Ideal Whitehead Graphs in Out(F_r) III: Achieved Graphs in Rank 3

0.51 MB·English

by  Catherine Pfaff

Out F
Ideal Whitehead Graphs in ( )
r
III: Achieved Graphs in Rank 3
Catherine Pfaff
3
1
0
2 Abstract

n
By proving precisely which singularity index lists arise from the pair of invariant foliations for a
a
J pseudo-Anosov, Masur and Smillie, in [MS93], determined a Teichmu¨ller flow invariant stratification of

9 the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to an
2 Out(F ) analog of the [MS93] theorem. Since the ideal Whitehead graphs of [HM11] give a strictly finer
r

invariant in the analogous Out(F ) setting of a fully irreducible outer automorphism, we determined
] r
R which of the twenty-one connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully
G irreducible outer automorphisms in Out(F ). It can be noted that our methods are valid in any rank.
3
.
h
t
a 1 Introduction
m
[
Our main theorem (Theorem 4.1) is motivated by the [MS93] theorem of Masur and Smillie listing

1 precisely which invariant singular measured foliation singularity index lists arise from pseudo-Anosov
v
mappingclasses. TheMasur-Smillietheoremwassignificantinitsdeterminingastratificationofthespace
0
8 of quadratic differentials invariant under the Teichmu¨ller flow. For several results on the stratification
0
of the space of quadratic differentials and on the Teichmu¨ller flow, one can see, for example, [KZ03],
7
. [Lan04], [Lan05], [AB06], [Ath06], [EM08], and [Zor10]. This paper is the first step to proving an analog
1
to the [MS93] theorem for outer automorphism groups of free groups.
0
3 For a free group F of rank r, we denote the outer automorphism group by Out(F ). In this paper we
r r
1 analyze outerautomorphismsbytopological representatives: LetR bether-petaled rose. Given agraph
r
:
v Γ with no valence-one vertices, we can assign to Γ a marking via a homotopy equivalence R → Γ. One
r
i
X calls such a graph, together with its marking, a marked graph. Each outer automorphism φ ∈ Out(Fr)
r can be represented by a homotopy equivalence g: Γ → Γ of a marked graph, where φ= g∗ is the induced
a
map of fundamental groups. Analogous to pseudo-Anosov mapping classes are fully irreducible (iwip)
outer automorphisms, i.e. those such that no representative of a power leaves invariant a subgraph with
a nontrivial component. Thus, the analog theorem would involve fully irreducible outer automorphisms.
FI will denote the set of all fully irreducible elements in Out(F ).
r r
A beauty in studying the groups Out(F ) is how they are actually richly more complicated than
r
mapping class groups. A particularly good example of this arises when trying to generalize the Masur-
Smillie pseudo-Anosov index theorem. Unlike in the surface case where one has the Poincare-Hopf index
equality i(ψ) = χ(S), for a pseudo-Anosov ψ on a surface S, Gaboriau, Jaeger, Levitt, and Lustig proved
in[GJLL98]thatthereisinsteadanindexsuminequalityi(φ) ≥ 1−rforthefullyirreducibleφ ∈ Out(F ).
r
The index lists of geometric (induced by homeomorphisms of compact surfaces with boundary) fully
irreducibles are understood by the Masur-Smillie theorem, but complexity of the nongeometric case
prompted the following question [HM11]:
1
Question 1.1. Which index types, satisfying i(φ) > 1−r, are achieved by nongeometric, fully irreducible
φ∈ Out(F )?
r
Unlike in the surface case, the ideal Whitehead graph IW(φ) for a fully irreducible φ ∈ Out(F ) (see
r
[HM11], [Pfa12a], or Subsection 2.2) gives a strictly finer outer automorphism invariant than just the
corresponding index list. Indeed, for an ageometric φ ∈ FI , the index of a component in IW(φ) is
r
simply 1− k, where k is the number of vertices in the component. One can think of an ideal Whitehead
2
graph as describing the structure of singular leaves, in analog to the boundary curves of principle regions
in Nielsen theory [?].
Since an ideal Whitehead graph is a strictly finer invariant than a singularity index list, the deeper,
more appropriate question was thus:
Question 1.2. Which isomorphism types of graphs occur as IW(φ) for fully irreducible φ?
