End Invariants for $SL(2,C)$ characters of the one-holed torus

a r

X i

v

:m

a

t h /

5

1

1

6

2

1

v

1

[

m

a

t h

.

G

T

]

2

5

N o

v

2

5

END INV ARIANTS FOR SL(2,C )CHARACTERS OF THE ONE-HOLED TORUS SER PEOW TAN,YAN LOI WONG,AND YING ZHANG Abstract.We de?ne and study the set E (ρ)of end invariants of a SL(2,C )character ρof the one-holed torus T .We show that the set E (ρ)is the entire projective lamination space PL of T if and only if (i)ρcorresponds to the dihedral representation,or (ii)ρis real and corresponds to a SU(2)represen-tation;and that otherwise,E (ρ)is closed and has empty interior in PL .For real characters ρ,we give a complete classi?cation of E (ρ),and show that E (ρ)has either 0,1or in?nitely many elements,and in the last case,E (ρ)is either a Cantor subset of PL or is PL itself.We also give a similar classi?cation for “imaginary”characters where the trace of the commutator is less than 2.Finally,we show that for discrete characters (not corresponding to dihedral or SU(2)representations),E (ρ)is a Cantor subset of PL if it contains at least three elements.1.Introduction and statement of results Let T be the one-holed torus,and πits fundamental group which is free on two generators X,Y .The SL(2,C )character variety of T is the set X of equivalence classes of representations ρ:π→SL(2,C ),where the equivalence classes are ob-tained by taking the closure of the orbits under conjugation by SL(2,C ).In this paper we de?ne and study the set of end invariants associated to the SL(2,C )char-acters of the one-holed torus.To simplify the exposition,by abuse of notation,we use ρinstead of [ρ]to denote the characters in X in the rest of the paper,there should be no confusion,as we will be mostly interested in the trace function which is invariant under conjugation.Let PL be the projective lamination space of T and C ?PL the set of (free homotopy classes of)essential simple closed curves on T .De?nition 1.1.(End invariants)An element X ∈PL is an end invariant of the character ρif there exists K >0and a sequence of distinct elements X n

∈C such

that X n →X and |tr ρ(X n )|

2SER PEOW TAN,YAN LOI WONG,AND YING ZHANG

classify the possible structure of E(ρ)for reducible,real,imaginary and discrete

characters(Theorems1.4,1.5,1.6and1.7).

The set E(ρ)gives information about the dynamics of the action of the mapping class groupΓof T on the characterρ,and is closely related to the study of Kleinian groups,dynamical systems,see for example[5]or[8],and also certain problems in mathematical physics,see[8].

The character variety X strati?es into relative character varieties:forκ∈C,the κ-relative character variety is the set of equivalence classesρsuch that

trρ(XY X?1Y?1)=κ

for one(and hence any)pair of generators X,Y ofπ.Denote by Xκtheκ-relative character variety.By classical results of Fricke(see for example[4]or[11]),we have the following identi?cations:

X～=C3,Xκ～={(x,y,z)∈C3|x2+y2+z2?xyz?2=κ},

with the identi?cation given by

ι:ρ→(x,y,z):=(trρ(X),trρ(Y),trρ(XY)),

where X,Y is a?xed pair of generators ofπ.The topology on X and Xκwill be

that induced by the above identi?cations.

√

A characterρ∈Xκsuch thatι(ρ)is a permutation of the triple(0,0,±

END INV ARIANTS FOR SL(2,C)CHARACTERS3

C are connected by an edge if and only if X,Y have geometric intersection number one in T.C(T)can be realized as the completion of the Farey triangulation F of the hyperbolic plane H2.In this way,C is naturally identi?ed with?Q:=Q∪{∞}, and the projective lamination space PL of T is identi?ed with the projective real line?R:=R∪{∞},the boundary of the hyperbolic plane H2.The mapping class groupΓacts on these sets and C(T)in a natural way,this action is realized via the isomorphism ofΓwith SL(2,Z),which acts on the upper half-plane as a model of H2.

We now give the exact statements of our results.The?rst result describes all charactersρfor which E(ρ)=PL,and shows that otherwise,E(ρ)has empty interior.

Theorem 1.2.The set of end invariants E(ρ)is equal to PL if and only if (i)ρis dihedral;or(ii)ρcorresponds to a SU(2)representation.Furthermore,if E(ρ)=PL,then E(ρ)has empty interior in PL.

The above can be thought of as the opposite extreme of the following theorem, characterizing the characters for which E(ρ)is empty,which is a consequence of results in[1](Theorem2),[11](Theorem2.3,Proposition2.4)and[10](Theorem 1.6);we will give a sketch of the proof in§4.

Theorem1.3.(Bowditch,Tan-Wong-Zhang)The set of end invariants E(ρ)is empty if and only ifρsatis?es

(i)trρ(X)∈(?2,2)for all X∈C;

(ii)|trρ(X)|≤2for only?nitely many(possibly no)X∈C.

We call conditions(i)and(ii)in Theorem1.3the extended BQ-conditions.

The reducible characters(κ=2)are somewhat special;the following result classi?es E(ρ)for such characters.

Theorem1.4.(End invariants for reducible characters)Forρ∈X2,E(ρ)={X0} or PL.Furthermore,in the?rst case,if X0∈C,then trρ(X0)∈[?2,2]and trρ(X)∈[?2,2]for all X∈C\{X0},while if X0∈C,then trρ(X)∈[?2,2]for all X∈C;and in the second case,trρ(X)∈[?2,2]for all X∈C.

Note that in particular,E(ρ)is never empty in this case,so that a reducible character never satis?es the extended BQ-conditions.

Denote by X R and X Rκthe real SL(2)character variety and relative character varieties respectively.We have the following classi?cation of E(ρ)forρ∈X Rκ, together with the description of the correspondingρ;we exclude the caseκ=2 which was covered in the preceding theorem.

Theorem1.5.(End invariants for real characters)Supposeρ∈X Rκ,withκ=2. Then exactly one of the following must hold:

(a)E(ρ)=?,andρsatis?es the extended BQ-conditions.

(b)E(ρ)={?X}where?X∈C,ρis a SL(2,R)representation,trρ(?X)∈(?2,2),

and trρ(X)∈(?2,2)for all X∈C\{?X}.

(c)E(ρ)is a Cantor subset of PL,ρis a SL(2,R)representation,trρ(X)∈

√(?2,2)for at least two distinct X∈C,and trρ(Y)∈(?2,2)∪{±

4SER PEOW TAN,YAN LOI WONG,AND YING ZHANG

Furthermore,case(a)occurs only whenκ∈(?∞,2)∪[18,∞);case(b)when κ∈[6,∞);case(c)whenκ∈(2,∞);and case(d)whenκ∈[?2,2)∪(2,∞].

Forρ∈X R,Theorem1.5implies that if E(ρ)has more than one element,then E(ρ)is either a Cantor set or all of PL.Furthermore,if E(ρ)has only one element X(andκ=2),then X is rational,that is,corresponds to a simple closed curve. These results are not true for general complex characters,for example,punctured torus groups with two geometrically in?nite ends have two end invariants,and those with one geometrically in?nite end have an end invariant which is irrational.Note also that condition(ii)of the extended BQ-conditions follows from condition(i) in the case of real characters withκ=2;this will follow from the proof of the theorem.

