Spectral Rigidity and Subgroups of Free Groups

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by  Brian Ray

SPECTRAL RIGIDITY AND SUBGROUPS OF FREE GROUPS
BRIANRAY
4
1
0 Abstract. A subset Σ⊂FN of the freegroup ofrank N iscalledspectrally
2 rigid if whenever trees T,T′ in Culler-Vogtmann Outer Space are such that
kgkT = kgkT′ for every g ∈ Σ, it follows that T = T′. Results of Smillie,
n Vogtmann,Cohen,Lustig,andSteinerprovethat(forN ≥2)nofinitesubset
a ofFN isspectrallyrigidinFN. Weprovethatif{Hi}ki=1isafinitecollectionof
J subgroups,eachofinfiniteindex,andgi∈FN,then∪ki=1giHiisnotspectrally
8 rigidinFN. TakingHi =1,werecover theresultsabout finite sets. We also
prove that any coset of a nontrivial normal subgroup H ⊳FN is spectrally
] rigid.
R
G
.
h
t 1. Introduction
a
m
The Culler-VogtmannOuterSpace(denotedcv )wasintroducedbyCuller and
N
[ Vogtmann[9] as a free groupanalogofTeichmuller Space. While the latter admits

1 an action by the Teichmuller Modular Group, Outer Space admits an action by
v Out(FN), the outer automorphism group of FN. The space cvN has proven to be
2 a key tool in studying Out(F ).
N
6
Formally, cv is the space of all free, minimal, discrete, isometric actions of
N
8 F on R-trees, T, considered up to F -equivariant isometry. One may think of a
1 N N
. pointinthespaceasatriple(Γ,τ,l)whereΓisafinite graphequippedwithbotha
1 markingisomorphism,τ: F →π (Γ),andafunctionl: E(Γ)→R whichassigns
0 N 1 e >0
4 a length to each edge. Given such a triple, one sets T = Γ so that T is equipped
1 with a covering space action of FN ∼= π1(Γ) which is free, minimal, discrete, and
: isometric. One obtains (Γ,τ,l) from T by considering the quotient graph Γ =
v
i FN\T. To each T we associate a length function k·kT: FN → R≥0 defined by
X
kgk =inf {d(x,gx)}. Equivalently, kgk is the lengthofthe cyclicallyreduced
T x∈T T
r loop(inΓ)whichrepresentstheconjugacyclassofτ(g). Theclosure,cv ,ofcv is
a N N
knowntoconsistofthesocalledverysmallactionsofF onR-trees[1,8]. Thefact
N
that k·k determines T [6] gives an injection ℓ: cv →(R )FN; we call ℓ(T) the
T N ≥0
markedlengthspectrum ofT. Letℓ=ℓ| andnotethatℓisalsoinjective. Morally
cvN
speaking, a subset Σ⊂F is spectrally rigid (resp. strongly spectrally rigid) if one
N
canreplacethetargetofℓ(resp. ℓ)by(R )Σ andstillretaininjectivity. Theterm
≥0
was recently coined (in the free group setting) by Kapovich [15], though similarly
spirited work dates back to the early 1990’s (see [7, 24]).
Definition 1.1 ((Strongly) Spectrally rigid). We say Σ ⊆ F is (strongly) spec-
N
trally rigid if whenever T ,T ∈ cv (resp. cv ) are such that kgk = kgk for
1 2 N N T1 T2
every g ∈Σ, then T =T in cv (resp cv ).
1 2 N N
2000 Mathematics Subject Classification. 20F65.
Theauthor acknowledges supportfromNationalScienceFoundation grantDMS-0904200.
1
2 BRIANRAY
For N ≥3 Smillie and Vogtmann [24] prove that no finite set is spectrally rigid
in F . More specifically, they show that given a finite set, Σ, of conjugacy classes
N
there exists a one parameter family of trees, T ∈ cv , whose translation length
t N
functionsallagreeonΣ. ForN =2,Cohen,LustigandSteiner[7]provideasimilar
argument as to why no finite set is spectrally rigid in F . We prove the following.
N
Theorem A. Let N ≥ 2. Let {H }k be a finite collection of finitely generated
i i=1
subgroups H < F . Let {g }k be a finite collection of elements g ∈ F . Let
i N i i=1 i N
H=∪k g H . Then the following are equivalent.
i=1 i i
(1) For every i, [F : H ]=∞.