We focusontheindexsum3/2−r,theclosest possibletothatof 1−r,achieved bygeometrics, without
being achieved by any geometric outer automorphism. As in [Pfa12c], we denote the set of connected
(2r − 1)-vertex simplicial graphs by PI . Our partial answer (Theorem 4.1) to Question 1.2
(r;(3/2−r))
completely answers the following subquestion posed in person by Mosher and Feighn:
Question 1.3. Which of the twenty-one graphs in PI are the ideal Whitehead graph IW(φ) for
(3;(−3/2))
a fully irreducible φ∈ Out(F )?
3
The complete answer to Question 1.3 is our main theorem, Theorem 4.1:
Theorem. Exactly eighteen of the twenty-one connected, simplicial five-vertex graphs are the ideal
Whitehead graph IW(φ) for a fully irreducible outer automorphism φ ∈Out(F ).
3
The twenty-one graphs in PI ([CP84]) are:
(3;(−3/2))
I II III IV V VI
VII VIII IX X XI XII
XIII XIV XV XVI XVII XVIII
XIX XX XXI
The graphs in PI that are not ideal Whitehead graphs for fully irreducible φ ∈Out(F ) are:
(3;(−3/2)) 3
Outline:
2
Recall [BF94] that, for a train track g: Γ → Γ, a periodic Nielsen path (pNp) is a nontrivial path ρ in
Γ such that, for some k, gk(ρ) ≃ ρ rel endpoints. Also [GJLL98], an outer automorphism is ageometric
whose stable representative, in the sense of [BH92], has no pNp’s (closed or otherwise). We use AFI to
r
denote the subset of FI comprised of its ageometric elements.
r
Feighn and Handel defined rotationless train tracks and outer automorphisms in [FH11]. Recall
[HM11]: Let a φ ∈ AFI be such that IW(φ) ∈ PI , then φ is rotationless if and only if the
r (r;(3/2−r))
vertices of IW(φ) ∈ PI are fixed by the action of φ.
(r;(3/2−r))
Finally, recall [Pfa12c] that φ ∈ AFI with IW(φ) ∈ PI have pNp-free representatives of
r (r;(3/2−r))
a rotationless power whose Stallings fold decomposition [Sta83] consists entirely of proper full folds of
roses. We call such representatives ideally decomposed. The first ingredient in our Theorem 4.1 proof is
[Pfa12c] Proposition 3.3, implying ideal decomposition existence:
Proposition. Let φ ∈ FI be ageometric with IW(φ) ∈ PI . There exists a pNp-free train
r (r;(3/2−r))
track on the rose representing a rotationless power ψ = φR and decomposing as Γ −g→1 Γ −g→2 ··· −g−n−−→1
0 1
Γn−1 −g→n Γn, where:
(I) the index set {1,...,n} is viewed as the set Z/nZ with its natural cyclic ordering;
(II) each Γ is a rose with an indexing {e ,e ,...,e ,e } of the edge set such that:
k (k,1) (k,2) (k,2r−1) (k,2r)
(a) one can index the edge set of Γ with E(Γ) = {e1,e2,...,e2r−1,e2r} where, for each t with 1 ≤t ≤ 2r,
g(e ) = e ...e where (g ◦···◦g )(e ) = e ...e ;
t i1 is n 1 0,t n,i1 n,is
(b) for some i ,j with e 6= (e )±1
k k k,ik k,jk
e e for t = i
gk(ek−1,t):= k,t k,jk k
(cid:26)ek,t for all ek−1,t 6= e±k−11,jk; and
(c) for each e ∈ E(Γ) such that t 6= j , we have Dh(d )= d , where d = D (e ).
t n t t t 0 t
The next proof ingredient is the “lamination train track (ltt) structures” of [Pfa12c]. Using smooth
paths in ltt structures (see [Pfa12b]), we “construct” subgraphs of the ideal Whitehead graphs using
the construction compositions of [Pfa12b]. To determine which construction compositions to compose,
we use the “ideal decomposition (ID) diagrams” of [Pfa12c]. Recall that, if there is φ ∈ AFI with
r
IW(φ) ∼= G, where G ∈ PI(r;(3/2−r)), then there is a loop in the ID diagram for G corresponding to an
ideally decomposed representative of some φk (Proposition 2).
InSection 3we describethethreemain categories of strategies we usedtoproducetherepresentatives
for the main theorem.
Finally, in order to show that our maps represent φ∈ AFI , we use the “Full Irreducibility Criterion
r
(FIC)” proved in [Pfa12b] (Lemma 4.1):
Lemma. (The Full Irreducibility Criterion) Let g: Γ → Γ be a pNp-free, irreducible train track repre-
sentative of φ ∈ Out(F ). Suppose that the transition matrix for g is Perron-Frobenius and that all the
r
local Whitehead graphs are connected. Then φ is fully irreducible.