There is also an intriguing connection between the end invariants of real char-acters and the dynamical spectrum of the almost periodic Schr¨o dinger operator; see[1]and[8]for details.Roughly speaking,consider a one(real)parameter family of charactersρE∈X Rκ(parametrized by the energy E∈R)such that ι(ρE)=(2,E?V0,E?V1),where V0=V1are?xed constants andκ=2+(V0?V1)2; and an irrational elementλ∈PL\C.Then the set of values of E for which λ∈E(ρE)corresponds to the dynamical spectrum of a discrete almost periodic Schr¨o dinger operator,and the conjecture is that this set is always a Cantor set.

Denote by X I and X Iκthe imaginary character variety and relative character va-rieties respectively.Recall that dihedral characters are not in X I by our convention. Note thatκ∈R in this case.We classify E(ρ)forκ<2:

Theorem1.6.(End invariants for imaginary characters)

(i)κ=?2:Forρ∈X I?2,E(ρ)is either a Cantor subset of PL,or consists of

a single element X in C.In the latter case,trρ(X)=0andρis equivalent

under the action of the modular groupΓto a character corresponding to the triple(0,x,ix)where x∈R satis?es|x|≥2.

(ii)?14<κ<2:Forρ∈X Iκ,E(ρ)is either a Cantor subset of PL,or consists of a single element X in C.

(iii)κ≤?14:Forρ∈X Iκ,E(ρ)is a Cantor subset of PL;consists of a single element X in C;or is empty.

We chose in the statement of Theorem1.6above to emphasize the caseκ=?2 since the results are somewhat sharper than for general?14≤κ<2,and the case is itself of independent interest.

Our?nal result is for discrete characters.We say a characterρis discrete if the set of values{trρ(X)|X∈C}is a discrete subset of C.Denote the set of discrete characters by X disc.We have:

Theorem1.7.Forρ∈X disc,if E(ρ)has at least three elements and E(ρ)=PL, then E(ρ)is a Cantor set.

The above can be rephrased as follows:For discrete characters not corresponding to dihedral or SU(2)representations,E(ρ)is a Cantor subset of PL if it contains at least three elements.Note that a discrete character may also have0,1or2 elements in E(ρ).

The de?nition of an end invariant given generalizes the de?nition of a(geomet-rically in?nite,or degenerate)end for the case where the representation is type-preserving(κ=?2),and discrete and faithful,which is the subject of intensive

END INV ARIANTS FOR SL(2,C)CHARACTERS5 study in the last decade,especially in relation to Thurston’s Ending Lamination Conjecture,proven by Minsky(for the punctured torus)in[7].Our de?nition is motivated by that given by Bowditch in[1],where end invariants were de?ned for type-preserving but not necessarily discrete or faithful representations.In fact,this work was very much inspired by[1]and grew out of our attempt to study and develop the thread of ideas presented in§5of[1].

Note however that our de?nition di?ers slightly from that used in[1];the dif-ference is that accidental parabolics were(isolated)end invariants there,whereas they are not in ours.This slight variation simpli?es the statements of the results above.It also allows us to state the following conjecture,of which the preceding results can be regarded as supporting evidence.

Conjecture1.8.(The dendrite conjecture)Suppose that E(ρ)has more than two elements.Then either E(ρ)=PL or E(ρ)is a Cantor subset of PL.

The above conjecture is a re?nement and generalization of the suggestion by Bowditch in[1]that for a genericρ∈X?2not satisfying the BQ-conditions,E(ρ) should be a Cantor set.The“convex hull”of E(ρ)is a subtree of the dual treeΣof C(T);the above conjecture says that this tree should look like a dendrite,in the sense that if it has more than two ends,than there should be in?nite branching at any end and all ends are not isolated.The statement would have been somewhat more complicated,with several exceptions,if accidental parabolics are considered to be end invariants.

Note that in the cases considered by Bowditch,he did not really have to worry about the case E(ρ)=PL since forκ=?2this only occurs for the trivial dihedral character corresponding to the quaternionic representationρwithι([ρ])=(0,0,0).

Bowditch did not produce examples of characters for which E(ρ)was a Cantor set.Theorem1.5produces many such examples for real characters withκ>2and Theorems1.6and1.7produces many non-real examples for generalκ,for example the characterρ∈X?2withι(ρ)=(0,1,i)has E(ρ)a Cantor set by either theorem. To the best of our knowledge,these are the?rst examples for which E(ρ)is known to be a Cantor set.

There is also a generalization of the Ending Lamination Conjecture for SL(2,C) characters(as Bowditch conjectured for theκ=?2case)which can be stated as follows:

Conjecture1.9.Suppose thatρ,ρ′∈Xκare such that E(ρ)=E(ρ′),E(ρ)has at least two elements,and E(ρ)=PL.Thenρ=ρ′.

We end this introduction with a few words about the generalizations to arbitrary surfaces.The de?nition of E(ρ)can be extended without much di?culty.The case of the four-holed sphere is similar and the techniques given here should give similar results in that case,although the analysis is generally more di?cult.In other cases, PL is homeomorphic to the sphere S n for some n≥2and a possible generalization of Theorem1.2is that E(ρ)has either full measure or measure zero.A possible generalization of Conjecture1.8would be that E(ρ)is perfect,if it contains more than two elements.However,these are just speculations and we do not have any insights into these more general cases.

The rest of the paper is organized as follows.In§2we give the notation and basic de?nitions to be used in the rest of the paper.In§3we state three key lemmas

6SER PEOW TAN,YAN LOI WONG,AND YING ZHANG

used for the proofs of the theorems.In§4,we prove Theorems1.2and1.3.In§5 we consider reducible characters and prove Theorem1.4.In§6we prove Theorems 1.5and1.7and in§7,we prove Theorem1.6.Finally,in the Appendix,we give a brief description of theτ-reduction algorithm of Goldman-Stantchev in[5]for the imaginary characters,which generalizes that used by Bowditch in[1]and which is used in a crucial way in the proof of Theorem1.6.

Acknowledgements.Part of this work was carried out while the?rst named author was visiting the University of Maryland,College Park,the University of Warwick, and the Tokyo Institute of Technology,he would like to thank his hosts Bill Gold-man,Caroline Series and Sadayoshi Kojima and these institutions for their hospi-tality.He would also like to thank Rich Schwartz,Rich Brown,George Stantchev, Javier Aramayona,John Parker,Juan Souto,Brian Bowditch and especially Bill Goldman,Caroline Series,Makoto Sakuma and Greg McShane for many stimu-lating and useful conversations.He would also like to thank Shigeru Mizushima for help with a computer program to help visualize the characters and their end invariants.

2.Notation and definitions

In this section we introduce the notation and de?nitions to be used in the rest of this paper.As in the introduction,let T denote the one-holed torus,that is,a torus with an open disk removed.Its fundamental groupπis freely generated by two elements X,Y corresponding to two simple closed curves on T with intersection number one.