N i
(2) H⊂F is not spectrally rigid in F .
N N
Intheabovetheorem,werecovertheresultsofSmillie,Vogtmann,Cohen,Lustig,
andSteiner by setting H ={1}. The proofby Smillie andVogtmann uses the fact
i
that ina finite set ofconjugacy classes,there is a universalbound on the exponent
with which a given primitive element may occur. Their argument, however, works
for any set Σ which satisfies the following property: there exists a triple (A,a,M)
with A a free basis, a ∈ A, and M ≥ 1, so that for any g ∈ Σ, if ak occurs as a
subword of the cyclically reduced form (over A) of g, then |k|≤M (see Definition
2.1 and Theorem 2.2). To prove Theorem A, we use laminations associated to a
fully irreducible automorphism. Introduced by Bestvina, Feighn, and Handel in [2]
theselaminationsaredefined–foragivenfullyirreducibleϕ∈Out(F )–interms
N
ofa traintrack representativef: Γ→Γ. Leaves ofthe lamination are “generated”
by edgepaths of the form fn(e) for any e ∈ E(Γ) and n ∈ N (see [2, 12]). It is a
theoremofBestvina,Feighn,andHandel[2]thatonlyfinitely generatedsubgroups
of finite index may “carry a leaf” of such a lamination. It follows that, for finitely
generatedinfinite index subgroups, edgepaths of the form fn(e) (for suitably large
n) can not be “read” in H. Thus if a primitive loop contains such an fn(e), then
it cannot be a subword of any h∈H (compare Definition 2.1).
Theorem A fails if we allow infinite unions. Indeed, consider H = ∪ g{1}.
g∈FN
Then H =F and so is spectrally rigid. Since translation length functions satisfy
N
kgnk = nkgk, any finite index subgroup is spectrally rigid. More generally, if
π: FN →G is a presentation homomorphism (G∼=FN/kerπ) for a torsion group,
G, then kerπ is spectrally rigid. Note that this applies to (normal) subgroups
of infinite index (in the case G is infinite), and these subgroups are necessarily
infinitely generated. Thus the finite generation assumption in Theorem A is also
essential.
Carette, Francaviglia, Kapovich, and Martino prove [10] that if H < Aut(F )
N
is such that the image of H in Out(F ) contains an infinite normal subgroup,
N
then for any g ∈ F (so long as N 6= 2 or g not conjugate to a power of [a,b]
N
in F = F(a,b)) the orbit Hg is spectrally rigid. This shows, for example, that
2
the commutator subgroup, [F ,F ], of F is spectrally rigid, as it contains the
N N N
orbitAut(F )[g,h]forasuitablecommutator[g,h]. Moregenerally,anynontrivial
N
verbal or marginal subgroup is spectrally rigid since such groups are known to be
characteristic [23, §2.3].
Motivatedbytheaboveresultregardingcharacteristicsubgroups,IlyaKapovich
asked the following question: Is it true that for any nontrivial normal subgroup
H ⊳F , H is a spectrally rigid set? We prove the following.
N
SPECTRAL RIGIDITY AND SUBGROUPS OF FREE GROUPS 3
Theorem B. Let H⊳F be a nontrivial normal subgroup. Then for any g ∈F ,
N N
the coset gH is strongly spectrally rigid.
The proof uses geodesic currents on free groups (see [14, 17, 18, 20]). These are
positiveRadonmeasureson∂2F =∂F ×∂F \∆(where∆isthediagonal)which
N N N
are invariantunder the diagonalaction of F and under the map which exchanges
N
the coordinates. The space of all such currents is denoted Curr(F ). The spaces
N
Curr(F )andcv arerelatedbyanintersectionform [17],h·,·i: cv ×Curr(F )→
N N N N
R . The main feature of the intersection form that we use is as follows: given a
≥0
conjugacy class g ∈ F , there is a canonically associated current η ∈ Curr(F )
N g N
such that for any T ∈ cv , we have hT,η i = kgk . Now given any nontrivial
N g T
g,r ∈ F , we show that the normal closure, ncl(r), of r contains a sequence of
N
words w such that large powers of g exhaust w . Linearity of the intersection
n n
form then reveals that the associated counting currents converge to the counting
currentofg,andthatiftwotreesT,T′ agreeonncl(r)(andhenceonthew ),then
n
they must agree on g. Since g is arbitrary, we conclude T = T′. Our result then
follows from the fact that any nontrivial normalsubgroupcontains ncl(r) for some
nontrivial r.
