Let 1 → H → G → Z → 1 be an exact sequence of hyperbolic groups induced by an automorphism Φ of the free group H. Let H1(⊂ H) be a finitely generated distorted subgroup of G. Then there exist N &gt; 0 and a free factor K of H such that the conjugacy class of K is preserved by ΦN and H1 contains a finite index subgroup of a conjugate of K. This is an analog of a Theorem of Scott and Swarup for surfaces in hyperbolic 3-manifolds

Mitra, Mahan

English

Publications of the IAS Fellows

ON A THEOREM OF SCOTT AND SWARUP
MAHAN MITRA
Abstract. Let 1 → H → G → Z → 1 be an exact sequence of
hyperbolic groups induced by an automorphism φ of the free group
H . Let H1(⊂ H) be a finitely generated distorted subgroup of G.
Then there exist N > 0 and a free factor K of H such that the
conjugacy class of K is preserved by φN and H1 contains a finite
index subgroup of a conjugate ofK. This is an analog of a Theorem
of Scott and Swarup for surfaces in hyperbolic 3-manifolds.
1. Introduction
In [13] , Scott and Swarup prove the following theorem:
Theorem [13] Let 1 → H → G → Z → 1 be an exact sequence
of hyperbolic groups induced by a pseudo Anosov diffeomorphism of a
closed surface with fundamental group H. Let H1 be a finitely generated
subgroup of infinite index in H. Then H1 is quasiconvex in G.
In this paper we derive an analogous result for free groups (see Sec-
tion 2 below or [3] [2] [7] for definitions).
We note at the outset that hyperbolic stands for two notions. When
qualifying manifolds, they indicate spaces of constant curvature equal
to -1. When qualifying groups or metric spaces, we use hyperbolic in
the sense of Gromov [8]. It will be clear from the context which of
these meanings is relevant.
Theorem 3.4 Let 1 → H → G → Z → 1 be an exact sequence
of hyperbolic groups induced by an aperiodic automorphism of the free
group H. Let H1 be a finitely generated subgroup of infinite index in
H. Then H1 is quasiconvex in G.
In fact we prove the following more general theorem
Theorem 3.7 Let 1 → H → G → Z → 1 be an exact sequence of
hyperbolic groups induced by a hyperbolic automorphism φ of the free
group H. Let H1(⊂ H) be a finitely generated distorted subgroup of
G. Then there exist N > 0 and a free factor K of H such that the
Research partly supported by Alfred P. Sloan Doctoral Dissertation Fellowship,
Grant No. DD 595
AMS subject classification = 20F32, 57M50.
1
2 MAHAN MITRA
conjugacy class of K is preserved by φN and H1 contains a finite index
subgroup of a conjugate of K.
In fact the methods of this paper can be used to give a new proof
of the Theorem of Scott and Swarup mentioned above. We sketch
this proof for closed surfaces first. Let M be a closed hyperbolic 3-
manifold fibering over the circle with fiber F . Let F˜ and M˜ denote
the universal covers of F and M respectively. Then F˜ and M˜ are
quasi-isometric to H2 and H3 respectively. Now let D2 = H2 ∪ S1
∞
and
D
3 = H3 ∪ S2
∞
denote the standard compactifications. In [5] Cannon
and Thurston show that the usual inclusion i of F˜ into M˜ extends
to a continuous map iˆ from D2 to D3. Cannon and Thurston further
show that iˆ identifies precisely those pairs of points which are boundary
points of an ending lamination. Since a leaf of the stable (or unstable)
lamination is dense in in the whole lamination, it cannot be carried by a
(perhaps immersed) proper sub-surface (one can see this, for instance,
by using the fact that surface groups are LERF [12] ). The subgroup
corresponding to the fundamental group of such a subsurface must
therefore be quasiconvex in G.
This idea goes through for free groups. We give a brief sketch for
aperiodic automorphisms. In this case, Bestvina, Feighn and Handel
[3] have shown that any leaf of the stable (or unstable) lamination
‘fills’ H , i.e. it cannot be carried by a finitely generated subgroup
H1 of infinite index in H . We combine this with the description of
boundary identifications given in [9] to show that no pair of points on
the boundary of H1 are identified. Thus H1 must be quasiconvex in G.