To apply the criterion we use the [Pfa12b] method for identifying ideally decomposed train track
representative pNps.
We proved Graphs II, V, and VII are unachievable in [Pfa12c].
Acknowledgements
The author expresses gratitude to Lee Mosher for incredible patience and generosity, invaluable discus-
sions, and answers to her endless questions; Mark Feighn for numerous meetings and question answers,
3
as well as recommendations for uses of her methods; Michael Handel for meeting with her and recommen-
dations, particularly on how to finish proving the FIC; Yael Algom-Kfir and Mladen Bestvina for all they
patiently taught her; Arnaud Hilion for his advice; and Martin Lustig for his continued interest in her
work. She also extends her gratitude to Bard College at Simon’s Rock and the CRM for their hospitality.
2 Preliminary definitions and notation
We use this section to establish notation and to remind the reader of background used throughout this
document. One familiar with [Pfa12c] and [Pfa12b] may simply skip to Section 3.
We continue with the introduction’s notation. Additionally, unless otherwise stated, we
assume throughout this document that outer automorphism representatives are train track
(tt) representatives in the sense of [BH92]. Further, unless otherwise specified, g: Γ → Γ
will represent φ ∈ Out(F ).
r
2.1. Directions and turns
In general we usedefinitions of [BH92] and [BFH00] for discussing train tracks. We remind the reader
hereofadditionaldefinitionsandnotationgivenin[Pfa12c]. E+(Γ) := {E1,...,En} = {e1,e1,...,e2n−1,e2n}
will be the edge set of Γ with a prescribed orientation. E(Γ) := {E ,E ,...,E ,E }, where E denotes
1 1 n n i
Ei oppositely oriented. If an edge indexing {E1,...,En} (thus indexing {e1,e1,...,e2n−1,e2n}) is pre-
scribed, we call Γ edge-indexed. V(Γ) will denote the vertex set of Γ and D(Γ) := D(v), where
v∈V(Γ)
D(v) is the set of directions at v. For each e ∈ E(Γ), D0(e) will denote the initiaSl direction of e and
D γ := D (e ) for each path γ = e ...e in Γ. Dg will denote the direction map induced by g. We call
0 0 1 1 k
d∈ D(Γ) periodic if Dgk(d) = d for some k > 0 and fixed if k = 1.
T(v) will denote the set of turns at a v ∈ V(Γ) and Dtg the induced turn map. Sometimes we
abusively write {ei,ej} for {D0(ei),D0(ej)}. For a path γ = e1e2...ek−1ek in Γ, we say γ traverses
{e ,e } for each 1 ≤ i < k. Recall, a turn is called illegal for g if Dgk(d ) = Dgk(d ) for some k.
i i+1 i j
2.2. Ideal Whitehead graphs and lamination train track (ltt) structures.
IdealWhiteheadgraphsweredefinedin[HM11]andlamination traintrack structuresin[Pfa12a](and
[Pfa12c]). We recount relevant definitions here. Further expositions can be found in [HM11],[Pfa12a],
and [Pfa12c].
Let Γ be a marked graph, v ∈ Γ a singularity (vertex with at least three periodic directions), and
g: Γ → Γ a tt representing φ∈ Out(F ). The local Whitehead graph LW(g;v) for g at v has:
r
(1) a vertex for each direction d∈ D(v) and
(2) edges connecting vertices for d ,d ∈D(v) when gk(e), with e∈ E(Γ), traverses {d ,d }.
1 2 1 2
The local Stable Whitehead graph SW(g;v) is the subgraph obtained by restricting precisely to vertices
with periodic direction labels. For a rose Γ with vertex v, we denote the single local stable Whitehead
graph SW(g;v) by SW(g) and the single local Whitehead graph LW(g;v) by LW(g).