2.1.The(relative)character variety X(resp.Xκ).The SL(2,C)character variety is the set

X=Hom(π,SL(2,C))//SL(2,C)

where the quotient is the geometric invariant theory quotient by the conjugation action.By abuse of notation,denote byρ(instead of[ρ])the elements of X;we call them characters of T.Forκ∈C,theκ-relative character variety is the subset

Xκ={ρ∈X|trρ(XY X?1Y?1)=κ},

where X,Y are generators ofπ.By results of Fricke,it does not matter which pair of generators X,Y are used to de?neκ.We have

X～=C3,Xκ～={(x,y,z)∈C3|x2+y2+z2?xyz?2=κ};(1) the identi?cation is given by

ι:ρ→(x,y,z):=(trρ(X),trρ(Y),trρ(XY)),

where X,Y is a?xed pair of free generators ofπ.Conversely,ι?1(x,y,z)can be realized by the following representation(see[3]):

ρ(X)=A:= x1?10 ,ρ(Y)=B:= 0?ζ

ζ?1y ,

whereζ+ζ?1=z.

2.2.Topology of the(relative)character variety.The topology on X and Xκwill be that induced by the identi?cations de?ned in(1)respectively.

END INV ARIANTS FOR SL(2,C)CHARACTERS7

2.3.Action of the mapping class groupΓ.The mapping class group

Γ:=π0(Homeo+(T))～=SL(2,Z)

acts onπand hence on X;the action is given by

φ(ρ)=ρ?φ?1,

whereφ∈Γandρ∈X.The action is not e?ective,the kernel is generated by the elliptic involution corresponding to?I∈SL(2,Z),so the e?ective action is by SL(2,Z)/±I=PSL(2,Z).The quantityκ=trρ(XY X?1Y?1)is preserved under the action ofΓby results of Nielsen(see[3]);henceΓalso acts on the relative varieties Xκ.With the identi?cation of X and Xκwith the complex varieties in(1), the action ofΓis realized via polynomial maps on these varieties;it is generated by the cyclic permutation

c:(x,y,z)→(z,x,y)(2) and the involution

s:(x,y,z)→(y,x,xy?z)(3) corresponding to 1?110 , 01?10 ∈PSL(2,Z)respectively.

2.4.Sign change automorphisms.There is a Z/2×Z/2action on X(resp. Xκ)generated by simultaneously changing the signs of two of the entries ofι(ρ)= (x,y,z).Two characters are equivalent under this action if and only if they cor-respond to lifts of the same representation ofπinto PSL(2,C).The large scale behavior of the action ofΓon X and the end invariants ofρare not a?ected by these sign change automorphisms;nonetheless,it will be convenient to use them for some local trace reduction arguments later.

2.5.The pants graph C(T)and the projective lamination space PL.Let

C denote the set of free homotopy classes of essential(nontrivial,non-boundary) simple closed curves on T,which can be regarded as the set of vertices of the pants graph C(T)of T,where two vertices are connected by an edge if and only if the corresponding curves have geometric intersection number one.The pants graph C(T)can be concretely realized as the completionˉF of the Farey tessellation F of the upper half-plane H2in H2∪?R(see Figure1);recall that F is the tessellation of H2by ideal triangles where the edges are the translates of the in?nite geodesic (0,∞)by PSL(2,Z).Note that C(T)is not locally?nite,and every vertex has in?nite degree.The space PL of projective laminations on T can be regarded as the completion of C;in this way,PL can be identi?ed with?R,where the irrational points of R correspond to projective laminations which are not closed, and C is identi?ed with?Q,once we?x an identi?cation of X,Y and XY with0,∞and1,where X,Y are a?xed pair of generators ofπ.There is a natural topology on PL induced from the topology of?R which agrees with the usual topology on PL regarded as the completion of C.There is also a natural orientation induced on PL from this identi?cation,where we use the usual orientation of?R.It is also convenient to use the conformal unit disk model of the hyperbolic plane to visualize all this;in this way,C(T)is a triangulation of the unit disk D,and PL is identi?ed with the unit circle S1～=?R,where the anti-clockwise direction is positive.We use upper case letters to denote elements of C,and more generally,elements of PL. Occasionally,we will also useλto denote an element of PL\C.

8SER PEOW TAN,YAN LOI WONG,AND YING ZHANG

2.6.The dual trivalent treeΣ.The dual graph of F(or C(T))is an in?nite trivalent treeΣ.Geometrically,we may choose the vertices ofΣas the incenters of the ideal triangles in the tessellation F and the edges as the geodesic arcs connecting the incenters of pairs of adjacent ideal triangles in F,where two ideal triangles are said to be adjacent if they share a common side.Σis trivalent since for each ideal triangle in F there are exactly three others adjacent to it;see Figure1.Note thatΣis properly imbedded in the hyperbolic plane and all its ends form the whole ideal boundary of the hyperbolic plane.Denote by V(Σ),E(Σ)the sets of vertices and edges ofΣrespectively,and we use the notation v,e to represent elements of V(Σ) and E(Σ)respectively.A complementary region ofΣis the closure of a connected component of the complement ofΣin H2;we denote the set of complementary regions ofΣby?(Σ).

2.7.Generating pairs,triples and quadruples.For X,Y,Z,Z′∈C:

?The(unordered)pair(X,Y),is a generating pair if X and Y are connected by an edge of C(T).We say that X and Y are neighbors(in C(T));

?The(unordered)triple(X,Y,Z)is a generating triple if X,Y and Z are the vertices of a triangle in C(T);and

?(X,Y;Z,Z′)(where each of the?rst and second pair in the quadruple is un-ordered)is a generating quadruple if(X,Y,Z)and(X,Y,Z′)are generating triples(note that the pair Z,Z′is determined uniquely by X,Y).

We shall denote the sets of generating pairs,triples and quadruples by GP,GT and GQ respectively.

2.8.Correspondences between various sets of objects.There are natural correspondences between the following sets,which are self-evident(see Figure2): V(Σ)←→GT;v→(X,Y,Z)(4)

E(Σ)←→GP←→GQ;e?(X,Y)?(X,Y;Z,Z′)(5)

?(Σ)←→C←→?Q.(6) In the second correspondence,(X,Y)de?nes an edge in F which is dual to e.In the last correspondence,we will use the same letters X,Y,Z to denote the elements of all three sets,namely,?(Σ),C and?Q.Indeed,we shall use the correspondence freely,so that the same symbol X may denote an element of?(Σ),C or?Q;it should be clear from the context which one we mean.We will also use the symbols X(p/q)to indicate that X∈C(or?(Σ))corresponds to p/q∈?Q.

2.9.Directed edges ofΣ.Let E(Σ)denote the set of directed edges ofΣ.Denote by e the elements of E(Σ),where the direction of the arrow goes from the tail to the head.We use the notation e?(X,Y;Z→Z′)to indicate that the directed edge e corresponds to the generating quadruple(X,Y;Z,Z′)with the direction of the arrow pointing from Z towards Z′.We shall also use the orientation convention that X,Y,Z are in clockwise order as points on?R(as the boundary of H2),so that if e?(X,Y;Z→Z′),then? e?(Y,X;Z′→Z)where? e is the directed edge which is directed in the opposite direction of e and has the same underlying undirected edge e.In other words,directed edges correspond to ordered generating pairs,and ordered generating quadruples;see Figure2.