2. Ending Laminations
Let G be a hyperbolic group in the sense of Gromov [8] . Let H
be a hyperbolic subgroup of G. Choose a finite generating set of G
containing a finite generating set of H . Let ΓG and ΓH be the Cayley
graphs of G, H with respect to these generating sets. Let i : ΓH → ΓG
denote the inclusion map.
Definition : [7] [6] If i : ΓH → ΓG be an embedding of the Cayley
graph of H into that of G, then the distortion function is given by
disto(R) = DiamΓH (ΓH∩B(R)),
where B(R) is the ball of radius R around 1 ∈ ΓG.
If H is quasiconvex in G the distortion function is linear and we
shall refer to H as an undistorted subgroup. Else, H will be termed
distorted.
ON A THEOREM OF SCOTT AND SWARUP 3
For distorted subgroups, the distortion information is encoded in a
certain set of ending laminations defined below.
Definition : If λ is a geodesic segment in ΓH then λ
r, a geodesic
realization of λ, is a geodesic in ΓG joining the end-points of i(λ).
Now consider sequences of geodesic segments λi ⊂ ΓH such that
1 ∈ λi and λ
r
i ∩ B(i) = ∅, where B(i) is the ball of radius i around
1 ∈ ΓG. Take all bi-infinite subsequential limits (in the Hausdorff
topology) of all such sequences {λi} and denote this set by Σ.
Let th denote left translation by h ∈ H . Let Γ̂H and Γ̂G denote the
Gromov compactifications of ΓH and ΓG respectively. Further let ∂ΓH
and ∂ΓG denote the boundaries of ΓH and ΓG respectively [8] .
Definition : The set of ending laminations Λ = Λ(H,G) is given
by
Λ = {(p, q) ∈ ∂ΓH × ∂ΓH |p 6= q and p, q are the end-points of th(λ)
for some λ ∈ Σ}
Lemma 2.1. H is quasiconvex in G if and only if Λ = ∅
Proof : Suppose H is quasiconvex in G. Then any geodesic realiza-
tion λr of a geodesic segment λ ⊂ ΓH lies in a bounded neighborhood
of ΓH and hence of λ as H is hyperbolic. Hence Λ = ∅ .
Conversely, if H is not quasiconvex in G, there exist λi ⊂ ΓH and
pi ∈ λi such that λ
r
i ∩Bpi(i) = ∅, where Bpi(i) denotes the ball of radius
i around pi in ΓG. Translating by p
−1
i and taking subsequential limits,
we get Σ 6= ∅ and hence Λ 6= ∅. 2
Definition : A Cannon-Thurston map for the pair (H,G) is a
map iˆ : Γ̂H → Γ̂G which is a continuous extension of i : ΓH → ΓG.
Note that if such a continuous extension exists, it is unique. We get
a simplified collection of ending laminations when a Cannon-Thurston
map exists.
Definition : ΛCT = {(p, q) ∈ ∂ΓH × ∂ΓH |p 6= q and iˆ(p) = iˆ(q)}.
Lemma 2.2. If a Cannon-Thurston map exists, Λ = ΛCT .
Proof: Let (p, q) ∈ Λ. After translating by an element of H if
necessary assume that a bi-infinite geodesic λ passing through 1 has
p, q as its end-points. By definition of Λ there exist geodesic segments
λi ⊂ ΓH converging to λ in the Hausdorff topology such that λ
r
i∩B(i) =
∅. Since a Cannon-Thurston map exists, there exists z ∈ ∂ΓG such that
λri → z in the Hausdorff topology on Γ̂G and iˆ(p) = z = iˆ(q). Hence
Λ ⊂ ΛCT .
4 MAHAN MITRA
Conversely, let (p, q) ∈ ΛCT . After translating by an element of H
if necessary assume that a bi-infinite geodesic λ passing through 1 has
p, q as its end-points. Choose pi, qi ∈ ΓH such that pi → p and qi → q.
