley
Note that G acts on G,
−1
graph, and that the stabilizer of the cone-point gji Hi is exactly gji Hi gji
. Moreover, this
action is cocompact: there are finitely many orbits of original vertices and edges since
G is finitely generated and there are finitely many orbits of cone-points since the set I
is finite. Finally, we saw in some examples that Gb is quasi-isometric to a graph whose
vertex set is the set of cone-points. In fact, this is true in general, and in light of this
fact, there is no need to consider specific generating sets and coned-off Cayley graphs.
Accordingly, in [Bo97], Bowditch gives the following definition of weak hyperbolicity.
Definition 2.12 (Weak hyperbolicity 2). Let G be a group and {Hi } a conjugacyinvariant collection of subgroups. Then G is weakly hyperbolic relative to {Hi } if there
exists a connected graph Γ such that
(1) G acts cocompactly on Γ (i.e. there are finitely many orbits of edges).
(2) Each vertex-stabilizer contains some Hi as a finite-index subgroup, and each Hi
stabilizes some vertex.
(3) There exists δ such that Γ is δ-hyperbolic.
For example, consider Z2 , together with the cyclic subgroups generated by the elements
(1, 1) and (1, 0). Define a graph Γ as follows: there is a set D of “diagonal” vertices,
R = {(1, 1) + (n, 0) : n ∈ Z} and a set V of “vertical” vertices, V = {(1, 0) + (n, n) :
n ∈ Z}. Edges are defined by the fact that every line of slope 1 in R2 crosses every
vertical line, i.e. Γ is the complete bipartite graph on the sets D t V . The Z2 action on
Γ is given by (x, y) ((1, 1) + (n, 0)) = (1, 1) + (n + x − y, 0) and (x, y) ((1, 0) + (n, n)) =
(1, 0)+(n+y, n+y). Clearly the stabilizer of a diagonal vertex is the subgroup generated
by (1, 1) and the stabilizer of a vertical vertex is the subgroup generated by (1, 0).
Moreover, the action is cocompact, since the quotient G/Γ is a single edge. Finally,
complete bipartite graphs have diameter 2 and are thus quasi-isometric to points.
2.2.2. Fine graphs. As in Section 2.1, we wish to exclude from our definition of relative hyperbolicity any pair (G, {Hi }) where the peripheral subgroups fail to form an
almost malnormal collection. In particular, we need to exclude the preceding example.
Note that in an infinite complete bipartite graph, each edge has the property that it is
BEYOND RELATIVE HYPERBOLICITY
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contained in infinitely many simple cycles of length 4. The correct combinatorial version of the bounded coset penetration condition is, under some additional hypotheses, a
generalization of this fact.
Definition 2.13. A simple cycle γ in a graph Γ is a closed combinatorial path that
passes through each vertex and each edge of Γ at most once. Γ is fine if for all n ≥ 0
and for all edges e, there are finitely many simple cycles in Γ of length n that contain e.
In Bowditch’s definition of relative hyperbolicity, fineness of a G-graph Γ plays the
role of bounded coset penetration. Accordingly, fineness had better imply almost malnormality of the collection of peripheral subgroups. Let H1 , H2 be distinct peripheral
subgroups, so that for 1 ∈ {1, 2}, there is a vertex Vi of Γ stabilized by Hi . Suppose that
V1 , V2 are chosen so that H1 ∩ H2 is infinite, and let V1 and V2 be a closest pair with this
property, in the sense that if γ ⊂ Γ is a geodesic segment joining V1 and V2 , no interior
vertex of γ is stabilized by infinitely many elements of H1 or H2 . The endpoints of γ
are fixed by H1 ∩ H2 , but, assuming that G acts on Γ with finite edge-stabilizers, there
are infinitely many distinct translates gγ with g ∈ H1 ∩ H2 , none of which has interior
points in common with γ. Thus any edge of γ is contained in infinitely many length-2|γ|
cycles of the form γ(gγ)−1 , contradicting fineness of Γ. Hence, under the conditions of
Definition 2.14, fineness of Γ implies almost malnormality of the collection of peripheral
subgroups, mirroring bounded coset penetration.
