Table Of Content8
0 Operator algebras and conjugacy problem for the
0
2 pseudo-Anosov automorphisms of a surface
r
p
A Igor Nikolaev
∗
5
April 15, 2008
1
]
T
G
Abstract
.
h The conjugacy problem for the pseudo-Anosov automorphisms of a
at compact surface is studied. To each pseudo-Anosov automorphism φ, we
m assign an AF-algebra Aφ (an operator algebra). It is proved that the
[ assignment is functorial, i.e. every φ′, conjugate to φ, maps to an AF-
algebraAφ′,whichisstablyisomorphictoAφ. Thenewinvariantsofthe
4 conjugacy of the pseudo-Anosov automorphisms are obtained from the
v knowninvariantsofthestableisomorphismsoftheAF-algebras. Namely,
7 themaininvariantisatriple(Λ,[I],K),whereΛisanorderintheringof
5 integersinarealalgebraic numberfieldK and[I]an equivalenceclass of
1
theideals in Λ. Thenumerical invariants includethedeterminant ∆ and
5
the signature Σ, which we compute for the case of the Anosov automor-
0
phisms. Aquestionconcerningthep-adicinvariantsofthepseudo-Anosov
6
automorphism is formulated.
0
/
h
t Key words and phrases: mapping class group, AF-algebras
a
m
: AMS (MOS) Subj. Class.: 46L, 57M.
v
i
X
1 Introduction
r
a
A. Conjugacy problem. Let Mod (X) be the mapping class group of a
compact surface X, i.e. the groupof the orientationpreserving automorphisms
of X modulo the trivial ones. Recall that φ,φ Mod (X) are the conjugate
′
∈
automorphisms,wheneverφ =h φ h 1foranh Mod(X). Itisnothardto
′ −
◦ ◦ ∈
seethattheconjugationisanequivalencerelationwhichsplitsthemappingclass
group into the disjoint classes of conjugate automorphisms. The invariants of
the conjugacyclassesinMod(X)is animportantanddifficultproblemstudied
∗PartiallysupportedbyNSERC.
1
by Hemion [8], Mosher [11] and others. Knowing such invariants leads to a
topological classification of the three-dimensional manifolds, which fiber over
the circle with the monodromy φ Mod (X) [14].
∈
B.Thepseudo-Anosovautomorphisms. Itisknownthatanyφ Mod(X)
∈
is isotopic to an automorphism φ, such that either (i) φ has a finite order, or
′ ′
(ii) φ is a pseudo-Anosov(aperiodic)automorphism,orelse (iii) φ is reducible
′ ′
by a system of curves Γ surrounded by the small tubular neighborhoods N(Γ),
suchthat onX N(Γ) φ satisfies either (i) or(ii). In the case φ Mod(X)is
′
− ∈
apseudo-Anosovautomorphism,thereexistapairofthestable andunstable
s
F
mutuallyorthogonalmeasuredfoliationsonthesurfaceX,suchthatφ( )=
u s
F F
1 andφ( )=λ ,whereλ >1iscalledadilatationofφ. Thefoliations
λφFs Fu φFu φ
, areminimal, uniquely ergodic anddescribe the automorphismφ up to a
s u
F F
power. In the sequel, we shall focus on the conjugacy problem for the pseudo-
Anosov automorphisms of a surface X.
C. The AF-algebras. The C -algebra is an algebra A over C with a norm
∗
a a and an involution a a such that it is complete with respect to the
∗
7→|| || 7→
norm and ab a b and a a = a2 for all a,b A. The C -algebras
∗ ∗
|| ||≤|| || || || || || || || ∈
have been introduced by Murray and von Neumann as the rings of bounded
operators on a Hilbert space and are deeply intertwined with the geometry
and topology of manifolds [3], 24. Any simple finite-dimensional C -algebra
∗
§
is isomorphic to the algebra M (C) of the complex n n matrices. A natural
n
×
completion of the finite-dimensional semi-simple C -algebras (as n ) is
∗
→ ∞
known as an AF-algebra [6]. The AF-algebra is most conveniently given by an
infinite graph,which recordsthe inclusionof the finite-dimensionalsubalgebras
into the AF-algebra. The graph is called a Bratteli diagram. In an important
special case when the diagram is periodic, the AF-algebra is called stationary.