For a pNp-free g, the ideal Whitehead graph IW(φ) of φ is isomorphic to SW(g;v).
singularitiesv∈Γ
In particular, when Γ is a rose, IW(φ) ∼= SW(g). F
Let g: Γ → Γ be a pNp-free tt on a marked rose with vertex v. Recall from [Pfa12c] the definition
of the lamination train track (ltt) structure G(g) for g: The colored local Whitehead graph CW(g) at v
is LW(g), but with the subgraph SW(g) colored purple and LW(g)−SW(g) colored red (nonperiodic
direction vertices are red). Let Γ = Γ − N(v) where N(v) is a contractible neighborhood of v. For
N
each E ∈ E+, add vertices D (E ) and D (E ) at the corresponding boundary points of the partial edge
i 0 i 0 i
E −(N(v)∩E ). The lamination train track (ltt) structure G(g) for g is formed from Γ CW(g) by
i i N
identifying vertex di in ΓN with vertex di in CW(g). Vertices for nonperiodic directions are rFed, edges of
4
Γ black, and all periodic vertices purple.
N
G(g) is given a smooth structure via a partition of the edges at each vertex into two sets: E (the
b
black edges of G(g)) and E (the colored edges of G(g)). A smooth path will mean a path alternating
c
between colored and black edges.
Several notions of abstract lamination train track structures are defined in [Pfa12a] and [Pfa12c]. We
recall now what is necessary for this paper:
A lamination train track (ltt) structure G is a pair-labeled colored train track graph (black edges will
be included, but not considered colored) satisfying:
ltt1: Vertices are either purple or red.
ltt2: Edges are of 3 types (E comprises the black edges and E comprises the red and purple edges):
b c
(Black Edges): A single black edge connects each pair of (edge-pair)-labeled vertices. There are
no other black edges. In particular, each vertex is contained in a unique black edge.
(Red Edges): A colored edge is red if and only if at least one of its endpoint vertices is red.
(Purple Edges): A colored edge is purple if and only if both endpoint vertices are purple.
ltt3: No pair of vertices is connected by two distinct colored edges.
We denote the purple subgraph of G (from SW(g)) by PI(G) and, if G ∼= PI(G), say G is an ltt
Structure for G. An (r;(3 −r)) ltt structure is an ltt structure G for a G ∈ PI such that:
2 (r;(3−r))
2
ltt(*)4: G has precisely 2r-1 purple vertices, a unique red vertex, and a unique red edge.
We consider ltt structures equivalent thatdiffer by an ornamentation-preserving homeomorphism and
refer the reader to the Standard Notation and Terminology 2.2 of [Pfa12c]. In particular, in abstract and
nonabstract ltt structures, [d ,d ] is the edge connecting a vertex pair {d ,d }, [e ] denotes the black edge
i j i j i
[d ,d ] for e ∈ E(Γ), and C(G) denotes the colored subgraph (from LW(g)). Purple vertices are periodic
i i i
and red vertices nonperiodic. G is admissible if birecurrent as a train track structure (i.e has a locally
smoothly embedded line traversing each edge infinitely many times as R→ ∞ and as R→ −∞).
For an (r;(3 −r)) ltt structure G for G, additionally:
2
1. du labels the unique red vertex and is called the unachieved direction.
2. eR = [tR] denotes the unique red edge, da labels its purple vertex, thus tR = {du,da} (eR = [du,da]).
3. da is contained in a unique black edge, which we call the twice-achieved edge.
4. da will label the other twice-achieved edge vertex and be called the twice-achieved direction.
5. If G has a subscript, the subscript carries over to all relevant notation. For example, in G , du will
k k
label the red vertex and eR the red edge.
k
We call a 2r-element set of the form {x ,x ,...,x ,x }, elements paired into edge pairs {x ,x }, a
1 1 r r i i
rank-r edge pair labeling set (we write x = x ). We call a graph with vertices labeled by an edge pair
i i
labeling set a pair-labeled graph, and an indexed pair-labeled graph if an indexing is prescribed.
Anlttstructure,indexpair-labeledas agraph,isanindexed pair-labeled lttstructureiftheblack edge
vertices are indexed by edge pairs. Index pair-labeled ltt structures are equivalent that are equivalent as
ltt structures via an equivalence preserving the indexing of the vertex labeling set.
Byrank-rindexpair-labelingan(r;(3−r))lttstructureGandedge-indexingtheedgesofanr-petaled
2
rose Γ, one creates an identification of the vertices in G with D(v), where v is the vertex of Γ. With this
identification, we say G is based at Γ. In such a case we may use the notation {d1,d2,...,d2r−1,d2r} for
the vertex labels. Additionally, [e ] denotes [D (e ),D (e )] = [d ,d ] for each edge e ∈E(Γ).
i 0 i 0 i i i i
To resolve a reoccurring point of confusion, we remind the reader that the Whitehead graphs used
here (defined in [HM11]) differ from Whitehead graphs mentioned elsewhere in the literature. In general,
Whitehead graphs record at turns taken by immersions of 1-manifolds into graphs. In our case, the 1-
manifold is a set of lines, the attracting lamination. In much of the literature the 1-manifolds are circuits
representing conjugacy classes of free group elements. For example, for the Whitehead graphs of [CV86],
edge images are viewed as cyclic words. This is not true of ours.