Without additional hypotheses, the converse is not true: there exist weakly hyperbolic
groups whose peripheral structure is malnormal, such that the corresponding graph is not
fine. For example, the Baumslag-Solitar group B ∼
= ha, b | ab = a2 i is weakly hyperbolic
relative to A = hai, and the latter subgroup is malnormal, but B is not hyperbolic
relative to A‘[Fa98]. An interesting exercise is to produce a Bowditch graph for the pair
(B, A) and verify that it is not fine.
2.2.3. Relative hyperbolicity.
Definition 2.14 (Relatively hyperbolic group [Bo97]). The group G is hyperbolic relative
to the collection {Hi } of infinite, finitely generated subgroups if
(1) G acts cocompactly with finite edge-stabilizers on the graph Γ.
(2) Each Hi stabilizes a vertex of Γ and each vertex-stabilizer is equal to some Hi .
(3) There exists δ such that Γ is δ-hyperbolic.
(4) Γ is fine.
Remark 2.15 (Equivalence of the definitions). Definitions 2.11 and 2.14 are equivalent
for finitely generated groups2.
Definition 2.14 is almost, but not quite, the same as “weakly hyperbolic via an action
on a fine graph”. The differences are that, in addition to the conditions on Γ given
in Definition 2.7, relative hyperbolicity requires that the edge-stabilizers be finite and
that the vertex stabilizers by equal to, rather than virtually equal to, the peripheral
subgroups.
The condition on edge-stabilizers is already met for coned-off Cayley graphs, since in
that situation they are finite. The main point is that the existence of a fine Γ is equivalent
2In fact, Chris Hruska has shown the definitions to be equivalent for all countable groups.
BEYOND RELATIVE HYPERBOLICITY
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to fineness of the coned-off Cayley graph, which is in turn equivalent to bounded coset
penetration.
We saw in Example 2.5 that Z2 is weakly hyperbolic relative to a cyclic subgroup,
but not relatively hyperbolic since the coned-off Cayley graph is not fine. On the other
hand, Example 2.6 shows that F2 is hyperbolic relative to a cyclic subgroup. More
generally, a word hyperbolic group is hyperbolic relative to any malnormal collection of
quasiconvex subgroups, and any peripheral structure for a word-hyperbolic group has
these properties; this fact is due to Bowditch.
The following is a somewhat random example of a group that is hyperbolic relative to
a free abelian subgroup.
Example 2.16. Consider the HNN extension of Z2 ∗ Z presented by
G∼
= ha, b, c, t | [a, b], at = aci.
Geometrically, G is the fundamental group of the space X, shown in Figure 5, obtained
from T 2 ∧ S 2 by attaching S 1 × [−1, 1] along the indicated paths. Let A = ha, bi ≤ G be
Figure 5. The space X.
the fundamental group of the torus. Then A is malnormal. This can be verified combie of X splits as a tree of spaces whose
natorially. More elegantly, the universal cover X
e
vertex-spaces Xv are themselves trees of planes and edges corresponding to conjugates
e are “strips” – copies of the universal cover of the
of ha, b, ci ≤ G. The edge-spaces in X
e
cylinder – attached to the various Xv along lines corresponding to a and ac. Since the
line corresponding to ac spends a finite amount of time, namely one edge, in each plane,
the overlap between two planes, each corresponding to a conjugate of A, is finite. This
divergence of the planes from one another characterizes malnormality.
e with two vertices joined by an
Let Γ be the graph whose vertices are the planes of X,
edge if the corresponding planes intersect the same strip. (In other words, we are using
the Bass-Serre tree as our Γ). Then Γ is hyperbolic and fine, being a tree.
