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

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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.

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