In the addition to a regular isomorphism =, the C -algebras A,A are called
∼ ∗ ′
stably isomorphic whenever A = A , where is the C -algebra of
⊗K ∼ ′ ⊗ K K ∗
compact operators.
D. Main idea. Let φ Mod (X) be a pseudo-Anosov automorphism. The
∈
main idea of the present paper is to assign to φ an AF-algebra, A , so that for
φ
every h Mod (X) the following diagram commutes:
∈
conjugacy
φ - φ =h φ h 1
′ −
◦ ◦
? ?
isomorphism
-
Aφ Aφ′
⊗K ⊗K
2
(In other words, if φ,φ are the conjugate pseudo-Anosov automorphisms, then
′
theAF-algebrasAφ,Aφ′ arestablyisomorphic.) Forthesakeofclarity,weshall
consider an example illustrating the idea in the case X =T2 (a torus).
E. Model example. Let φ Mod (T2) be the Anosov automorphism given
∈
by anon-negativematrixA SL (Z). Considera stationaryAF-algebra,A ,
φ 2 φ
∈
given by the following periodic Bratteli diagram:
a a a
11 11 11
b b b b ...
(cid:0)@ (cid:0)@ (cid:0)@ (cid:0)
(cid:0)b a12@(cid:0) a12@(cid:0) a12@(cid:0) A = a11 a12 ,
@ a (cid:0)@ a (cid:0)@ a (cid:0)@ φ (cid:18)a21 a22(cid:19)
21 21 21
@b(cid:0) @b(cid:0) @b(cid:0) @b ...
a a a
22 22 22
Figure 1: The AF-algebra A .
φ
where a indicate the multiplicity of the respective edges of the graph. (We
ij
encourage the reader to verify that F : φ A is a correctly defined function
φ
7→
on the set of Anosov automorphisms given by the hyperbolic matrices with the
non-negative entries.) Let us show that if φ,φ Mod (T2) are the conjugate
′
∈
Anosov automorphisms, then Aφ,Aφ′ are the stably isomorphic AF-algebras.
Indeed, let φ′ = h φ h−1 for an h Mod (X). Then Aφ′ = TAφT−1 for a
◦ ◦ ∈
matrix T SL (Z). Note that (A )n =(TA T 1)n =TAnT 1, where n N.
∈ 2 ′φ φ − φ − ∈
We shall use the following criterion ([6], Theorem 2.3): the AF-algebras A,A
′
are stably isomorphic if and only if their Bratteli diagrams contain a common
block of an arbitrary length. Consider the following sequences of matrices:
A A ...A
φ φ φ
n
T A|φAφ.{.z.AφT}−1,
n
| {z }
whichmimictheBrattelidiagramsofAφ andAφ′. Lettingn ,weconclude
→∞
that Aφ⊗K∼=Aφ′ ⊗K.
F. The invariants oftorus automorphismsobtainedfrom the operator
algebras. The conjugacy problem for the Anosov automorphisms can now be
recastintermsoftheAF-algebras: findtheinvariantsofthestableisomorphism
classes of the stationary AF-algebras. One such invariantis due to Handelman
[7]. Consider an eigenvalue problem for the hyperbolic matrix A SL (Z):
φ 2
∈
A v = λ v , where λ > 1 is the Perron-Frobenius eigenvalue and v =
φ A A A A A
(v(1),v(2)) the corresponding eigenvector with the positive entries normalized
A A
3
so that v(i) K = Q(λ ). Denote by m = Zv(1) +Zv(2) a Z-module in the
number fiAeld∈K. RecallAthat the coefficient ringA, Λ, ofAmodule m consists of
the elements α K such that αm m. It is known that Λ is an order in K
∈ ⊆
(i.e. a subring of K containing 1) and, with no restriction, one can assume
that m Λ. It follows from the definition, that m coincides with an ideal,
⊆
I, whose equivalence class in Λ we shall denote by [I]. It has been proved by
Handelman, that the triple (Λ,[I],K) is an arithmetic invariant of the stable
isomorphism class of Aφ: the Aφ,Aφ′ are stably isomorphic AF-algebras if and
only if Λ=Λ,[I]=[I ] and K =K . It is interesting to compare the operator
′ ′ ′
algebra invariants with those obtained in [16].