More examples shall appear where they are relevant. Many finite-volume hyperbolic
manifolds with boundary are hyperbolic relative to their boundary components. If is
useful to bear this example in mind in the later discussion of Dehn filling; we shall
return to this type of example in Section 4.
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Equivalence of the Farb and Bowditch definitions is discussed in [Bo97], [Bu04] and [Da03].
2.3. Osin’s notion of relative hyperbolicity. Osin’s definition of relative hyperbolicity, in terms of relative isoperimetric functions, is shown to be equivalent to Definitions 2.11 and 2.14 in [Os04a]. The concept of a relative presentation of a group is central
to this definition, and also to the definition of a hyperbolically embedded subgroup, so
that much of the following material is used throughout these notes. The discussion in
this section is based on Osin’s book [Os04a].
2.3.1. Relative presentations. In this section, G is a group, {Hi } is a collection of subgroups, and X ⊂ G
S is some symmetric collection of elements. G is generated by X
relative to {Hi } if ( i Hi ) ∪ X is a generating set for G.
Suppose G is generated by X relative to {Hi }. Denote by F (X) the free group
generated by X, and consider the free product
b i ∗ F (X),
F = ∗i H
b i is a copy of Hi ≤ G, so that G is a quotient of F .
where for each i, the group H
b i , so that X ∪ H is a
More precisely, Let H be the set of nonidentity elements of ∪i H
set of generators of G, leading to the following presentation. Let Si be the set of words
b i − {1} that represent the identity in F .3 This gives a presentation
in the alphabet H
∼
F = hX, H | s, s ∈ ∪i Si i. Next, define an epimorphism φ : F → G that is the identity
b i → Hi .
on X and extends each of the isomorphisms H
Definition 2.17 (Relative presentation). In the context of the previous discussion, we
say that G has the relative presentation
(†)
hX, H | s, s ∈ ∪i Si ; , Ri
if R is a set of words, closed under inverses and cyclic permutations, in the alphabet
X ∪ H, whose normal closure in F is ker φ.
In any interesting examples, the sets Si are infinite, so the relative presentation for
G is finite if X and R are finite. Geometrically, Definition 2.17 is describing the following presentation complex. For each i, consider the multiplication table presentation
bi ∼
b i , h1 h2 = h3 }i. The universal cover of
b i − {1} | {h1 h2 h−1 : h1 , h2 , h3 ∈ H
H
= hH
3
the associated presentation complex is the two-skeleton of the (infinite) simplex whose
b i ; let Yi be its quotient by the Hi -action by left multiplivertices are the elements of H
cation. Let W be a wedge of circles, with a circle for each x ∈ X and a circle for each
h ∈ ∪i (Hi − {1}). For each i, we attach to Z and Yi the mapping cylinder corresponding
b i , and then we attach 2-cells for R, resulting in a space Z
to the isomorphism Hi ∼
=H
whose fundamental group is G, and whose 2-cells are simplices in Yi corresponding to
b i or “cylinder cells” corresponding to the identifications of Hi and H
b i , or
relators in H
R-cells corresponding to the kernel of φ.
Relative presentations appear readily in nature. For example, consider A, B ≤ H and
and isomorphism φ : A → B. The HNN extension G ∼
= hH, t | tat−1 = φ(a), a ∈ Ai
3Observe that we are using the presentation of H
b i given by its multiplication table.
BEYOND RELATIVE HYPERBOLICITY
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is a relative presentation of G relative to H, with X = {t}. If A is finitely generated,
then φ is determined by its effect on finitely many elements of A, so that the relative
presentation is finite.
2.3.2. Relative lengths and relative areas. Given the relative presentation (†) and an
element g ∈ G, let |g| be the length of the shortest word in X ∪ H representing g – this
is the relative length. The metric d(g, h) = |gh−1 | is the relative distance function on
G. While this depends on the choice of X, the distance functions arising from any two
finite relative generating sets (relative to the same collection of peripheral subgroups)
are Lipschitz-equivalent.