5
The invariance of the ideal Whitehead graph is explained in [Pfa12a], as is its connection to the
expanding lamination for a fully irreducible outer automorphism.
2.3. Ideal decompositions.
Recall that tt’s satisfying (I)-(II) of [Pfa12c] Proposition 3.3 (see the introduction) are called ideally
decomposable (ID) with an ideal decomposition (ID). When we additionally require φ ∈ AFI and
r
IW(φ) ∈ PI , we will say g has type (r;(3 −r)). (By saying g has type (r;(3 −r)), it will be
(r;(3−r)) 2 2
2
implicit that, not only is φ∈ AFI , but φ is ideally decomposed, or at least ideally decomposable.)
r
Again we denote ek−1,jk by epku−1, denote ek,jk by euk, denote ek,ik by eak, and denote ek−1,ik−1 by epka−1.
D will denote the set of directions at the vertex of Γ and E := E(Γ ). Further recall [Pfa12c] that, for
k k k k
a (r;(3/2−r)) tt g: Γ → Γ, G(g) is an (r;(3/2−r)) ltt structure with base Γ. We denote the ltt structure
G(f ) by G where f := g ◦···◦g ◦g ◦···◦g : Γ → Γ ,
k k k k 1 n k+1 k k
g := gk ◦···◦gi: Γi−1 → Γk if k > i and
k,i
(cid:26)
g ◦···◦g ◦g ◦···◦g if k < i.
k 1 n i
C(G ) will denote the subgraph of G , from LW(f ), containing all colored (red and purple) edges of G .
k k k k
Sometimes PI(G ) will be used to denote the purple subgraph of G from SW(f ).
k k k
In [Pfa12c] we proved D (eu) = du, D (ea) = da, D (epu ) = dpu , and D (epa ) = dpa . As
0 k k 0 k k 0 k−1 k−1 0 k−1 k−1
described in [Pfa12c], for any k,l, there exists a direction map Dg , induced turn map Dgt , and
k,l k,l
induced ltt structure map DgkT,l: Gl−1 7→ Gk. The restriction of DgkT,l to C(Gl−1) is denoted DgkC,l.
2.4. Extensions and switches.
As in [Pfa12c], a triple (gk,Gk−1,Gk) is an ordered set of three objects where gk: Γk−1 → Γk is a
proper full fold of roses and, for i = k −1,k, G is an ltt structure with base Γ . Recall from [Pfa12c]
i i
that each triple (gk,Gk−1,Gk) in an ideal decomposition of a representative of (r;(3/2−r)) type is either
a “switch” or an “extension.”
A generating triple (GT) is a triple (gk,Gk−1,Gk) where
(gtI) gk :Γk−1 → Γk is a proper full fold of edge-indexed roses defined by
a. gk(ek−1,jk)= ek,ikek,jk where dak = D0(ek,ik), duk = D0(ek,jk), and ek,ik 6= (ek,jk)±1 and
b. gk(ek−1,t)= ek,t for all ek−1,t 6= (ek,jk)±1;
(gtII) G is an indexed pair-labeled (r;(3 −r)) ltt structure with base Γ for i = k−1,k; and
i 2 i
(gtIII) The induced map of based ltt structures DT(gk) :Gk−1 → Gk exists and, in particular, restricts
to an isomorphism from PI(Gk−1) to PI(Gk).
Thetriplewillbecalledadmissible ifGk andGk−1 arebothbirecurrent(andthusareactually indexed
(edge-pair)-labeled (r;(32 −r)) admissible ltt structures) and if either duk−1 = dk−1,jk or duk−1 = dk−1,ik.
In this case g will also be considered admissible.
k
We call Gk−1 the source ltt structure and Gk the destination ltt structure. We sometimes write
gk: epku−1 7→ eakeuk for gk, write dpku−1 for dk−1,jk, and write ek−1,ik for epka−1. If Gk and Gk−1 are indexed
pair-labeled (r;(3/2−r)) ltt structures for G, then (gk,Gk−1,Gk) will be a GT for G.