END INV ARIANTS FOR SL(2,C )CHARACTERS 9

.................................................................................................................................................................................

0

113242531

3354757E d d d d d d

X Y Z ′Z e Figure 2.The directed edge e ?(X,Y ;Z →Z ′)

2.10.Tri-coloring of C ,?(Σ)and E (Σ).The set C (or ?(Σ))can be naturally partitioned into three equivalence classes,R ,G and B (we use the colors Red,Green and Blue to denote each class)in such a way that any generating triple (X,Y,Z )contains exactly one element in each class.Similarly,E (Σ)admits a partition into three equivalence classes E r ,E g and E b ,where if e ?(X,Y ),then e ∈E r if X,Y ∈R and so on.

2.11.Subsets of C and PL .Denote by [X,Y ]the set of points in PL in the closed interval from X to Y ,going in the anti-clockwise/positive direction,so for example [X (0),X (1)]consists of all points in PL corresponding to real numbers in the interval [0,1]and [X (1),X (0)]correspond to points in ?R

outside the open interval (0,1).Denote similarly the half open and open subsets of PL by [X,Y ),(X,Y ]and (X,Y )(note that we use the same notation for a generating pair but it should be clear from the context if we mean a generating pair,or the corresponding open interval in PL ).Denote by C [X,Y ](and similarly for the other intervals)the

10SER PEOW TAN,YAN LOI WONG,AND YING ZHANG

intersection C∩[X,Y].To each directed edge e?(X,Y;Z→Z′),we associate the subset of C,called the tail of e de?ned by tail( e):=C[X,Y].Note that this is the intersection of C with the closed interval in PL with end points X and Y which contain Z.

2.12.The Fricke trace map induced by a characterρ.A characterρ∈Xκinduces a map

φ:=φρ:C→C

given byφ(X)=trρ(X).Equivalently,φcan be thought of as a map from?(Σ) to C,from the correspondence between C and?(Σ).We adopt the convention that the corresponding lower case letters denote the values ofφ,that isφ(X)=x,φ(Y)=y,φ(Z)=z and so on.This will simplify notation considerably.This map is called a Marko?map by Bowditch in[1]whenκ=?2,and a generalized Marko?map by the authors in[11]for generalκ(where the map was de?ned from?(Σ)to C);we call this the Fricke map or Fricke trace map in this paper.At any rate,φsatis?es the following vertex and edge relations:

x2+y2+z2?xyz?2=κ,(7) for any generating triple(X,Y,Z)∈GT;and

z+z′=xy,(8) for any generating quadruple(X,Y;Z,Z′)∈GQ(recall we are using the corre-sponding lower case letters to denote the values ofφ).

The vertex relation(7)holds for every generating triple if it holds for one by the edge relation(8),that is,it propagates along the treeΣ.These relations arise from the corresponding Fricke trace identities:

(tr A)2+(tr B)2+(tr AB)2?tr A tr B tr AB?2=tr[A,B],(9)

tr AB+tr AB?1=tr A tr B.(10) Note that the sign-change automorphism described in§2.4corresponds to chang-ing the signs ofφ(X)for all X in two of the classes in the tri-coloring of C described in§2.10,while keeping the entries in the third class?xed.

2.1

3.The extended BQ-conditions.Letφbe a Fricke trace map and S?C. We say thatφsatis?es the extended BQ-conditions on S if

(i)φ(X)∈(?2,2)for all X∈S;

(ii)|φ(X)|≤2for only?nitely many(possibly no)X∈S.

If S=C,we say thatφsatis?es the extended BQ-conditions.If C=S1∪···∪S n, then clearlyφsatis?es the extended BQ-conditions if and only if it does on all the S i,i=1,···,n.

√

2.14.Dihedral characters.Note thatι?1(0,0,±

κ+2.We call any character for which two of the entries ofι(ρ) are zero dihedral characters;they are generated by two order two elliptics A and B as above,and contain the cyclic subgroup ζ00ζ?1 as a normal subgroup

END INV ARIANTS FOR SL(2,C)CHARACTERS11 of index two.By(8),ifρis a dihedral character,φ(X)takes only the values0,±√

d),where d<0is a square-free integer.

2.18.The?ow onΣassociated to a characterρ.Associated to a characterρis a?ow on the treeΣ,that is,a map f:=fρ:E(Σ)→ E(Σ)de?ned as follows: For e?(X,Y;Z,Z′),f(e)= e?(X,Y;Z→Z′)if|z|≥|z′|,that is,the?ow goes from the larger absolute value ofφto the smaller one alongΣ.If|z|=|z′|,the

?ow is de?ned arbitrarily for that edge;this ambiguity does not have any serious consequences on the subsequent developments.A vertex v∈V(Σ)is called a sink if f(e)is directed towards v for all the three edges e∈E(Σ)meeting at v.A?nite subtreeΣ0ofΣis called an attractor for the?ow if f(e)is directed towardsΣ0for all e∈Σ\Σ0.Note that a sink v?(X,Y,Z)may or may not be an attractor for the?ow;however,it is if in addition|x|,|y|,|z|>2.

12SER PEOW TAN,YAN LOI WONG,AND YING ZHANG

2.19.End invariants.End invariants for a characterρare de?ned as in De?nition 1.1in the introduction;the set of end invariants is denoted by E(ρ).We shall see in§6that there are other equivalent de?nitions which we will use.The relation between the end invariants and(non-)properness of the action ofΓonρcan be described as follows.X∈PL is an end invariant ofρ∈Xκif and only if there exists a sequence{θn}?Γ～=SL(2,Z)with|trθn|→∞such thatθn(ρ)stays in a?xed compact subset S of Xκ(independent ofρ),and the repelling?xed points μ?n∈PL ofθn approach X as n→∞.

3.Basic results

Much of the ensuing discussion in this paper hinges on three fairly elementary but fundamental results on quasi-convexity(Lemma3.1),escaping orbits(Lemma 3.4)and behavior of neighbors around X(Lemma3.5)which are easy to prove but play key roles in controlling the large scale behavior of the action ofΓon the characterρ,and in the proof of the theorems.They were?rst proved by Bowditch in[1]forκ=?2;the generalization to arbitraryκcan be found in[11].

For the rest of this section,we?x aρ∈Xκand let the corresponding Fricke trace map beφ:=φρ,which we take to be a map from?(Σ)to C.We also assume κ=2.Recall that we adopt the conventionφ(X)=x,φ(Y)=y,and so on.

3.1.Quasi-convexity:Connectedness of Cφ(K)for K≥2.We say that a subset S?C is connected if the subgraph spanned by S in the pants graph C(T) is connected.For K>0,let C(K):=Cφ(K)={X∈C||φ(X)|≤K},and we de?ne?(K)similarly.We then have

Lemma3.1.(Quasi-connectivity)For all K≥2,C(K)(equivalently,?(K))is connected.

Lemma3.1can be deduced easily from the following results.

Proposition3.2.Suppose that K≥2and(X,Y;Z,Z′)is a generating quadruple such that Z,Z′∈C(K).Then either X or Y(or both)is in C(K).

Proof.This follows directly from the edge relation(8).