Given a relative presentation G ∼
= hX, H | R, Si, the relative Cayley graph Γ =
Γ(G, X, H) is the Cayley graph of G with respect to the generating set X ∪ H. Note that
the path-metric on Γ restricts to d on the 0-skeleton, and that Γ is not in general locally
finite, even if the relative presentation is finite, since generally at least one of the Hi is
infinite. Γ is the “important” part of the 1-skeleton of Z, i.e. loops in Γ correspond to
words representing the identity that are homotopically trivial in Z via a disc made out
of 2-cells that are not simplices of some Yi .
Osin builds a theory of relative van Kampen diagrams and isoperimetric functions
analogous to the usual theory for presentations, but it is now important to consider
which type of 2-cell is being counted in area computations. The idea is to ignore the
2-simplices in the Yi .
Definition 2.18. An R-cell in Z is a 2-cell whose boundary path is labelled by a word
in R, and an S-cell is a 2-cell whose boundary path is labeled by a word in ∪i Si = S.
Let w be a word over X ∪H representing the identity in G, and let D be a disc diagram
in Z whose boundary path is labeled by w. Then NR (D) is the number of R-cells in D
and NS (D) is the number of S-cells. We define Area(D) = NR (D) + NS (D).
Definition 2.19 (Relative area). Let w be a loop in Γ representing the identity in G.
The relative area of w is
Arearel (w) = inf{NR (D) : ∂D = w}.
Here D varies over disc diagrams in Z and ∂D means “the label of the boundary path
of D”. Note that a word in H that already represents the identity in some Hi has relative
area 0; relative area is the minimal cost of nullhomotopy of a close based path, where
we can homotop across discs in the Yi and in the mapping cylinders for free.
Definition 2.20 (Relative Dehn function). The function f : N → N is a relative isoperimetric function for (†) if for all n and for any closed based path w in Γ of length at most
n, we have
Arearel (w) ≤ f (n).
Let I be the set of relative isoperimetric functions for (†). Then the relative Dehn
function ∆ of the relative presentation is ∆(n) = inf{f (n) : f ∈ I}.
In general, ∆ is not defined. However, when it exists, up to the equivalence relation
∼, the relative Dehn function is an invariant of the pair (G, {Hi }), provided some finite
relative presentation exists:
BEYOND RELATIVE HYPERBOLICITY
12
Lemma 2.21. Let P1 and P2 be finite relative presentations for (G, {Hi }) and let ∆i
be the relative Dehn function associated to Pi . Then, if ∆1 is well-defined, we have that
∆1 ∼ ∆2 and, in particular, ∆2 is well-defined.
This leads to the definition of relative hyperbolicity.
Definition 2.22 (Relative hyperbolicity [Os04a]). G is hyperbolic relative to {Hi } if
the pair (G, {Hi }) admits a finite relative presentation with linear Dehn function. The
group G is relatively hyperbolic if it is hyperbolic relative so some collection of peripheral
subgroups.
3. Hyperbolically embedded subgroups
3.1. Lecture 1 - October 6. Note that a group G acts on its Cayley graph with
trivial vertex- and edge-stabilizers, and the action is cocompact if G is finitely generated.
Moreover, the trivial subgroup if malnormal and quasiconvex, so that if G is wordhyperbolic, then Bowditch’s theorem implies that G is hyperbolic relative to the trivial
subgroup. Hence hyperbolic groups are relatively hyperbolic. On the other hand, the flat
plane theorem implies that the group in Example 2.16 is not hyperbolic, so the property
of being relatively hyperbolic is a real generalization of word-hyperbolicity. (We shall
see many more examples of relatively hyperbolic groups that are not hyperbolic. The
motivating example is the fundamental group of a hyperbolic 3-manifold with boundary;
by JSJ and Thurston, such animals are everywhere.)
In this section, we discuss a generalization of relatively hyperbolic groups, namely
groups with hyperbolically embedded subgroups. Throughout, G is a group, H ≤ G and
X ⊂ G is a relative generating set for the pair (G, {H}). The relative Cayley graph is
Γ = Γ(G, X ∪ H) and let ΓH be the Cayley graph of H with respect to the (infinite)
generating set H.