For a purple edge [dak,dk,l] in Gk, the extension determined by [dak,dk,l], is the GT (gk,Gk−1,Gk) for
G satisfying:
(extI): Therestriction ofDT(gk)toPI(Gk−1)isdefinedbysending,foreach j, thevertex labeleddk−1,j
to the vertex labeled d and extending linearly over edges.
k,j
(extII): duk−1 = dpku−1, i.e. dpku−1 = dk−1,jk labels the single red vertex in Gk−1.
(extIII): dak−1 = dk−1,l.
6
dku-1=dk-1,jk=dkp-u1 epu Gk-1 e gk e e duk=dk,jk eku Gk
er k-1 k-1,jk k,ik k,jk er
k-1 da epu eaeu k
k-1 k-1 k k
ea
dk-1,l k dka=dk,i dk,l
k
The switch determined by a purple edge [dak,d(k,l)] in Gk is the GT (gk,Gk−1,Gk) for G where:
(swI): DT(gk) restricts to an isomorphism from PI(Gk−1) to PI(Gk) defined by
PI(Gk−1) −d−pk−u−−17→−−dak−=−d−k−,i→k PI(Gk)
pu
(dk−1,t 7→ dk,t for dk−1,t 6= dk−1) and extended linearly over edges.
(swII): dpa = du .
k−1 k−1
(swIII): dak−1 = dk−1,l.
dkp-u1=dk-1,j Gk-1 gk dku=dk,j eu Gk
k e e e k k
k-1,jk k,ik k,jk
a
dk-1 epu eaeu
epu k-1 k k
dku-1=dkk-1-1,ik=dkp-a1 dk-1,l eka-1 eak dak=dk,ik dk,l
Recall from [Pfa12c] that admissible switches and extensions satisfy the “admissible map properties”
AM I-VII of [Pfa12c] and that the converse holds by [Pfa12c] Proposition 7.8. These facts motivated
our defining “ideal decomposition diagrams” (see the final subsection of this section) in [Pfa12c].
2.5. Construction paths.
As in [Pfa12b], to ensure the entire ideal Whitehead graphs are realized, we use “building block”
compositions of extensions, “construction compositions:”
Definition 2.1. A preadmissible composition (gi−k,...,gi,Gi−k−1,...,Gi) for a G ∈ PI(r;(3−r)) is a
2
sequence of proper full folds of (edge-pair)-indexed roses, Γi−k−1 −g−i−−→k Γi−k··· −g−i−−→1 Γi−1 −g→i Γi, with
associated sequence of (r;(32 −r)) ltt structures for G, Gi−k−1 −D−T−(−g−i−−k→) Gi−k −D−T−(−gi−−−k+−1→) ··· −D−T−(−g−i−−1→)
Gi−1 −D−T−(−g→i) Gi, where, for each i−k−1 ≤ j < i, (gj+1,Gj,Gj+1) is an extension or switch for G.
The Definition 2.1 notation is standard. A composition is admissible if each Gj is. We call gi,i−k the
associated automorphism, Gi−k−1 the source ltt structure, and Gk the destination ltt structure.
If each (gj,Gj−1,Gj) with i −k < j ≤ i is an admissible extension and (gi−k,Gi−k−1,Gi−k) is an
admissible switch, then we call (gi−k,...,gi;Gi−k−1,...,Gi) an admissible construction composition for
G. We call gi,i−k a construction automorphism. Leaving out the switch, gives a purified construction
automorphism gp = gi ◦···◦gi−k+1 and purified construction composition (gi−k+1,...,gi;Gi−k,...,Gi).
Recall from [Pfa12b]:
Lemma. Let (g1,...,gn,G0,...,Gn) be an ID for a G ∈ PI(r;(3−r)) and (gi−k,...,gi;Gi−k−1,...,Gi) a
2
construction composition. Then [dui,dai,dai,di,dai−1,di−1,...,dai−k+1,di−k+1,dai−k]= [dui,dai,dai,dai−1,
...,da ,da ] is a smooth path in the ltt structure G .
i−k i−k i
7
We called the path in the lemma the construction path for (gi−k,...,gi;Gi−k−1,...,Gi) and denoted
it γ .
gi,i−k
We now remind the reader of the definition of a “potential construction path,” for which we must
define a “construction subgraph.”
LetGbeanadmissible(r;(3−r))lttstructurewiththestandardnotation. Theconstruction subgraph
2
G is constructed from G via the following procedure:
C
1. Remove theinterior of theblack edge [eu], the purplevertex du, and theinterior of any purpleedges
containing the vertex du. Call the graph with these edges and vertices removed G1.