Proposition3.3.Suppose X,Y,Z∈?(Σ)meet at a vertex v∈V(Σ),and that the arrows on the edges X∩Y and X∩Z arising from the?ow fρboth point away from v.Then either|x|≤2,or y=z=0.

Proof.Let Z′and Y′be the regions opposite to Z and Y respectively,from the vertex v.By the assumption of the direction of the arrow on the edge X∩Y,we have2|z|≥|z|+|z′|≥|z+z′|=|xy|.Similarly,2|y|≥|xz|.Adding,we get 2(|z|+|y|)≥|x|(|y|+|z|)from which the conclusion follows.

Sketch of proof of Lemma3.1.We prove connectedness of?(K).Suppose that this is not connected.Take a minimal path inΣconnecting two of the components. If this path consists of only one edge inΣ,we get a contradiction by Proposition 3.2.If it consists of more than one edge,then by the construction,the?ow at edges on the two ends of the path point outwards,so that there is a vertex inside this path where two of the arrows are pointing outwards,and we get a contradiction by Proposition3.3.

END INV ARIANTS FOR SL(2,C)CHARACTERS13

3.2.Escaping orbits.

Lemma3.4.(Escaping orbits)Suppose that{ e n},n∈N is an in?nite directed path inΣwith the head of e n equal to the tail of e n+1,and such that fρ(e n)= e n.Furthermore,suppose that{ e n}does not limit to a rational point of PL. Then there exists in?nitely many X∈C(2)such that the path{ e n}intersects the boundary of the corresponding complementary regions X∈?(Σ).

Proof.See[1]for the caseκ=?2and[11]for the extension to the general case whereκ=2.

3.3.Neighbors around X.For each X∈C,let Y n,n∈Z be the consecutive neighbors of X,so that(X,Y n,Y n+1)∈GT is a generating triple for all n.For example,if X corresponds to∞∈?Q,then we can take Y n to correspond to n∈Z. Let x=λ+λ?1where|λ|≥1.Note that|λ|=1if and only if x∈[?2,2]?R.If x=2,then from the vertex relation(7)and edge relation(8),y n+1=y n±√

κ?2,but this time,y n+1+y n=?(y n+y n?1),hence the±sign alternates in n.If x=±

√

κ+2}then there are(non-zero)constants A,B∈C\{0}with AB=(x2?κ?2)/(x2?4)such that y n=Aλn+Bλ?n.Hence we deduce that the following holds.(This is Corollary3.3in[1]in the caseκ=?2.)

Lemma3.5.(Neighbors of X)Suppose that X∈C has consecutive neighbors Y n, n∈Z.Letρ∈Xκ,κ=2,with corresponding Fricke trace mapφ.

(a)If x/∈[?2,2]∪{±

√

q πfor some p/q∈Q,and quasi-periodic otherwise,

that is,for any n∈Z and?>0,there exists in?nitely many indices n k with|y n?y n k|

(c)If x=2,then either y n=y0+n

√κ?2 for all n.In particular,since we assume thatκ=2,|y n|grows linearly in |n|.

(d)If x=?2,then either y n=(?1)n(y0+n

√

κ?2)for all n.

(e)If x=±

√

14SER PEOW TAN,YAN LOI WONG,AND YING ZHANG

We shall see later (Proposition 4.3)that if in addition,x,y =±√

κ+2}.

Proof.It is clear from parts (b)and (e)of Lemma 3.5that X ∈E (ρ)if x ∈

(?2,2)∪{±

√κ+2}.Then by parts (a),(c)and (d)of Lemma 3.5,for all K >0,there exists N ∈N such that |y n |>K for all n >N and n 0,by the remark preceding Proposition 4.2,there exists a neighborhood N (X )of X (depending on K )such that |z |>K for all Z ∈N (X )\{X }.We conclude that X ∈E (ρ).

Proposition 4.4.For any character ρ∈X ,the set of end invariants E (ρ)is closed in PL .

Proof.We show that PL \E (ρ)is open.Suppose ?rst that X ∈C is not an end invariant.Then by Proposition 4.3and Lemma 3.5,if Y n are the neighbors of X ,|y n |→∞as n →±∞.Hence,by Propositions 4.2,4.3,there exists an open neighborhood U of X such that U ∩E (ρ)=?.Now suppose λ∈PL \C is not an end invariant.Let {e n }be a path in Σlimiting to λand let (X n ,Y n )∈GP be the generating pair corresponding to the edge e n ,oriented so that λ∈[X n ,Y n ].Since λ∈E (ρ),we have |x n |,|y n |→∞;so we may assume say that |x 1|=K >2,and both |x N |and |y N |>K for some N ∈N .By Lemma 3.1,|z |>K for all Z ∈[X N ,Y N ],so that C [X N ,Y N ]satis?es the extended BQ-conditions.Hence,(X N ,Y N )∩E (ρ)=?by Proposition 4.1.We conclude that PL \E (ρ)is open.Proof of Theorem 1.2.If ρis dihedral,then x ∈{0,±√

κ+2}.Suppose say

that x =±√

κ+2∈[?2,2].Then either y =z =0in which case ρis a dihedral character,or y,z =0,in which case by part (e)of Lemma 3.5and Proposition 4.3,there exists a neighbor Y n of X such that Y n ∈I but Y n ∈E (ρ),which gives a contradiction.Hence,we may suppose that x,y,z =±√κ+2(see for example [3])and so W ∈E (ρ).Since x ∈(?2,2),

END INV ARIANTS FOR SL(2,C)CHARACTERS15 by Lemma3.5(b),the neighboring values around X are either periodic or quasi-periodic;it follows that there exists W n arbitrarily close to X(hence in I)with W n∈E(ρ).The contradiction completes the proof. Proof of Theorem1.3.Suppose thatρsatis?es the extended BQ-conditions.It was shown in[10]that in this case,there exists a?nite subtree ofΣwhich is an attractor for the?ow fρassociated toρ,and that the generalized McShane’s identity holds.This implies that for any K>0,the set{X∈C||x|≤K}is?nite. It follows that E(ρ)=?.Note that Lemmas3.1and3.4played essential roles in the proof,in particular,if the extended BQ-conditions are satis?ed,there cannot be an escaping orbit in the sense of Lemma3.4,which is a crucial step towards showing the existence of the attractor.

Conversely,ifρdoes not satisfy the extended BQ-conditions,then either there exists some X∈C with x∈(?2,2)or the set C(2)={X∈C||x|≤2}is in?nite. In the?rst case,X∈E(ρ),and in the second case,C(2)has an accumulation point in PL which lies in E(ρ).

5.Reducible characters

We consider the reducible characters in this section and prove Theorem1.4. As pointed out in§2.16,we may use representations into diagonal matrices of SL(2,C)to represent the reducible characters.Note thatρ(XY X?1Y?1)=I for all generating pairs X,Y ofπ,where I is the identity matrix.Hence we may think ofρas corresponding to a representation of the fundamental group of the torus T (without boundary)and use the homology classes[X m Y n]where m,n are relatively prime and n≥0to represent the elements of C.Letπ(T)= X,Y|XY X?1Y?1= I ,then

ρ(X)= α1/20

0α?1/2 ,ρ(Y)= β1/20 0β?1/2 ,

whereα,β∈C,and x=trρ(X)=α1/2+α?1/2,y=trρ(Y)=β1/2+β?1/2.There

are two cases to consider,(i)p·log|α|+q·log|β|=0for some integers p,q(not all

zero);and(ii)otherwise.In the?rst case,we have|αp·βq|=1.By using the action ofΓ,we might as well assume that in fact p=1and q=0,so that|α|=1,and

x∈[?2,2].If|β|=1as well,then y∈[?2,2]and similarly z=trρ(XY)∈[?2,2].