2. Preliminaries
The N-Rose, R , is the graph with one vertex and N topological edges. If p
N
is an edgepath in a graph, G, we let [p] denote the reduced edgepath homotopic
(rel. endpoints)to p. If p is a closedpath, we let [[p]]denote the cyclically reduced
edgepath freely homotopic to p. If g ∈ F and A is a free basis, then [g] (resp
N A
[[g]] )denotesthe freelyreduced(resp. cyclicallyreduced)formofg overthebasis
A
A. If v,w are edgepaths in G, we write v ⋐w to indicate that v is a subpath of w.
If g,h∈F and a free basis, say A, is specified, then g ⋐h means the edgepath g
N
occurs as a subpath of the edgepath h in the graph G = R , the rose with petals
A
labelled by the elements of A.
2.1. Property W. As mentioned in the introduction, the proof by Smillie and
Vogtmann [24] that no finite set is spectrally rigid in F (for N ≥ 3) admits a
N
generalization. Specifically,theirargumentproceedsverbatimsolongasthesubset
Σ⊂F satisfies the following property.
N
Definition 2.1 (Property W). Let Σ ⊆ F . We say Σ has property W if there
N
exist a free basis A of F , a∈A, and M ≥1 so that for any σ ∈Σ, if ak ⋐[[σ]]
N A
then |k|≤M.
In [21] it is verified that the argument provided by Cohen, Lustig, and Steiner
[7] (for N =2) also works under the assumption of property W.
Theorem2.2(CLS,R,SV[7,21,24]). ForN ≥2,ifasubsetΣ⊂F hasProperty
N
W, then Σ is not spectrally rigid.
2.2. Stallings Subgroup Graphs. Given a point T = (Γ,τ,l) ∈ cv , and a
N
finitely generated subgroup H ≤F , we consider the (possibly infinite) cover of Γ
N
correspondingtothesubgroupτ(H),denoted(XT) . Itscore,denotedXT,isfinite
H r H
(since H is finitely generated)andrepresentsthe conjugacyclassofτ(H) inπ (Γ).
1
NotethatXT =(XT) (i.e. XT isE(Γ)-regular)ifandonlyif[F :H]<∞ifand
H H r H N
onlyifXT →Γisacovering. IfH isofinfiniteindex,thenoneobtains(XT) from
H H r
4 BRIANRAY
XT byattaching“hangingtrees”atverticeswhicharenotE(Γ)-regular. Notethat
H
varying the basepoint in (XT) corresponds to conjugation of τ(H) by elements
H r
of π (Γ). In the event the basepoint, b, in (XT) lies outside XT, we let (XT)
1 H r H H b
denote the graph XT ∪[XT,b], where [XT,b] is the bridge between XT and b in
H H H H
(XT) . For more information see, e.g., [4, 19, 25].
H r
2.3. Train Track Maps. By agraph map, wemeana functionf: V(Γ)∪E(Γ)→
V(Γ′)∪P(Γ′) sending vertices to vertices and edges to edgepaths and for which
the notions of incidence and inverse are preserved. If A is a free basis of F and
N
ϕ ∈ Out(F ), then we can always think of ϕ as a graph map R → R by
N N N
pairing E(R ) with A. By a marked graph we mean a pair (Γ,τ), where Γ is a
N
graph and τ: R →Γ is a homotopy equivalence. In the event Γ = R , we write
N N
R =R , with A the free basis determined by the pairing E(R ) with A. Given
N A N
ϕ ∈ Out(F ), and a marked graph (Γ,τ) we say that a graph map f: Γ → Γ is
N
a topological representative of ϕ with respect to τ if f is a homotopy equivalence
and the outer automorphism determined by τ ◦f ◦τ−1: R → R is equal to
N N
ϕ: R →R .
N N
Definition 2.3 (Train track map). A graph map f: Γ →Γ is called a train track
map if for every e∈E(Γ), the edgepath fn(e) is not a vertex and is reduced.