2. Given Gj−1, recursively define Gj: Let {αj−1,i} be the set of vertices in Gj−1 not contained in any
colored edge of Gj−1. Gj is obtained from Gj−1 by removing all black edges containing a vertex
αj−1,i ∈ {αj−1,i}, as well as the interior of each purple edge containing a vertex αj−1,i.
3. G = ∩Gj.
C
j
A potential construction path in G of an admissible (r;(3 −r)) ltt structure G is a smooth oriented
C 2
path [du,da,da,x2,x2,...,xn−1,xn,xn] in G that:
1. starts with the red edge of G (oriented from du to da);
2. is entirely contained in G after the initial red edge and subsequent black edge;
C
3. and satisfies the following: Each G is an ltt structure (and, in particular, is birecurrent), where G
t t
is obtained from G by moving the red edge of G to be attached at x .
t
Example 2.2. A potential composition construction path is given by the numbered colored edges and
black edges between in:
a
0
b 2 c
4
5 3
b 1 c
2.6. Switch paths and sequences.
Switch sequences and switch paths were introduced in [Pfa12b] as useful tools in representative con-
struction.
Definition 2.3. An admissible switch sequence for a (r;(3 −r)) graph G is an admissible composition
2
(gi−k,...,gi;Gi−k−1,...,Gi) for G such that
(SS1) each (gj,Gj−1,Gj) with i−k ≤ j ≤ i is a switch and
(SS2) da = du 6= du = da and da 6= du = da for all i ≥ n > l ≥ i−k.
n+1 n l l+1 l n n+1
We call the associated automorphism gi,i−k = gi ◦···◦gi−k a switch sequence automorphism.
Definition 2.4. Let (gj,...,gk;Gj−1,...,Gk) be an admissible switch sequence. Its switch path is a
path in the destination ltt structure G traversing the red edge [du,da] from its red vertex du to da, the
k k k k k
black edge [dak,dak] from dak to dak, what is the red edge [duk−1,dak−1] = [dak,dak−1] in Gk−1 (purple edge in
G ) from da = du to da , the black edge [da ,da ] from da to da , continues as such through
k k k−1 k−1 k−1 k−1 k−1 k−1
the red edges for the G with j ≤ i≤ k (inserting black edges between), and ends by traversing the black
i
edge [da ,da ] from da to da , what is the red edge [du,da] = [da ,da] in G (purple edge in G ),
j+1 j+1 j+1 j+1 j j j+1 j j k
and then the black edge [da,da] from da to da. In other words, a switch path alternates between the red
j j j j
edges (oriented from du to da) for the G (for descending j) and the black edges between.
j j j
8
The following lemma was proved in [Pfa12b] and shows that switch paths are indeed smooth paths
in destination LTT structures. It is important to note that this only holds when (SS1) and (SS2) hold.
Lemma. Let (g1,...,gn,G0,...,Gn) be an ID for a G ∈ PI(r;(3−r)) and (gi−k,...,gi;Gi−k−1,...,Gi) a
2
switch sequence. Then the associated switch path forms a smooth path in the ltt structure G .
k
Example 2.5. In the ltt structure G , we number the colored edges of a switch path:
i
a a
b0 2 c
1
b c
The switch sequence constructed from the switch path is:
G G G
k-2 k-1 k
c c b bc b b a ab a a
a b g c a g b c
k-1 k
a b c a b c
The red edge erk in Gk is (0), the red edge erk−1 in Gk−1 is (1), and the red edge erk−2 in Gk−2 is (2).
2.7. Ideal decomposition (ID) diagrams.
As in [Pfa12c] and [Pfa12b], we use in this paper the fact that type (r;(3 −r)) representatives can
2
be realized as loops in“ideal decomposition diagrams.”
Recall from [Pfa12c] that, for a G ∈ PI , the preliminary ideal decomposition diagram for G is
(r;(3−r))
2
the directed graph where:
1. the nodes correspond to equivalence classes of admissible indexed pair-labeled (r;(3 −r)) ltt struc-
2
tures for G and
2. for each equivalence class of an admissible GT (gi, Gi−1, Gi) for G, there exists a directed edge
E(gi,Gi−1,Gi) from the node [Gi−1] to the node [Gi].
We called the disjoint union of the maximal strongly connected subgraphs of the preliminary ideal de-
composition diagram for G the ideal decomposition diagram for G (or ID(G)). [Pfa12a] gives a procedure
for constructing ID diagrams (there called “AM Diagrams”).