Sinceκ=2,ρcorresponds to a SU(2)representation and E(ρ)=PL in this case.If|β|=1,then|z|∈[?2,2]for all other Z∈C\{X},since the exponent of Y is non-zero for all the other elements.Furthermore,since|αnβ|=|β|is bounded,X∈E(ρ).For any other element Z∈PL,if Z n is any sequence of distinct elements approaching Z,then the exponent of Y in the homology class of Z n approaches±∞,so that log|Z n|→±∞.Hence Z∈E(ρ).It follows that in this case,E(ρ)={X}.Now we consider case(ii),where log|α|

16SER PEOW TAN,YAN LOI WONG,AND YING ZHANG

6.Proofs of Theorems1.5and1.7

We?rst give an equivalent de?nition for the set E(ρ)in terms of the(pro-jectivized)ends of a certain subtree H ofΣ;this is the variation of the original de?nition given by Bowditch in[1].

Supposeρ∈X does not satisfy the extended BQ-conditions.We de?ne a subtree H:=HρofΣas follows:Suppose that e∈E(Σ)corresponds to the generating pair(X,Y).Then e?H if both C[X,Y]and C[Y,X]does not satisfy the extended BQ-conditions.

For edges e∈E(Σ)\H,exactly one of the sets C[X,Y],C[Y,X]does not satisfy the extended BQ-conditions(sinceρdoesn’t);we de?ne a direction d(e)∈ E(Σ)so that tail of d(e)satis?es the extended BQ-conditions.Hence,d(e)points towards H for all e∈H.Note that d(e)is not to be confused with the?ow f(e)de?ned in §2.18.It is easy to see from the de?nition that any vertex v∈V(Σ)has0,2or3 edges of H adjacent to it.A vertex v∈H is called a node of H if there are three edges of H adjacent to it.Furthermore,if v is not a vertex of H,then for the three edges incident at v,d(e)points towards v for two of the edges and away from v for the third.If v has2edges of H adjacent to it,then d(e)points towards v for the remaining edge.

There is a natural projection map from the ends ofΣto PL which is one to one onto irrational points and two to one to rational points.If H is empty then d(e)all point towards a uniqueλ∈PL and E(ρ)={λ};otherwise,E(ρ)is the image of the ends of H under the projection map to PL.

For an irrationalλ∈E(ρ),the following result states that we can take K=2in the de?nition of E(ρ).

Proposition6.1.Suppose thatλ∈E(ρ)andλ∈C.Then there exists a sequence of distinct elements Z n∈C such that Z n→λand|z n|≤2for all n.

Proof.If H is empty,then d(e)points towardsλfor all e∈E(Σ).Choose any path{e n}limiting atλand let(X n,Y n)be the generating pair corresponding to e n,ordered so thatλ∈[X n,Y n].Thenρdoes not satisfy the extended BQ-

conditions on C[X

n ,Y n]

,and hence,there exists Z n∈C[X n,Y n]such that|z n|≤2.

Since X n,Y n→λ,by passing to a subsequence if necessary,we obtain a sequence of distinct Z n→λwith|z n|≤2.The same argument works ifλis the end of a path in H:we just choose a path in H ending atλ.

Proof of Theorem1.5.Letρ∈X Rκwithφthe corresponding Fricke trace map, and suppose thatκ=2.We?rst prove the following claim:

Claim.If|E(ρ)|>1,then|E(ρ)∩C|=∞,and if E(ρ)={X},then X∈C. Furthermore,if|E(ρ)|>1,then E(ρ)is perfect(that is,every element of E(ρ)is an accumulation point of E(ρ)).

First note that x=φ(X)∈R for all X∈C by the edge relation(8).Hence by Proposition4.3,X∈E(ρ)if|x|<2.If there exists two distinct X,Y∈C with |x|,|y|<2,then by looking at the behavior ofφabout X and Y respectively and applying Lemma3.5(b),we see that there are in?nitely many Z∈C∩E(ρ)and both X and Y are accumulation points in E(ρ).Similarly,if there exists X∈C∩E(ρ) with x=±

√κ+2>2,then by Lemma3.5(e),there are in?nitely many Z∈C∩E(ρ)and X is an accumulation point of E(ρ)(in fact,it is a one

END INV ARIANTS FOR SL(2,C)CHARACTERS17 sided limit point of E(ρ)ifρis not dihedral).Now suppose thatλ∈E(ρ)with λ∈C.By Proposition6.1,|C(2)|=∞and since C(2)is connected,we can?nd a sequence{X n}?C(2)such that X n→λand(X n,X n+1)is a generating pair

for all n.If there are in?nitely many X k in the sequence such that|x k|<2,we are done.Otherwise,re-indexing if necessary,we may suppose that x n=±2for all n∈N.Note that X k?1and X k+1are neighbors of X k.If(X k?1,X k,X k+1) is a generating triple,then using the sign-change automorphism(§2.4),we may assume that(x k?1,x k,x k+1)=(2,2,2)or(?2,?2,?2).In the?rst case,κ=2 which contradicts our assumption.The second case corresponds to the holonomy representation of the thrice punctured sphere andφsatis?es the extended BQ-conditions,so E(ρ)=?,again a contradiction.Hence,(X k?1,X k,X k+1)is not a generating triple and there exists Z k,a neighbor of X k lying between X k?1and X k+1.Again,by the sign change automorphism,we may assume that x k=2. Now by Lemma3.5(c),x k?1and x k+1have opposite signs,say x k?1=?2and x k+1=2,which forces|z k|<2.This produces a sequence of elements Z k∈E(ρ)∩C approachingλwhich completes the proof of the claim.