We say that ϕ ∈ Out(F ) is reducible if there is a free factorization F =
N N
F0 ∗···∗Fl−1 ∗H, with l ≥ 1 and 1 ≤ rk(F0) < N, so that ϕ permutes the
conjugacy classes of the Fi; we allow H =1. Otherwise, we say ϕ is irreducible.
Theorem 2.4 (Bestvina and Handel [3, Theorem 1.7]). Every irreducible auto-
morphism is topologically represented by a train track map.
If all positive powers of ϕ are irreducible, then we say that ϕ is irreducible with
irreducible powers (iwip for short). Thus ϕ is fully irreducible if and only if for all
k ≥ 1, ϕk does not preserve the conjugacy class of a proper free factor. In what
follows, we use train track maps f: Γ → Γ representing iwip automorphisms in
Out(F ). The existence of such automorphisms – for every rank N ≥ 2 – is well
N
known (see, for example, [11, 22])
2.4. Bestvina-Feighn-Handel Laminations. Fix a free basis A of F . Let
N
∂F be the Gromov boundary of the word hyperbolic group, F ; that is, ∂F :=
N N N
{a a ··· | a ∈ A±,a−1 6= a }. The double boundary of F , denoted by ∂2F ,
1 2 i i i+1 N N
is defined by ∂2F := (∂F ×∂F )\∆, where ∆ ⊂ ∂F ×∂F consists of those
N N N N N
pairs (ζ ,ζ ) for which ζ = ζ . More generally, given a marked graph (Γ,τ), one
1 2 1 2
e
defines ∂2Γ by considering instead one sided infinite reduced edgepaths in Γ.
Definition 2.5 (Bestvina-Feighn-Handel lamination). Let ϕ ∈ Out(F ) be a
N
fully irreducible automorphism equipped with a train track map f: Γ → Γ. The
Bestvina-Feighn-Handellamination associatedtoϕ∈Out(F ),denotedbyL (ϕ,f,Γ),
N BFH
e
is the set of pairs (ζ ,ζ ) ∈ ∂2Γ which have the following property: for every fi-
1 2
nite subpath [z ,z ] ⋐ (ζ ,ζ ) there exists an e ∈ E(Γ) and an n ≥ 1 so that
1 2 1 2
fn(e)⋑π([z ,z ]). Hereπ: ∂2Γe →Γis the “labelling”mapwhich,oneachcoordi-
1 2
e
nate, coincides with projection from the universalcovering, π: Γ→Γ. Such a pair
(ζ ,ζ ) is called a leaf of the lamination.
1 2
SPECTRAL RIGIDITY AND SUBGROUPS OF FREE GROUPS 5
If f: Γ → Γ is a train track representing an iwip, then given any pair of edges
e,e′ ∈E(Γ),thereexistsann≥1sothateitherfn(e)⋑e′orfn(e−1)⋑e′. Inwhat
follows, we will always assume that we have passed to a power of ϕ so that this is
indeed the case. Definition 2.5 can be formulated(equivalently) in terms of a fixed
edge e∈E(Γ). Furthermore, the leaves of the Bestvina-Feighn-Handel lamination
satisfy a certain “recurrence” property, which we now formulate precisely.
Definition 2.6 (Quasiperiodicity). A leaf (ζ ,ζ ) of L (ϕ,f,Γ) is said to be
1 2 BFH
quasiperiodic if for every L>0, there exists L′ > L so that the following holds: if
[z ,z ],and[w ,w ]aresubpathsof(ζ ,ζ )forwhich|[z ,z ]|>L′ and|[w ,w ]|<
1 2 1 2 1 2 1 2 1 2
L, then π([w ,w ])⋐π([z ,z ]) (here π: Γe →Γ is the labeling map).
1 2 1 2
Proposition 2.7 (Bestvina, Feighn, and Handel [2, Proposition 1.8]). Every leaf
(ζ ,ζ ) of L (ϕ,f,Γ) is quasiperiodic. (cid:3)
1 2 BFH
Definition 2.8 (Carrying a leaf). Let H ≤ F be a finitely generated subgroup.