We use the following to prove that, for a G ∈ PI , representatives form loops in ID(G),
(r;(3/2−r))
satisfying certain properties, are indeed tt representatives of φ∈ AFI with IW(φ) ∼= G.
r
Lemma. ([Pfa12b], Lemma 4.2) Suppose G ∈ PI(r;(3/2−r)) and L(g1,...,gk;G0,G1...,Gk−1,Gk) =
E(g1,G0,G1)∗···∗E(gk,Gk−1,Gk) is a loop in ID(G) satisfying:
1. Each purple edge of G(g) corresponds to a turn taken by some gk(E ) where E ∈ E(Γ);
j j
2. for each 1 ≤ i,j ≤ q, there exists some k ≥ 1 such that gk(E ) contains either E or E¯ ; and
j i i
3. g has no periodic Nielsen paths.
Then g: Γ → Γ is a train track representative of an ageometric φ ∈ FI such that IW(φ) = G.
r
3 Representative construction strategies
We describe here three categories of strategies for constructing (r;(3 −r)) tt representatives. Different
2
strategies work better in different circumstances. For example, if most ltt structures G with PI(G) = G
are birecurrent, then category IIand IIIstrategies are better suited (ID(G) may belarge andimpractical
to construct). On the other hand, if only a few ltt structures G with PI(G) = G are birecurrent, then
9
constructing ID(G) is simpler than using “guess and check” strategies, so category I strategies are often
more practical.
Before actually describing the strategies, we establish in Subsection 3.1 some additional terminology,
proveausefulfact(Lemma3.1),andgive(inExample3.5)amethodusedrepeatedlyforcheckingwhether
the entire ideal Whitehead graph is “achieved.”
3.1 Preliminary definitions and tools
Thefollowinglemmagives apreconditionfora(r;(3−r))lttstructuretobebirecurrent(thusadmissible).
2
A valence-1 edge will mean an edge with a valence-1 vertex. A G ∈ PI will be called edge-pair
(r;(3−r))
2
(index)-labeled if its vertices are labeled by a 2r − 1 element subset of the rank r (indexed) edge pair
labeling set.
Lemma 3.1. If a (r;(3 −r)) ltt structure G is birecurrent, then C(G) can have at most one valence-1
2
edge-pair-(labeled) edge [x ,x ].
i i
Proof. Suppose, for contradiction’s sake, G is a birecurrent with some birecurrent line l. Without gener-
ality loss assume x had valence-1 in C(G). Since l must be birecurrent with both orientations, we can
i
focus on the situation where l traverses [x ,x ] oriented from x to x . Since l is smooth, l must traverse a
i i i i
black edge after [x ,x ]. But the only black edge at x is [x ,x ]. To be smooth, after traversing the black
i i i i i
edge [x ,x ], it must traverse a colored edge containing x . Since x had valence-1 in C(G), this would
i i i i
imply l would traverse [x ,x ] again. Inductively, one sees that l will get caught in this loop formed by
i i
the colored edge [x ,x ] and black edge [x ,x ], never again traversing a colored edge other than [x ,x ]
i i i i i i
as it heads toward this end, violating birecurrency.
Definition 3.2. Motivated by the above lemma, the edge-pair labeling will be considered preadmissible
if G contains no more than one (edge-pair)-labeled edge.
Remark 3.3. IftheLemma3.1condition is violated, regardless oftherededge, theC(G) forany (r;(3−
2
r)) ltt structure G with PI(G) ∼= G, has at least one (edge-pair)-labeled edge, violating birecurrency.
The different strategies we describe here frequently requirewe track our progress in ensuringall edges
of G are actually in the ideal Whitehead graph for a given representative gG. In this section we give
methods and terminology we use for this purpose. We start by establishing the notion of a “preimage
subgraph:”
Definition 3.4. For an admissible map (g(k,m); Gm−1, ..., Gk), the preimage subgraph under (g(k,m);
Gm−1, ..., Gk) for a subgraph H ⊂ PI(Gi) will be denoted H−gk,m. It is obtained from H by replacing
each edge of H with its preimage under the isomorphism from PI(Gm−1) to PI(Gk).
Example 3.5. Consider a subgraph H of an ltt structure G :
i
a
b b c
On the left we show the preimage subgraph H−gi under the direction map Dg :a¯ 7→¯b for g : a¯ 7→¯ba¯:
i i
a a b a
b a c b b c
10