Observe that there is a combinatorial embedding ι : ΓH → Γ induced by the inclusion
H ,→ G. This induces a metric dH : H × H → [0, ∞]. We aren’t interested in measuring
distances between points in ΓH according to the distance of their images in Γ, since ΓH is
just the 1-skeleton of a simplex, and thus bounded. Instead we do something reminiscent
of “hatting the path” described in Definition 2.9.
Definition 3.1 (The metric dh ). Let g, h ∈ H. Let P be the set of paths ρ ⊂ Γ joining g
and h such that ρ does not contain any edges of ι(Γh ). Then dH (g, h) = inf{|ρ| : ρ ∈ P}.
4
Example 3.2.
(0) If H = G and X is empty, then Γ = ΓH so that dH (g, h) = ∞ for all g 6= h.
(1) If G = H × Z with Z generated by x and X = {x} then dh (g, h) ≤ 3 for
all g, h ∈ H. In fact, if g 6= h then distance is 3 and given by, say the path
g, xg, xh, h.
(2) If the previous example is replace with a free product (i.e. G = H ∗ Z), this gives
a tree-like structure. There are no loops back to distinct points g 6= h ∈ H so in
fact dH (g, h) = ∞.
4 Benjamin: can you add
some
examples/interpretation of
the above definition? This
should include the stuff
about: “which metrics on H
are realized as this type of
relative metric?” Osin gives
some examples where dH is
Lipschitz-equivalent to a
word metric on H and then
states that not all dH have
this property. I don’t think
Svetla wrote the example
properly, but you might find
it in the paper. Also, Dani
asked if this can be done for
a finitely presented G
(legitimately finitely
presented, too, not just
finitely presented relative to
H). Let’s talk about this for
a bit.
BEYOND RELATIVE HYPERBOLICITY
13
(2 12 ) Suppose G = HAt =B , X = {t}. In this case, dH is Lipschitz equivalent to a word
metric with respect to A ∪ B (this is an easy exercise).
A natural question is; given H, which functions d : H × H → [0, ∞] can be
realized as dH for some G and X? In all the above examples, dH is Lipschitz
equivalent to a word metric (in a trivial way). Another natural question is can
we realize a metric dH which is not Lipschitz equivalent to some word metric?
The answer is yes as demonstrated in the next example.
(3) There exists a group G having generating set X with subgroup H ∼
= Z (i.e.
cyclically generated by some word) such that dH (m, n) is Lipschitz equivalent to
log(|m − n| + 1). Having this shows that dH is not equivalent to any word metric.
In example (3), the G is not a finitely presented group. Dani Wise asked, can this be
done with a finitely presented group?
Proposition 3.3. Let Γ = Γ(G, X ∪ H) be hyperbolic. Then there exists a word-metric
on H that is Lipschitz-equivalent to dH .
Definition 3.4. A subgroup H ≤ G is hyperbolically embedded with respect to X, denoted H ,→h (G, X), if
(1) Γ is hyperbolic.
(2) (H, dH ) is locally finite.
If there exists X such that H ,→h (G, X), then we say H is hyperbolically embedded in
G and write H ,→h G.
The first condition in this definition is essentially weak hyperbolicity. The second is
intended to generalize bounded coset penetration.
Looking back at our previous examples we find
Example 3.5. The numbers in this example correspond to the previous example.
(0) We have that G ,→h (G, ø) and if H is finite, then H ,→h (G, G). These are
called the degenerate cases.
(1) If H is infinite, then H is not hyperbolically embedded in G. The previous
example (1) shows that (H, dH ) is not locally finite since H is entirely contained
in the ball of radius 3 with respect to dH .
(2) Here H ,→h (H ∗ hxi, {x}) is a hyperbolic embedding. We saw that Γ is a tree
and (H, dH ) has all distinct points, infinitely far away.