G. The AF-algebra A (pseudo-Anosov case). Denote by the stable
φ φ
F
foliation of a pseudo-Anosov automorphism φ Mod (X). For brevity, we
∈
assume that is an oriented foliation given by the trajectories of a closed
φ
F
1-form ω H1(X;R). Let v(i) = ω, where γ ,...,γ is a basis in the
∈ γi { 1 n}
relativehomologyH (X,Sing ;ZR) anddenote byθ =(θ ,...,θ )a vector
1 φ 1 n 1
with the coordinates θ = v(i+F1)/v(1). Consider the (infinite) Ja−cobi-Perron
i
continued fraction [2] of θ:
1 0 1 0 1 0
= lim ... ,
(cid:18)θ(cid:19) k (cid:18)I b1(cid:19) (cid:18)I bk(cid:19)(cid:18)I(cid:19)
→∞
where b = (b(i),...,b(i) )T is a vector of the non-negative integers, I the unit
i 1 n 1
matrix and I =(0,...,−0,1)T. By the definition, A is an AF-algebra given by
φ
0 1
the Bratteli diagram, whose incidence matrices coincide with B =
k (cid:18)I bk(cid:19)
for k =1,..., .
∞
H. Main results. For a matrix A GL (Z) with the positive entries, we let
n
∈
(1) (n)
λ be the Perron-Frobenius eigenvalue and (v ,...,v ) the corresponding
A A A
normalizedeigenvectorwithv(i) K =Q(λ ). Thecoefficient(endomorphism)
A ∈ A
ringofthemodulem=Zv(1)+...+Zv(n) willbedenotedbyΛ. Theequivalence
A A
class of ideals in the ring Λ generated by the ideal m, we shall write as [I].
Finally,wedenoteby∆=det(a )andΣthedeterminantandsignatureofthe
ij
symmetric bilinear form q(x,y) = n a x x , where a = Tr (v(i)v(j)) and
i,j ij i j ij A A
Tr( ) the trace function. Our mainPresults can be expressed as follows.
•
Theorem 1 A is a stationary AF-algebra.
φ
Theorem 2 Let F be a mapping acting by the formula φ A . Then:
φ
7→
(i) F is a functor, i.e. F maps the conjugate pseudo-Anosov automorphisms
to the stably isomorphic AF-algebras;
(ii) Ker F = [φ], where [φ] = φ Mod (X) (φ)m = φn, m,n N is
′ ′
{ ∈ | ∈ }
the commensurability class of the pseudo-Anoov automorphism φ.
4
Corollary 1 The following are the invariants of the conjugacy classes of the
pseudo-Anosov automorphisms:
(i) triples (Λ,[I],K);
(ii) integers ∆ and Σ.
I. Structure of the paper. The proofofmainresults canbe foundin section
4. The sections 2 and 3 are reserved for some helpful lemmas necessary to
provethe mainresults. Finally, insection5 someexamples,openproblems and
conjectures are considered. Since the paper does not include a formal section
on the preliminaries, we encourage the reader to consult [3], [6], [10] (operator
algebras&dynamics),[9],[15](measuredfoliations)and[2],[12](Jacobi-Perron
continued fractions).