N
Let ϕ∈Out(F ) be a fully irreducible automorphism equipped with a train track
N
map f: Γ → Γ (the marking on Γ being τ: R → Γ). Let T = (Γ,τ,l). Let
N
XT be the Stallings subgroup graph corresponding to the (conjugacy class of the)
H
subgroup τ(H) ≤ π (Γ,τ(∗)). We say that (the conjugacy class of) H carries the
1
leaf (ζ ,ζ ) of L (Φ,f,Γ) if for every finite subpath [z ,z ] of (ζ ,ζ ), the map
1 2 BFH 1 2 1 2
π| : [z ,z ]→Γ
[z1,z2] 1 2
factors through XT as a map [z ,z ]→XT →Γ.
H 1 2 H
Of particular interest to us is the following proposition.
Proposition2.9(Bestvina,Feighn,andHandel[2,Lemma2.4]). Letϕ∈Out(F )
N
be a fully irreducible automorphism equipped with a train track f: Γ → Γ. If a
finitely generated subgroup H ≤ F carries a leaf of L (ϕ,f,Γ), then [F :
N BFH N
H]<∞. (cid:3)
2.5. Stability of Quasi-Geodesics. We will also need some basic results about
stability of quasi-geodesics.
Proposition 2.10 (Bridson and Haefliger [5, III.H Theorem 1.7]). For all δ >
0,λ ≥ 1,ǫ ≥ 0 there exists a constant R = R(δ,λ,ǫ) with the following property:
If X is a δ-hyperbolic geodesic space, c is a (λ,ǫ)-quasi-geodesic in X and [p,q] is
a geodesic segment joining the endpoints of c, then the Hausdorff distance between
[p,q] and the image of c is less than R.
Proposition2.11. Let X,Y be δ-hyperbolic geodesic metricspaces equipped with a
(λ,ǫ)-quasi-isometry f: X →Y. Suppose A,B are collections of geodesic paths in
X withthefollowing property,thereexistsaconstantC suchthatforanyα∈Aand
β ∈B, diam (α∩β)<C. Let α′ =[f(α)],β′ =[f(β)] be geodesics in Y obtained
X
by reducing the images f(α),f(β), respectively. The there exists a constant C′ such
that for all α∈A,β ∈B, we have diam (α′∩β′)<C′.
Y
Proof. Let R = R(0,λ,ǫ) be the constant afforded by Proposition 2.10. Let
{α },{β } ⊂ X and and {α′},{β′} ⊂ Y be arbitrary sequences of geodesics (with
i i i i
α′,β′containedintheR-neighborhoodoff(α ),f(β ),respectively). Supposewith-
i i i i
outlossthatl (α ),l (β )→∞(the propositionis obviousotherwise). Since R is
X i X i
finite andsince f is a quasi-isometry,we must havethat l (α ),l (β )→∞. Sim-
X i X i
ilarly, the distance (in Y) between f(α ∩β ) and α′ ∩β′ is uniformly bounded
i i i i
6 BRIANRAY
(independent of i). Suppose for contradiction that no such C′ exists; that is,
diam (α′ ∩β′)→∞. Then (by passing to a subsequence and reindexing) we may
Y i i
find a sequence of points a ∈α ,b ∈ β with d (a ,α ∩β ),d (b ,α ∩β )→∞
i i i i X i i i X i i i
for which d (f(a ),f(b )) is uniformly bounded. This contradicts the fact that f
Y i i
is a quasi-isometry. (cid:3)
Corollary 2.12. Let (Γ,τ) be a marked graph. Suppose z is a loop in G which
represents a primitive element of π (Γ). Let A be an arbitrary collection of loops
1
in G. Suppose there exists a number M such that for all a∈A whenever zk ⋐a it
follows that |k|≤M. Then for any free basis B with z ∈B and a number M′ such
that for all a∈A whenever zk ⋐[a] it follows that |k|≤M′.