The classi?cation of the types for E(ρ)according to(a),(b),(c)or(d)of the theorem now follows from the above claim and Theorems1.2and1.3.It remains to show that each of these cases occur for the values ofκstated.We use here Goldman’s main Theorem in[3]which classi?es the action ofΓon X Rκ.The fact that case(a)occurs if and only ifκ<2or≥18follows from the fact thatΓacts properly only on characters in these ranges ofκ.The ranges for case(c)and(d)also follow from Goldman’s result,Theorem1.2,the above argument,and the fact that the dihedral character is real forκ>?2.It remains to show that case(b)occurs if and only ifκ≥6.It is easy to see that E(ρ)={X}ifι(ρ)=(0,a,a),where a≥2. Since in this caseκ=2a2?2≥6,we have case(b)occurs for allκ≥6.Conversely, we show that if case(b)occurs,that is,if E(ρ)={X},thenκ≥6.Let Y n,where n∈Z,be the neighbors of e443bf35a32d7375a4178060ing the sign change automorphism,we may assume that x≥0,so that x∈[0,2),x=e iθ+e?iθwhere0<θ≤π/2,and there exist A,B∈C with AB=(x2?κ?2)/(x2?4)such that y n=Ae inθ+Be?inθ.Since y n∈R for all n,we have B=ˉA so that y n=2Re(Ae inθ).Now since|y n|≥2for all n,θmust be a rational multiple ofπ,that is,θ=pπ/q for some rational p/q, and the set{Ae inθ}is the set of vertices of a regular q-gon in the complex plane centered at the origin with|2Re(Ae inθ)|≥2for all n.Re-indexing if necessary,we may assume that y0=2Re(A)>0;hence y0≥2,and y1=2Re(Ae iθ)<0,which implies y1≤?2.It follows thatκ=x2+y20+y21?xy0y1?2≥8?2=6which completes the proof of Theorem1.5. Proof of Theorem1.7.Suppose thatρis discrete,and E(ρ)has at least three elements but is not equal to PL.By Theorem1.2,it su?ces to prove that E(ρ) is perfect,that is every X∈E(ρ)is the limit of distinct Z n∈E(ρ).Note that in this case H has at least three distinct ends and at least one node.

Case1.X∈C∩E(ρ).By Proposition4.3and the discreteness ofρ,x= 2cos p

q∈Q,and by the periodicity ofφabout X(ρis stabilized by a reducible element of the mapping class groupΓ?xing X)and the fact that H has at least3ends,X is a limit of distinct Z n∈E(ρ).

Case2:X∈C.Choose a node v of H(which exists by the assumption)and the path{e n}from v to X and let(X n,Y n)be the generating pair corresponding to

18SER PEOW TAN,YAN LOI WONG,AND YING ZHANG

e n,ordered so that X∈[X n,Y n].By the connectedness o

f C(2)(Lemma3.1)and

the vertex relation(7),it is easy to see that there exists a universal constant K>0 depending only onκsuch that|x n|,|y n|

approaching X such that for all n∈N,x n=a,y n=b where a,b∈C are?xed constants.Let d be the distance from v to the edge e1?(X1,Y1).It follows that there is a node v n of H at distance d from each of e n?(X n,Y n).Since e n→X, v n→X,and hence X is the limit point of distinctλn∈E(ρ). Remark6.2.We end this section by remarking that ifρ∈X disc satis?es the con-ditions of Theorem1.7,and E(ρ)=PL,then the proof of Theorem1.7implies that in fact,the stabilizer ofρin the mapping class groupΓof T is relatively large, in particular,is not?nite or virtually cyclic.This answers a question posed by Bowditch in[1].Makoto Sakuma[9]has also independently obtained examples of characters whose stabilizer inΓis not?nite or virtually cyclic by considering the representations arising from the two bridge knot complements;in these cases there are at least two(hence in?nitely many)X∈C for whichφ(X)=0.

7.Proof of Theorem1.6

For the rest of this section,we?xρ∈X Iκ,whereκ<2,with corresponding

Fricke trace mapφ.It will be convenient and visually easier to regardφas a map from?(Σ)to C.Recall from§2.10that there is a partition of C and?(Σ)into three equivalence classes R,G and B.We use the letters Z,Y and X to denote the elements of R,G and B respectively,and W to denote a general element of C. Sinceρis imaginary,φtakes purely imaginary values on two of the equivalence classes and real values on the third;we may assume that it takes real values on Z∈R.For X∈B,Y∈G and Z∈R,we use the conventionφ(X)=ix,φ(Y)=iy andφ(Z)=z,where x,y,z∈R for the rest of this section(note that this is di?erent from the convention used earlier).Then if(X,Y,Z)∈GT and (X,Y;Z,Z′),(Y,Z;X,X′),(Z,X;Y,Y′)∈GQ,the vertex relation(7)and edge relation(8)can be rewritten as:

?x2?y2+z2+xyz?2=κ(13)

z+z′=?xy,y+y′=xz,x+x′=yz(14) Note that by our convention(§2.15)ρ∈X Iκis not dihedral.Hence,by Theorem 1.2,E(ρ)=PL,and in fact E(ρ)has empty interior in PL.We will be using the results of Goldman-Stantchev,in particular,the classi?cation result(Theorem A of[5])and theτ-reduction algorithm(§4of[5]).The following strengthening of Proposition6.1concerning irrational ends is key.

Proposition7.1.Supposeρ∈X Iκ,whereκ<2andλ∈E(ρ)withλ∈C.Then there exists a sequence of distinct elements{W n}?C∩E(ρ)such that W n→λ. Proof.By Proposition6.1,?(2)=?.Choose W0∈?(2)and let P={e n}be the minimal path inΣ(?)from W0limiting atλ.By Lemma3.1,we can?nd a sequence{W n}??(2)such that for all n≥1,W n is adjacent to P,W n,W n+1are neighbors and W n→λ(see Figure3).We consider two cases,?rst when W n∈R for in?nitely many n,and secondly,when W n∈R for all but?nitely many n.

END INV ARIANTS FOR SL(2,C )CHARACTERS 19

..............................................................................................................................................................................................................................................................................................................................................·......................................·................................·................................·.......................................·.............................·...............................................·...............................................·...............................................·.............................·..............................................·................................·..............................................···............................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................λW 0W 1W 2W 3W 4W 5W 6Figure 3.The sequence {W n }?→λalong P .

Case 1.W n ∈R for in?nitely many n .This gives a subsequence {Z n }?R approaching λsuch that |z n |≤2.If there are in?nitely many elements in this sequence with |z n |<2,these are in E (ρ)by Proposition 4.3,so we are done.Hence we might as well assume that z n =±2for all n .Then there exists in?nitely many k such that φ(W k )=±2in the sequence {W n }.Fix such a k and consider the triple W k ?1,W k ,W k +e443bf35a32d7375a4178060ing the sign-change automorphism (§2.4)we might as well assume that w k =2.W k ?1and W k +1are neighbors of W k (hence,not in R ),with φ(W k ?1)=iw k ?1,φ(W k +1)=iw k +1,where w k ?1,w k +1∈[?2,2],by assumption.If w k ?1=0(or w k +1=0),then W k ?1∈E (ρ)(resp.W k +1∈E (ρ)).If both w k ?1and w k +1have the same sign,then,since the values of the neighbors of W k grow linearly (Lemma 3.5(c)),the di?erence in the values of successive neighbors of W k is ci ,where 0?2or y <2.Now if we consider the generating quadruple (X,Y ;W k ,Z k ),we have |z k |=|?xy ?2|<2,so we obtain Z k ∈R ∩E (ρ).In all subcases,we obtain a subsequence {W n }?C ∩E (ρ)such that W n →λ,which completes the proof in this case.