2 The Jacobian of a measured foliation
Let be a measured foliation on a compact surface X [15]. For the sake of
F
brevity, we shall always assume that is an oriented foliation, i.e. given by
F
the trajectories of a closed 1-form ω on X. (The assumption is no restric-
tion – each measured foliation is oriented on a surface X, which is a double
cover of X ramified at the singular points of the half-integer index of the non-
e
orientedfoliation[9].) Let γ ,...,γ beabasisintherelativehomologygroup
1 n
{ }
H (X,Sing ;Z), where Sing is the set of singular points of the foliation
1
F F
. It is well known that n = 2g + m 1, where g is the genus of X and
F −
m= Sing ( ). The periods of ω in the above basis we shall write as:
| F |
λ = ω.
i
Z
γi
The λ are the coordinates of in the space of all measured foliations on X
i
F
(with a fixed set of the singular points) [5].
Definition 1 By a Jacobian Jac ( ) of the measured foliation , we under-
stand a Z-module m = Zλ +...+FZλ regarded as a subset ofFthe real line
1 n
R.
An importance of the Jacobians stems from an observation that although the
periods λ depend on the basis in H (X,Sing ;Z), the Jacobian does not.
i 1
F
Moreover,uptoascalarmultiple,theJacobianisaninvariantoftheequivalence
classofthefoliation . Weshallformalizetheobservationsinthefollowingtwo
F
lemmas.
Lemma 1 The Z-module m is independent of the choice of a basis
in H (X,Sing ;Z) and depends solely on the foliation .
1
F F
5
Proof. Indeed, let A=(a ) GL (Z) and let
ij n
∈
n
γi′ = aijγj
Xj=1
be a new basis in H (X,Sing ;Z). Then using the integration rules:
1
F
λ = ω = ω =
′i Rγi′ Z nj=1aijγj
nP
= n ω = a λ .
j=1 γj ij j
P R Xj=1
To prove that m=m, consider the following equations:
′
n n
m′ = ni=1Zλ′i = Z aijλj =
P Xi=1 Xj=1
= n ( n a Z)λ m.
j=1 i=1 ij j ⊆
P P
LetA 1 =(b ) GL (Z)beaninversetothematrixA. Thenλ = n b λ
− ij ∈ n i j=1 ij ′j
and P
n n
m = ni=1Zλi = Z bijλ′j =
P Xi=1 Xj=1
= n ( n b Z)λ m.
j=1 i=1 ij ′j ⊆ ′
P P
Since both m m and m m, we conclude that m =m. Lemma 1 follows. (cid:3)
′ ′ ′
⊆ ⊆
Recall that the measured foliations and are said to be equivalent, if there
′
F F
exists an automorphism h Mod (X), which sends the leaves of the foliation
∈
to the leaves of the foliation . Note that the equivalence deals with the
′
F F
topologicalfoliations(i.e. the projectiveclassesofmeasuredfoliations[15])and
does not preserve a transversalmeasure of the leaves.
Lemma 2 Let , betheequivalentmeasuredfoliationsonasurfaceX. Then
′
F F
Jac ( )=µ Jac ( ),
′
F F
where µ>0 is a real number.
Proof. Let h:X X be an automorphism of the surface X. Denote by h its
actiononH (X,S→ing( );Z)andbyh onH1(X;R)connectedbytheform∗ula:
1 ∗
F
ω = h (ω), γ H (X,Sing ( );Z), ω H1(X;R).
∗ 1
Z Z ∀ ∈ F ∀ ∈
h∗(γ) γ
6
Let ω,ω H1(X;R) be the closed 1-forms whose trajectories define the folia-
′
∈
tions and , respectively. Since , are equivalent measured foliations,
′ ′
F F F F
ω′ =µ h∗(ω)
for a µ>0.
Let Jac ( )=Zλ +...+Zλ and Jac ( )=Zλ +...+Zλ . Then:
F 1 n F′ ′1 ′n
λ = ω =µ h (ω)=µ ω, 1 i n.
′i Z ′ Z ∗ Z ≤ ≤
γi γi h∗(γi)
By lemma 1, it holds:
n n
Jac ( )= Z ω = Z ω.