B
Proof. Let Z = {zk | k ∈ Z}. Let f: Γe → Cay(F ,B) be a quasi-isometry with
N
B any free basis containing z. We may think of Z, A as defining a collection of
e
geodesicsinΓ. The assumptiononz affordsaconstant,C, suchthatforanyζ ∈Z
and any α ∈ A, we have diame(ζ ∩α) < C. We now apply Proposition 2.11 and
Γ
conclude that (in Cay(F ,B)) there is a bound, C′ on diam (ζ′,α′). Thus
N Cay(FN,B)
there is a bound M′ so that if zk ⋐[a] , then |k|≤M′. (cid:3)
B
2.6. Geodesic Currents. A geodesic current on F is a positive Radon measure
N
(a Borel measure that is finite on compact sets) on ∂2F which is invariant under
N
the diagonal action of F and under the map which interchanges the coordinates
N
of ∂2F . The space of all currents is denoted Curr(F ). If ν,µ ∈ Curr(F ) are
N N N
(nontrivial) currents such that ν = λµ for some λ ∈ R∗ then we write [ν] = [µ],
where[·]denotestheprojectiveclassofagivencurrent. Toeachrootfreeconjugacy
class g ∈ F we associate a counting curent, η as follows. Let A be a free basis
N g
of F . Write g as a cyclically reduced word over A. We can thus think of g as
N
the label of a directed graph, Γ , which is topologically homeomorphic to S1 and
g
has kgk edges (and hence kgk vertices), each edge labelled by the appropriate
A A
a ∈A. Nowletvbeanyfreelyreducedword. Bythenumberofoccurrencesofvor
i
v−1 in g, denoted n (v±), we mean the number of vertices of Γ at which one may
g g
read the word v or v−1 along Γ (going ‘with’ the oriented edges) without leaving
g
Γg. Note that nu(v±) = nu−1(v±) by definition. Now let ve be a lift of v to the
CayleytreeofF withrespecttoA. LetCyl(v)⊂∂2F be the setofall(ζ ,ζ )⊂
N N 1 2
∂2(F ) such that the bi-infinite geodesic representing (ζ ,ζ ) passes through ve.
N 1 2
Then η (Cyl(v)) = n (v±). If g = hk for h a root free conjugacy class, we define
g g
n (v±) = kn (v±). Currents of the form λη for λ > 0 and g ∈ F form a dense
g h g N
subset of Curr(F ) [13, Corollary 3.5]. For more information, see [14, 17, 18, 20].
N
We will need the notion of an intersection form h·,·i: cv ×Curr(F )→R.
N N
Proposition 2.13 (Kapovich and Lustig [17, Theorem A]). There exists a map
h·,·i: cv ×Curr(F ) → R which is continuous, Out(F )-invariant, and linear
N N N
with respect to the second argument. Furthermore for every g ∈ F , and T ∈ cv
N N
we have hT,η i=kgk .
g T
3. proof of theorem a
Inwhatfollows,wheneverwespeakofaniwipϕ∈Out(F )equippedwithtrain
N
track map f: Γ → Γ we assume that we have passed to a suitable power of ϕ so
that for any topological edge e in Γ, f(e) crosses each topological edge in Γ. We
will, however, continue to write ϕ,f.
SPECTRAL RIGIDITY AND SUBGROUPS OF FREE GROUPS 7
Proposition 3.1. Let H ≤ F be a finitely generated subgroup of infinite index.
N
Let ϕ∈Out(F ) be fully irreducible with train track map f: Γ→Γ on the marked
N
graph (Γ,τ). Let T = (Γ,τ,l) in cv . Let XT be the Stallings subgroup graph
N H
corresponding to the conjugacy class of τ(H) in π (Γ). Then there exists a power
1
m, such that for all n ≥ m, and for any e ∈ E(Γ), the edgepath fn(e) does not
factor through XT. Furthermore, given a basepoint b in (XT) , we may choose m
H H r
so that for all n≥m, fn(e) does not factor through (XT) .