Case 2.W n ∈R for all but ?nitely many n .Re-indexing if necessary,we might as well assume that W n ∈R for all n ,and {W n }=X 1,Y 1,X 2,Y 2,...,where X n ∈B ,Y n ∈G .The sequence {W n }must cross the path P in?nitely often,since the end of P is irrational,and W n ∈R for all n .Hence,renaming G and B if necessary,we obtain a subsequence of generating pairs {(X n ,Y n )}?GP and a nested sequence of closed intervals [X n ,Y n ]such that

λ∈···?[X n +1,Y n +1]?[X n ,Y n ]···?[X 1,Y 1],

X n ,Y n →λ,and |ix n |,|iy n |≤2for all n ≥1.Passing again to a subsequence,we may assume that x n →a ,y n →b ,where a,b ∈[?2,2].Let e n ∈E r ?E (Σ)be the edge of P corresponding to the generating pair (X n ,Y n )and v n ∈V (Σ)be the end of e n which is closer to λ.Note that the quantity τn =?z n z ′n associated to the edge e n ?(X n ,Y n ;Z n ,Z ′n )∈GQ approaches a ?xed constant c which depends only on a and b .We now consider the case where both a,b =0and the case where one of a,b =0separately,and perform the τ-reduction algorithm at each v n .

20SER PEOW TAN,YAN LOI WONG,AND YING ZHANG Subcase(i):a,b=0.Then,starting at v n,theτ-reduction algorithm of Goldman-Stantchev[5],see also the Appendix,produces a sequence in C ter-minating after a?nite number of steps at some Z n∈E(ρ).For n su?ciently

large,since a,b=0,the?rst(two)steps of the algorithm reducesτby at least some positive constant depending only on a and b,hence if n is su?ciently large, the algorithm starting at v n does not cross the edges e n?1?(X n?1,Y n?1)and e n+1?(X n+1,Y n+1).Hence,Z n∈[X n?1,Y n?1]\[X n+1,Y n+1].Again passing to a subsequence,we obtain a sequence of distinct Z n∈E(ρ)approachingλ.

Subcase(ii):One of a,b=0,say a=0.Then for any?>0,there exists N such that|ix n|N.Again,theτ-reduction algorithm starting at v n terminates,after a?nite number of steps,at some Z n∈C such that Z n∈E(ρ). Furthermore,if|ix n|is su?ciently small,by[5](see Appendix),Z n is in fact a neighbor of X n.Again it follows that for n su?ciently large,Z n∈[X n?1,Y n?1]\ [X n+1,Y n+1],and the conclusion follows as in Subcase(i).

Proof of Theorem1.6.The proof for Theorem1.6now follows easily,along similar lines to the proof of Theorem1.5.If|E(ρ)∩C|>1,it follows that|E(ρ)∩C|=∞by Lemma3.5(b),and furthermore,each W∈E(ρ)∩C is an accumulation point of E(ρ).If E(ρ)has an irrational endλ,it follows from Proposition7.1that |E(ρ)∩C|=∞andλis an accumulation point of E(ρ).Hence,E(ρ)is either empty,has one element W∈C or is a Cantor set.The fact that E(ρ)=?occurs only forκ≤?14and not for?14<κ<2follows from[5](actually,the case κ=?14was not covered,but as we saw earlier,the exampleρ∈X?14where ι(ρ)=(2,2i,?2i)has E(ρ)=?).It is fairly easy to construct,for allκ<2, examples where E(ρ)={X}:we useρwithι(ρ)=(0,yi,z)where y,z are chosen such that y,z>2and z2?y2=κ+2.In this case C(2)=E(ρ)={X}.Similarly, it is easy to construct examples where|E(ρ)|>1,and hence is a Cantor set:we just need to make sure that there are at least two element in C∩E(ρ).We leave this to the reader.Finally,to prove part(i)of the Theorem,we?rst show that forρ∈X?2, if E(ρ)={W},then W∈R.Suppose not,then we have some(X0,Y0,Z0)∈GT such that z0∈(?2,2).If z0=0,then ix0=iy0=0,a contradiction since E(ρ) has only one element.Hence,using the sign change automorphism if necessary,we may assume that z0∈(0,2).Use X n,Y n to denote the successive neighbors of Z0, with values ix n,iy n respectively.Then x n,y n lie on the ellipse

x2?z0xy+y2=z20,

with major axis y=x and minor axis y=?x,and intercepts(±z0,0)and(0,±z0). The values x n,y n are obtained by starting at the point(x0,y0)on the ellipse and taking the coordinates of the successive intersections of the ellipse with the up/down and left/right path;see Figure4(compare with[5]).There is at least one intercept which lies in either the second or the fourth quadrant,that is,there exist some successive neighbors X n,Y n(or Y n,X n+1)of Z0such that x n,y n have opposite signs and|x n|,|y n|≤z0<2.However,in this case,for the generating quadruple (X n,Y n;Z0,Z n)∈GQ,|z n|=|x n y n?z0|<2,contradicting the fact that E(ρ)has only one element.Hence we conclude that the single element W of E(ρ)is not in R;so it must be in G or B,which implies w=0.The conclusion then follows easily from the vertex relation(7)and the connectedness of C(2).

END INV ARIANTS FOR SL(2,C )CHARACTERS 21

.................................................................................................y ...............................................................................................................................................................................................................................x ....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................····........................................................................................................·z 0·

?z 0·z 0·?z 0(x 1,y 1)(x 1,y 0)(x 0,y 0)(x 0,y ?1)Figure 4.The ellipse x 2?z 0xy +y 2=z 20(z 0=1.5)and successive

values of x n ,y n .

8.Appendix:The τ-reduction algorithm

We give a brief description of the τ-reduction algorithm given by Goldman and Stantchev in [5].Bowditch’s algorithm in [1]for type-preserving imaginary Marko?maps (ρ∈X I ?2)is essentially a special case of this,although it was couched in a di?erent setting using the Jorgenson parameters,a =x/yz,b =y/xz and c =z/xy instead of x,y,z .We will describe the results and algorithm but refer the reader to [5]for detailed proofs.Fix a ρ∈X I κ,where κ<2,with corresponding φ.We adopt the notation in §7;in particular,we use X ,Y and Z for the elements of B ,G and R respectively (thought of as subsets of ?(Σ)),and write z =φ(Z ),ix =φ(X ),iy =φ(Y )where x,y,z ∈R .If e ∈E r ?E (Σ)corresponds to the generating quadruple (X,Y ;Z,Z ′),de?ne τ(e )=?zz ′.Similarly,for v ∈V (Σ),de?ne τ(v )=τ(e )where e is the unique edge in E r with one endpoint at v .Note that τis invariant under the sign-change automorphism.

The τ-reduction algorithm has as a starting point a vertex v 0∈V (Σ),and produces a (unique)?nite sequence of adjacent vertices v 0,v 1,···,v N such that either

(a)τ(v N )in minimal among all vertices v ∈V (Σ),v N is an attractor for the

?ow f ρ,φsatis?es the extended BQ-conditions and E (ρ)=?;or

(b)one of the three regions adjacent to v N is in E (ρ).

This also produces a sequence of elements W 1,W 2,···,W N ∈?(Σ),where W k is the (unique)region adjacent to v k which is not adjacent to v k ?1.The algorithm consists of two general types of moves,the ?rst type moves the vertices around the boundary of Z ∈R until we reach a point where τis minimum among all the vertices lying on the boundary of Z ,then the second type “?ips”the vertex across the edge e ∈E r to a vertex which is now adjacent to a di?erent Z ′∈R .The main point is that the algorithm is uniquely determined,does not backtrack,and

本文来源：https://www.bwwdw.com/article/zkte.html