(2 12 ) The corresponding example is a hyperbolic embedding whenever A is finite.
The main theorem of this section is
Theorem 3.6. G is hyperbolic relative to H if and only if there exists a finite set X
such that G ,→h (G, X).
Proof.
5
5add proof
It makes sense to look at infinite generating sets X, even if the group is finitely
generated. A major example of this (done later) is the mapping class group of a closed
space.
BEYOND RELATIVE HYPERBOLICITY
14
Observation. Suppose that H ,→h G. Then,
(1) H is finitely generated and undistorted (quasi-isometrically embedded) in G.
(2) For all g ∈ G\H , the intersection H g ∩ H is finite. Indeed if x ∈ H g ∩ H then
dH (1, x) ≤ 2|g| + 1 and local finiteness implies the claim.
The first observation prompts the question: If G is finitely presented and H ,→h G,
does it follows that H is finitely presented? This is true for relatively hyperbolic groups
(see [Da10]). The conjecture, in general, for this problem is no. Osin suggests this has
something to do with the Higman embedding theorem6.
The observations above result in the following:
6 Expand on this
Corollary 1. If the center of G is infinite, then no non-degenerate H is hyperbolically
embedded in G.
This follows from malnormality.
Corollary 2. If H is non-degenerate hyperbolically embedded in G then the index [G : H]
is infinite.
3.2. Lecture 2 - October 7. Suppose now that G acts by isometries on a metric space
(S, d). For g ∈ G define the length of g by l(g) = inf x∈S d(x, gx). As usual, the action is
cobounded if the quotient G \ S, with the induced metric, is bounded, or, equivalently, if
there exists a bounded set T ⊂ S such that S = ∪g∈G gT .
Theorem 3.7. Suppose S is a hyperbolic space, G acts on S isometrically and coboundedly. Suppose further that H ≤ G such that
(1) H is quasi-convex and the action of H on S is proper.
(2) For all > 0 there is a finite subset F of H such that for all x, y ∈ G − H with
max{l(x), l(y)} < then xHy ∩ H ⊆ F .
Then H is hyperbolically embedded in G.
Before proof we remark that condition (2) on H is referred to as the geometric almost
malnormality condition. Note that if x is chosen to be y −1 then this condition is that
H y ∩ H be finite.
Proof sketch. Since G acts coboundedly and isometrically, there is an X ⊆ G such that
Γ(G, X) is quasi isometric to S. Indeed, we follow the proof of the classical Svarc-Milnor
lemma. Let B be a ball of sufficient radius that S is covered by the G-translates of B.
Let X be the set of elements x ∈ G such that gB ∩ B 6= ∅. Then X is a (possibly infinite)
generating set. One checks that the map Γ(X, G) given by g 7→ gso is a quasi-isometry
for any fixed basepoint so ∈ S. 7
Γ(G, X ∪ H) is shown to be hyperbolic using the quasi convexity of H, while local
finiteness of (H, dH ) follows from geometric malnormality and properness of the H-action
on S.
Example 3.8. If G = M CGg,p is the mapping class group of a genug g surface with p
punctures. We assume 3g + p ≥ 5 to exclude the exceptional surfaces. There is a natural
action of G on the curve complex C defined as follows:
7Insert picture
BEYOND RELATIVE HYPERBOLICITY
15
Let the vertices V of C be the free homotopy types of simple essential closed curves
in Sg,p . Two vertices γ1 , γ2 are joined by an edge in C if they can be realized disjointly.
A theorem of Masur and Minsky states that C is hyperbolic.
Recall, that an element a ∈ G is pseudo-Anosov if the action of a on C. There is a
maximal virtually cyclic subgroup generated by any element a in G:
E(a) = Comm(hai) = {x ∈ G : haix ∩ hai =
6 {1}}
where Comm denotes the commensurator.
Corollary 1. If G = M CGg,p with 3g + p ≥ 5 and a ∈ G is pseudo-Anosov then E(a)
is hyperbolically embedded in G.