H b
Proof. By Proposition 2.9, H can not carry a leaf of L (ϕ,f,Γ). Thus there
BFH
e
is a finite subpath [z ,z ] (of a leaf (ζ ,ζ ) ⊂ ∂2Γ) which does not factor through
1 2 1 2
XT. Let L = |[z ,z ]|. Since leaves are quasiperiodic, there exists L′ so that for
H 1 2
any subpath [w ,w ] of length at least L′, we have [w ,w ] ⋑ [z ,z ]. Choose a
1 2 1 2 1 2
basepointv ∈Γandletbbethebasepointin(XT) sothattheimageofπ ((XT) )
H r 1 H b
in π (Γ,v) is equal to τ(H). Let l be the length of the bridge [XT,b]. Set L′′ =
1 H
L′+2l+2diam(XT)+1. Now let m≥1 be so that the length of fm(e) is at least
H
L′′ for any e∈E(Γ). Then for any n≥m, fn(e)⋑[z ,z ] for any e∈E(Γ). Thus
1 2
for any n ≥m, and for any e∈ E(Γ), we have that fn(e) does not factor through
XT, nor can it factor through (XT) since at least 2l+1 edges in fn(e) must lie
H H b
outside of XT. (cid:3)
H
Corollary 3.2. Let H = ∪H be a finite union of subgroups H < F of infinite
i i N
index. Then there exists a power m such that for all n≥ m and for any e∈E(Γ)
andforalli,theedgepathfn(e)doesnotfactor through(XT ) . Furthermore,there
Hi b
exists a primitive element z ∈π (Γ) so that z contains fm(e) and hence z does not
1
factor through (XT ) for any i.
Hi b
Proof. Fix a fully irreducible ϕ∈Out(F ) equipped with train track map f: Γ→
N
Γ. Fix an edge e ∈ E(Γ). Now apply Proposition 3.1 to each H and obtain a
i
collection {m }k of exponents for which the following holds for each i: for all
i i=1
n ≥ m the edgepath fn(e) does not factor through (XT ) . Set m = max {m }.
i Hi b i i
Thenforalln≥mandforalliwehavethatfn(e)doesnotfactorthrough(XT ) .
Hi b
Now let a be any primitive loop in π (Γ). Since ϕ is fully irreducible, z cannot
1
be a periodic conjugacy class, and hence l ([[fk(a)]]) → ∞. Thus we can find an
Γ
edge e′ in suitable iterate, fk(a), of a so that in all further iterates of a by f there
is no cancellation in the images of e′. Thus for all n ≥ k+m+1, we have that
fm(e)occursinfn(a)andhencefn(a)cannotfactorthrough(XT ) foranyi. Set
Hi b
z =fn(a) for some n≥k+m+1. (cid:3)
Proposition 3.3. Let H=∪H be a finite union of subgroups H <F of infinite
i i N
index. Then there exists a primitive element z and a free basis B with z ∈B and a
number M so that for all i and for all h∈H , if zk ⋐[h] , then |k|≤M.
i B
Proof. Let z be the primitive element affordedby Corollary3.2. Let Z ={zk |k ∈
Z}. Let B = {τ(h) | h ∈ H}. The previous proposition gives the following: there
exists a constant C so that for any ζ ∈ Z and for any β ∈ B and for any choice
e e e e e e
of lifts ζ,β in Γ, the intersection diam(ζ ∩β) < C. Let f: Γ → Cay(F ,B) be a
N
quasi-isometry to the Cayley graph of F with respect to a basis B with z ∈ B.
N
Now apply Proposition 2.10 using Z,B and f. We conclude there exists a bound
M so that for any h∈H, if zk ⋐[h] , then |k|≤M. (cid:3)
B
8 BRIANRAY
Theorem 3.4. Let H =∪g H be a finite collection of cosets of finitely generated
i i
subgroups of infinite index, H ≤F . Then the set H has property W.
i N
Proof. Letz,B be theprimitive elementandfreebasisaffordedby Proposition3.3.
Thus there exists an M′ such that for all i and all h ∈ H if zk ⋐ [h ] , then
i i B
|k| ≤ M′. Since {g }n is finite, there is M so that for all i, if zk ⋐ [g ] , then
i i=1 g i B
|k|≤M . Thus there is an M so that for any i and any h ∈H , if zk ⋐[[g h ]] ,
g i i i i B
then |k|≤M. Thus H has property W with respect to (z,B,M). (cid:3)
Since kgnk = nkgk it is clear that any (coset of a) subgroup of finite index is
spectrally rigid. We thus have the following.
Theorem A. Let N ≥ 2. Let {H }k be a finite collection of finitely generated
i i=1
subgroups H < F . Let {g }k be a finite collection of elements g ∈ F . Let
i N i i=1 i N
H=∪k g H . Then the following are equivalent.
i=1 i i
(1) For every i, [F : H ]=∞.
N i
(2) H⊂F satisfies property W.
N
(3) H⊂F is not spectrally rigid in F .