Let λi denote the subsegment of λ joining pi, qi. Then λ
r
i converges to
iˆ(p) = iˆ(q) in the Hausdorff topology on Γ̂G. Passing to a subsequence
if necessary we can assume that λri ∩B(i) = ∅. Hence ΛCT ⊂ Λ. 2
Remark: Suppose H1 is a hyperbolic subgroup of H . Let jˆ and
iˆ denote Cannon-Thurston maps for the pairs (H1, H) and (H,G) re-
spectively. Then the composition iˆ · jˆ is a Cannon-Thurston map for
the pair (H1, G). Further from Lemma 2.2 it follows that
Λ(H1, G) = Λ(H1, H) ∪ (jˆ)
−1
(Λ(H,G)).
3. Extensions by Free Groups
Let 1 → H → G → Z → 1 denote an exact sequence of hyperbolic
groups arising out of a hyperbolic automorphism φ of the hyperbolic
group H . The notion of a hyperbolic automorphism was defined in
[1] (see below) and shown to be equivalent to requiring that G be
hyperbolic.
Definition: Let φ be an automorphism of a hyperbolic group H
(equipped with the word metric |.|). Let λ > 1. Let S(φ, λ) = {h ∈ H :
|φ(h)| > λ|h|}. If h ∈ S(φ, λ), we say φ stretches h by λ. φ will be
called hyperbolic if for all λ > 1 there exists n > 0 such that for all
h ∈ H, at least one of φn or φ−n stretches h by λ.
φ and φ−1 induce bijections (also denoted by φ and φ−1 ) of the
vertices of ΓH .
A free homotopy representative of a word w ∈ H is a geodesic
[a, aw0] in ΓH where w0 is a shortest word in the conjugacy class of w
in H.
Given h ∈ H let Σ(h, n,+) (resp. Σ(h, n,−) be the (H-invariant)
collection of all free homotopy representatives of φn(h) (resp. φ−n(h))
in ΓH . The intersesction with ∂ΓH×∂ΓH of the union of all bi-infinite
subsequential limits (in the Hausdorff topology on Γ̂H ) of elements of
Σ(h, n,+) (resp. Σ(h, n,−) as n→∞ will be denoted by Λ+(h) (resp.
Λ−(h)).
Definition: The stable and unstable ending laminations are
respectively given by
Λ+ =
⋃
h∈HΛ+(h)
Λ− =
⋃
h∈HΛ−(h)
ON A THEOREM OF SCOTT AND SWARUP 5
Theorem 3.1. [11] Let 1 → H → G → Z → 1 denote an exact
sequence of hyperbolic groups arising out of a hyperbolic automorphism
φ of the hyperbolic group H. Then there exists a Cannon Thurston
map for the pair (H,G).
Theorem 3.2. [9] Let 1→ H → G→ Z → 1 denote an exact sequence
of hyperbolic groups arising out of a hyperbolic automorphism φ of the
hyperbolic group H. Then ΛCT = Λ+ ∪ Λ−.
Further, it is shown in [9] that only finitely many h’s need be con-
sidered in the definition of Λ+ or Λ−.
We turn now to the main focus of this paper, the case where H is
free and φ is a hyperbolic automorphism [1] . Such automorphisms
have been studied in great detail by Bestvina, Feighn and Handel [1] ,
[3] , [2] .
Definition: A non negative irreducible matrix is aperiodic if it has
an iterate that is strictly positive.
Definition: Let us assume for a start that transition matrices of φ
and φ−1 with respect to train-track representatives [see [4] for defini-
tions] are aperiodic. We shall refer to such automorphisms as aperiodic.
Note that in this definition, we require transition matrices of both φ
and φ−1 to be aperiodic.
In this case, the definitions of ending laminations here and in [3] co-
incide. To see this and for the sake of completeness we recall definitions
from [3] .
Let f : X → X be a train-track representative of an outer automor-
phism with aperiodic transition matrix. Endow X with the structure
of a marked R-graph so that f expands lengths of edges by a uniform
factor λ > 1. Let x ∈ X be an f -periodic point in the interior of some
edge. Let ǫ > 0 be small, and let U be the ǫ-neighborhood of x. Then
for some N > 0, U ⊂ fN(U). Choose an isometry l : (−ǫ, ǫ) → U and
extend it to the unique locally isometric immersion l : R → X such that
l(λN t) = fN(l(t)). We say l is obtained by iterating a neighborhood of
x. l will also be termed a leaf of the ending lamination.