Proof. This follows from our theorem and “Bowditch asphericity”.
Corollary 2. Suppose there is a non-degnerate H hyperbolically embedded in G. Then
for all n ∈ N there exists a subgroup Kn ≤ G such that Kn is isomorphic to the cartesian
product a free group on n generators with a finite group. Furthermore, each Kn is
hyperbolically embedded in G.
Proof. Let h ∈ H be such that dH (1, h) 1, take short ai ∈ H for each 1 ≤ i ≤ 4 and
let hi = hai . We have E(ai ) hyperbolically embedded in G. Consider x = hd1 hd2 and
y = hd3 hd4 for some sufficiently large d 1. Look at K2 = hx, yi. If G is torsion free,
then K2 = F2 which is hyperbolically embedded in G. We can show that K2 is quasi
convex and almost malnormal in Γ = Γ(G, X ∪H). Take the action of K2 on Γ and apply
Theorem 3.7. It follows that K2 ,→h G and that K2 is virtually free follows from some
isoperimetric inequalities. For higher values of n, one just includes more generators. We remark that the extra finite group in the Corollary 2 is unavoidable. For example, consider some hyperbolically embedded H in F (x, y) × Cn . Every hyperbolically
embedded subgroup is almost malnormal so necessarily contains the center. Hence Cn
is contained in H.
Theorem 3.9.
(1) Suppose that A ,→h G, B ,→h H and A ∼
= B. Then A is hyperbolically embedded in G ∗A=B H
(2) Suppose that {A, B} ,→h G where A, B ≤ G are isomorphic subgroups. Then
A ,→h GAt =B
Suppose now that N is a closed 3-manifold. Embed a knot K in N and drill it out
(i.e. cut a solid torus T ∼
= D2 × S 1 out of N ) to obtain a manifold M whose boundary is
homeomorphic to S 1 × S 1 . Now re-glue the torus back in via some map ϕ : ∂T → ∂M in
possibly a different way and let σ denote the homotopy class of ϕ(∂(D2 )) inside π1 (∂M ).
The resulting manifold M ∪ϕ T = M (σ) depends only on σ.
Theorem 3.10 (Thurston). If M − ∂M admits a complete finite volume hyperbolic
metric, then for all but finitely many σ ∈ π1 (∂M ), M (σ) is hyperbolic.
By the above theorem, π1 (∂M ) ,→ π1 (M ) , so we can think of σ ∈ π1 (M ). Then by
the Seifart-vanKampen theorem, it follows that π1 (M (σ)) ∼
= π1 (M )/ σ . Using
this, the above theorem can be restated algebraically.
BEYOND RELATIVE HYPERBOLICITY
16
Theorem 3.11 (Algebraic version). For all but finitely many σ ∈ π1 (∂M ), π1 (M (σ))/ σ is hyperbolic.
Now we proceed to generalize the algebraic version.
Theorem 3.12. If H ,→h (G, X), then there exists a finite subset F ⊆ H − {1} such
that for all normal subgroups N C H intersecting H trivially, the following conditions
hold:
(1) N G ∩ H = N where N G is the normal closure of N in G (i.e. H/N ,→ G/N G
in a natural way).
(2) H/N ,→h (G/N G , X). In particular, if G is hyperbolic relative to H, then G/N G
is hyperbolic relative to H/N .
(3) N G = ∗t∈T N t , for some T ⊆ G; for all g ∈ N G , either g is conjugate to g 0 in N
or the action of g on Γ(G, X ∪ H) is hyperbolic.
Example 3.13.
(1) If G = H ∗ Z, N C H, then G/N G ∼
= H/N ∗ Z. Notice that
π1 (∂M ) ,→h π1 (M ) which is H ,→h G and this is well known.