N N
Proof. Theorem3.4gives(1)implies(2). Theorem2.2gives(2)implies(3). Finally,
if at least one subgroup is of finite index (i.e. not (1)), then H is spectrally rigid
(i.e. not (3)). (cid:3)
4. proof of theorem b
Recall that the space, Curr(F ), of geodesic currents consists of positive radon
N
measureson ∂2F which are F and flip invariant. Also recallthat Curr(F ) and
N N N
cv arerelatedby anintersectionformh·,·iforwhichhη ,Ti=kgk (see§2.6and
N g T
Proposition 2.13).
Proposition 4.1 (Kapovich and Lustig [16, Lemma 2.10]). Let A be a free basis
of F . Then for any cyclically reduced words w ,w∈F , we have
N n N
η η
lim [η ]=[η ] if and only if lim wn = w
n→∞ wn w n→∞kwnkA kwkA
Proposition 4.2. For any nontrivial r ∈ F , the set ncl(r) is strongly spectrally
N
rigid.
Proof. Let u ∈ F be an arbitrary cyclically reduced word. We will show that
N
kukT = kukT′ from which it will follow that T = T′ (since k·k is constant on
conjugacyclasses). To that end, we first constructa sequence of cyclically reduced
words {w }⊂ncl(r) with the property that
i
[η ]→[η ]
wi u
To that end, let
w =αuiβrβ−1u−iα−1γuiδrδ−1u−iγ−1
i
Evidently w is a product ofconjugates ofr, so is inncl(r). It is clearthat we may
i
chooseα,β,δ,γ so that w is freely (andso cyclically, by inspection) reduced. Now
i
let v ∈F be arbitrary. By Proposition 4.1 it is enough to show that
N
n (v±) n (v±)
lim wi = u
i→∞ kwikA kukA
SPECTRAL RIGIDITY AND SUBGROUPS OF FREE GROUPS 9
Note that there is a uniform bound onthe number ofoccurrencesofv or v−1 inw
i
which do not occur entirely within the ui or u−i subwords of w . Furthermore the
i
difference between kw k and 4kuik is uniformly bounded. We now have that
i A A
lim nwi(v±) = lim 2nui(v±)+2nu−i(v±)+Di
i→∞ kwikA i→∞ 4ikukA+Ci
= lim 2inu(v±)+2inu−1(v±)+Di
i→∞ 4ikukA+Ci
2in (v±)+2in (v±)+D
u u i
= lim
i→∞ 4ikukA+Ci
4in (v±)+D n (v±)
u i u
= lim =
i→∞ 4ikukA+Ci kukA
as desired.
Now suppose that T,T′ ∈cv agree on ncl(r). Since [η ]→[η ] there exists a
N wi u
sequence λ such that
i
lim λ η =η
i wi u
i→∞
Then we have that
kuk =hT,η i=hT, lim λ η i= lim λ hT,η i
T u i wi i wi
i→∞ i→∞
= lim λikwikT = lim λikwikT′ = lim λihT′,ηwii
i→∞ i→∞ i→∞
=hT′, lim λiηwii=hT′,ηui=kukT′
i→∞
Recall that u was an arbitrary conjugacy class. Since length functions are class
functions, we conclude that k·kT =k·kT′ and so T =T′ in cvN. (cid:3)
Corollary 4.3. Let r ∈ F be nontrivial and consider gncl(r). Then gncl(r) is
N
strongly spectrally rigid.
Proof. Intheproofoftheaboveproposition,wemaychooseα,β,γ,δ sothatgw is
i
cyclicallyreduced. Then–withperhapsdifferentconstantsC ,D –weobtainthat
i i
[η ]→[η ]. We conclude as above that gncl(r) is strongly spectrally rigid. (cid:3)
gwi u
Theorem B. Let H⊳F be a nontrivial normal subgroup. Then for any g ∈F ,
N N
the coset gH is strongly spectrally rigid.
Proof. Since H is nontrivial and normal, H ⊃ncl(r) for some nontrivial r. By the
above proposition, for any g ∈ F , gncl(r) is strongly spectrally rigid. Thus so is
N
gH. (cid:3)
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DepartmentofMathematics,UniversityofIllinoisatUrbana-Champaign,1409West
GreenStreet, Urbana,IL61801,USA
E-mail address: [email protected]