Definitions : Two isometric immersions [a, b] → X and [c, d]→ X
are said to be equivalent if there is an isometry of [a, b] onto [c, d]
making the triangle commute.
A leaf segment of an isometric immersion R → X is the equivalence
class of the restriction to a finite interval.
Two isometric immersions l, l′ : R → X are (weakly) equivalent if
every leaf segment of l is a leaf segment of l′
6 MAHAN MITRA
Since f has an aperiodic transition matrix, l is surjective. Using
this, Bestvina, Feighn and Handel [3] show that any two leaves of the
ending lamination obtained by iterating neighborhoods of f -periodic
points are equivalent.
It is now clear ( see [3] Proposition 1.6, for instance ) that the def-
inition of stable laminations given before Theorem 3.1 above consists
of end-points of leaves belonging to the equivalence class obtained by
iterating a neighborhood of a periodic point in Bestvina, Feighn and
Handel’s construction above.
We can now use the results of [3] and [2] in our context.
Remark : Since any two leaves are (weakly) equivalent in the sense
of [3] above, the equivalence class can alternately be obtained by trans-
lating some (any) leaf by elements of the free group and taking Haus-
dorff limits. This is analogous to the case for surfaces where the stable
lamination of a pseudo anosov diffeomorphism is the closure of some
(any) leaf.
The following is a paraphrasing of Proposition 2.4 of [3] . It says
roughly that any leaf of the stable (or unstable) lamination of an ape-
riodic automorphism ‘fills’ the free group H .
Proposition 3.3. Let 1 → H → G → Z → 1 be an exact sequence of
hyperbolic groups induced by an aperiodic automorphism φ of the free
group H (i.e. φ and φ−1 have aperiodic transition matrices). If (p, q) ∈
Λ+ or Λ− lie in the boundary ∂ΓH1 ⊂ ∂ΓH for a finitely generated
subgroup H1 of H, then H1 is of finite index in H.
We are now in a position to prove the main theorem of this paper
for aperiodic φ .
Theorem 3.4. Let 1 → H → G → Z → 1 be an exact sequence of
hyperbolic groups induced by an aperiodic automorphism φ of the free
group H. Let H1 be a finitely generated subgroup of infinite index in
H. Then H1 is quasiconvex in G.
Proof: From Theorem 3.1 the pair (H,G) has a Cannon - Thurston
map. Further, from Theorem 3.2
ΛCT (H,G) = Λ(H,G) = Λ+ ∪ Λ−.
Let j : H1 → H and i : H → G denote inclusions. Since H1 is qua-
siconvex in H , Λ(H1, H) = ∅ (Lemma 2.1 ). Further from Proposition
3.3 above j−1(Λ(H,G)) = ∅.
Also from the remark following Lemma 2.2 , Λ(H1, G) = Λ(H1, H)∪
j−1(Λ(H,G)) = ∅.
ON A THEOREM OF SCOTT AND SWARUP 7
Hence from Lemma 2.1 H1 is quasiconvex in G. 2
As a second step we deal with automorphisms φ of H satisfying the
following:
There exists a decomposition H = K1 ∗ K2 ∗ · · · ∗ Kn of H into φ-
invariant factors Ki such that the restrictions φ|Ki = φi are aperiodic.
Further, assume φi has a train-track representaive with a fixed point
(this is always satisfied by large enough powers of φi).
Identifying the fixed points of the train-track representatives of φi
we obtain a particular representative of φ. Further, let Λi denote the
ending laminations of φi. It is easy to see from the representative we
have chosen that Λ ∩ ∂ΓKi = Λi. (See also Remark 1.4 of [3].)