(2) Let G = π1 (M ), H = π1 (∂M ) and N = hσi. By the second part of our theorem
π1 (M (σ)) is hyperbolic relative to H/N ∼
= Z ⊕ /Z/hσi, which is hyperbolic because its virtually cyclic, implying that π1 (M (σ)) is hyperbolic (since hyperbolic
relative to hyperbolic is hyperbolic)
(3) Let G be hyperbolic and g ∈ G be of infinite order. It is well known that
E(g) ,→h G where E(g) is the elementary subgroup. Take N = hg n i (if n is
chosen wisely, N will be normal. We can choose arbitrarily large n because this
subgroup is virtually cyclic). Then G1 = G/ g n is hyperbolic relative to
E(g)/ g n (this quotient is finite) and this implies G1 is hyperbolic. Note
that now g has finite order in G1 so we can use this idea to construct infinite
torsion groups. Let G = {1, g1 , g2 , . . . } and take quotients as above. This yields
the chain of quotients
g
n1
=1
g
n2
=1
G 1→ G1 2→ G2 → · · ·
The limit G∞ is an infinite torsion group (because G is infinitely presented group
so the sequence doesn’t end). Hence G = T ∗ Z where T is a torsion group.
(4) Let G = M CGg,p with 3g + p ≥ 5 and a ∈ G a pseudo-Anosov element. Then
E(a) ,→h G. Take normal subgroup N E E(a) to be cyclically generated by an .
We have previously shown there exists Kn ∼
= Fn × F ,→h G.
Corollary. han i is free and purely pseudo-Anosov. Thats is every non-trivial element is
pseudo-Anosov.
Take any countable group G, then G ,→ F2 /N (i.e. G embeds into a 2-generated
group) and we cant avoid short elements in F2 meaning we can avoid any finite subset.
Now we may apply our theorem to show G ,→ H/N ,→ G/N G . So G is SQ-universal.
Corollary. M CGg,p is SQ-universal. Furthermore, any subgroup of M CGg,p is either
virtually abelian or SQ universal.
BEYOND RELATIVE HYPERBOLICITY
17
4. Examples
Example 4.1 (A relatively hyperbolic knot complement). A large class of examples of
relatively hyperbolic groups consists of fundamental groups of hyperbolic knot complements. For instance, let K be the figure-eight knot complement, shown in Figure 6. Note
Figure 6. Viewing S 3 as a pair of 3-balls glued along their boundaries
gives this picture of a figure-eight knot complement.
that G = π1 (S 3 − N (K)) is not word-hyperbolic, since, by the loop theorem, the embedded toral boundary component T = ∂N (K) is π1 -injective. As a hyperbolic manifold,
M = S 3 − K acts properly on H3 , but not cocompactly; M has a cusp corresponding
to the torus T ⊂ M − N (K). FINISH AND ADD LATER IN DEHN FILLING
SECTION.
References
[Bo97] B.H. Bowditch. Relatively hyperbolic groups. Preprint, Southampton. 1997.
[Bu04] I. Bumagin. On the definition of relatyvely hyperbolic groups. Preprint, 2004.
http://xxx.lanl.gov/abs/math.GR/0402072.
[Da03] F. Dahmani. Les groupes relativement hyperboliques et leurs bords. PhD Thesis, 2003.
[Da10] F. Dahmani and V. Guirardel, Presenting parabolic subgroups, ArXiv 2010.
[Fa94] B. Farb. Relatively hyperbolic and automatic groups with applications to negatively curved manifolds. PhD Thesis, Princeton University, 1994.
[Fa98] B. Farb. Relatively hyperbolic groups. GAFA, Vol 8.(1998), 810-840.
[Gr87] M. Gromov. Hyperbolic groups. In Essays in Group Theory, MSRI Publ. 8, Springer, 1987 (Ed.
S.M. Gersten).
[Os04] D. Osin. Elementary subgroups of relatively hyperbolic groups and bounded generation. ArXiv
preprint, 2004.
[Os04a] D. Osin. Relatively hyperbolic groups: Intrinsic geometry, algebraic properties and algorithmic
problems. ArXiv, 2004.