Suppose H1 is a finitely generated subgroup of H that is distorted
in G. Since H1 is quasi-convex in H , there exists a pair (p, q) ∈ Λ =
Λ+ ∪ Λ− lying on the boundary ∂ΓH1 ⊂ ∂ΓH . Let l be a leaf of Λ
joining p, q. By Theorem 3.2 l lies in the Hausdorff limit of sequences of
segments obtained by iterating φ or φ−1 on some h ∈ H . By the pigeon-
hole principle there exists arbitrarily long segments of l contained in a
(fixed) conjugate of Kj for some i. For ease of exposition let us assume
that this is the trivial conjugate of Ki, i.e. Ki itself. Translating by
appropriate elements of H1 and taking a Hausdorff limit we obtain a
leaf of the ending lamination Λ whose endpoints lie in the intersection
∂ΓH1∩∂ΓKj . In particular, there exists a pair of points s, t ∈ ∂ΓH1 ∩Λj
since Λ ∩ ∂ΓKj = Λj .
Since intersection of quasiconvex subgroups is quasiconvex [14] it
follows that H1 ∩ Kj is quasi-convex in H . In particular H1 ∩ Kj is
finitely generated. Hence from Theorem 3.4 H1 ∩Kj is a finite index
subgroup of Kj.
We have shown :
Theorem 3.5. Let 1 → H → G → Z → 1 be an exact sequence of
hyperbolic groups induced by an automorphism φ of the free group H
satisfying the following :
There exists a decomposition H = K1 ∗ K2 ∗ · · · ∗ Kn of H into φ-
invariant factors Ki such that the restrictions φ|Ki = φi are aperiodic.
Further, assume φi and φi
−1 have train-track representatives with fixed
points.
Let H1 be a finitely generated subgroup of infinite index in H such
that H1 is distorted in G. Then there exist h ∈ H and Kj such that
H1 contains a finite index subgroup of h
−1Kjh.
We are now in a position to treat a general hyperbolic automorphism
φ.
8 MAHAN MITRA
We state a special case of a theorem due to Bestvina, Feighn and
Handel (Corollary 4.7 of [2] ).
Theorem 3.6. [2] Let φ be a hyperbolic automorphism of the free group
H. Then there exists an N > 0 satisfying the following :
There exists a decomposition H = K1 ∗ K2 ∗ · · · ∗ Kn of H into φ
N -
invariant factors Ki such that the restrictions φ
N |Ki and φ
−N |Ki are
aperiodic. Further, each of the restrictions have train-track represen-
tatives with fixed points.
Theorem 3.6 above allows us to reduce the general case to the case
treated by Theorem 3.5 .
Definition: A subgroup A of a group B is said to be a free factor
of B if there exists a group C (perhaps trivial) such that A∗C is equal
to B.
Theorem 3.7. Let 1 → H → G → Z → 1 be an exact sequence of
hyperbolic groups induced by a hyperbolic automorphism φ of the free
group H. Let H1(⊂ H) be a finitely generated distorted subgroup of
G. Then there exists N > 0 and a free factor K of H such that the
conjugacy class of K is preserved by φN and H1 contains a finite index
subgroup of a conjugate of K.
Proof: Taking a large enough power of φ we obtain an exact se-
quence 1 → H → G1 → Z → 1, induced by φ
N satisfying the conclu-
sions of Theorem 3.6. Note that G1 is a finite index subgroup of G.
Let H = K1 ∗K2 ∗ · · · ∗Kn be a decomposition of H into φ
N -invariant
factors Ki such that the restrictions φ
N |Ki and φ
−N |Ki are aperiodic.
Since H1(⊂ H) is a distorted subgroup of G and G1 is a finite index
subgroup of G, H1 is distorted in G1. Then by Theorem 3.5 there exist
h ∈ H andKj such thatH1 contains a finite index subgroup of h
−1Kjh.
Since Kj is a free factor of H , so are its conjugates. This proves the
Theorem. 2
Acknowledgements : The argument for the case of an aperiodic
automorphism had been outlined in [10] . The author would like to
thank his advisor Andrew Casson and John Stallings for their help.
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ON A THEOREM OF SCOTT AND SWARUP 9
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University of California at Berkeley, CA 94720
email: mitra@math.berkeley.edu
Current Address: Institute of Mathematical Sciences, C.I.T. Campus,
Madras (Chennai) - 600113, India.
email : mitra@imsc.ernet.in


On a theorem of Scott and Swarup

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