F Z Z
Xi=1 γi Xi=1 h∗(γi)
Therefore:
n n
Jac ( )= Z ω =µ Z ω =µ Jac ( ).
′ ′
F Z Z F
Xi=1 γi Xi=1 h∗(γi)
Lemma 2 follows. (cid:3)
3 Equivalent foliations are stably isomorphic
Let be a measured foliation on the surface X. We introduce an AF-algebra,
F
A , of the foliation the same way it was done in 1.G for the foliation .
φ
F F § F
The goal of present section is the commutativity of the following diagram:
equivalent
-
′
F F
? ?
isomorphic
-
A A ′
F ⊗K F ⊗K
Let us start with a lemma, which is a simple property of the Jacobi-Perron
fractions [2].
Lemma 3 Letm=Zλ +...+Zλ andm =Zλ +...+Zλ betwoZ-modules,
1 n ′ ′1 ′n
such that m =µm for a µ> 0. Then the Jacobi-Perron continued fractions of
′
the vectors λ and λ coincide except, may be, a finite number of terms.
′
7
Proof. Let m = Zλ +...+Zλ and m = Zλ +...+Zλ . Since m = µm,
1 n ′ ′1 ′n ′
where µ is a positive real, one gets the following identity of the Z-modules:
Zλ′1+...+Zλ′n =Z(µλ1)+...+Z(µλn).
One can always assume that λ and λ are positive reals. For obvious reasons,
i ′i
there exists a basis λ′′,...,λ′′ of the module m, such that:
{ 1 n} ′
λ =A(µλ)
′′
(cid:26) λ′′ =A′λ′,
whereA,A GL+(Z)arethematrices,whoseentriesarenon-negativeintegers.
′ ∈ n
In view of the Proposition 3 of [1]:
0 1 0 1
A = ...
(cid:18)I b1(cid:19) (cid:18)I bk(cid:19)
A = 0 1 ... 0 1 ,
′ (cid:18)I b′1(cid:19) (cid:18)I b′l(cid:19)
whereb ,b arenon-negativeintegervectors. Sincethat(Jacobi-Perron)contin-
i ′i
ued fraction for the vectors λ and µλ coincide for any µ > 0 [2], we conclude
that:
1 0 1 0 1 0 1 0 1 0
= ... ...
(cid:18)θ(cid:19) (cid:18)I b1(cid:19) (cid:18)I bk(cid:19)(cid:18)I a1(cid:19)(cid:18)I a2(cid:19) (cid:18)I(cid:19)
1 = 0 1 ... 0 1 0 1 0 1 ... 0 ,
(cid:18)θ′(cid:19) (cid:18)I b′1(cid:19) (cid:18)I b′l(cid:19)(cid:18)I a1(cid:19)(cid:18)I a2(cid:19) (cid:18)I(cid:19)
where
1 0 1 0 1 0
= lim ... .
(cid:18)θ′′(cid:19) i (cid:18)I a1(cid:19) (cid:18)I ai(cid:19)(cid:18)I(cid:19)
→∞
In other words, the continued fractions of the vectors λ and λ coincide, except
′
a finite number of terms. (cid:3)
Lemma 4 Let , betheequivalentmeasuredfoliationsonasurfaceX. Then
′
F F
the AF-algebras A ,A ′ are stably isomorphic.
F F
Proof. Notice that lemma 2 implies that the equivalent measured foliations
, have the proportional Jacobians, i.e. m =µm for a µ>0. On the other
′ ′
F F
hand, by lemma 3 the continued fraction expansion of the basis vectors of the
proportional Jacobians must coincide, except a finite number of terms. Thus,
the AF-algebras A and A ′ are given by the Bratteli diagrams, which are
F F
identical, except a finite part of the diagram. It is well known ([6], Theorem
2.3) that the AF-algebras, which have such a property, are stably isomorphic.
(cid:3)
8
4 Proofs
4.1 Proof of Theorem 1
Let φ Mod (X) be a pseudo-Anosovautomorphismof the surfaceX. Denote
∈
by theinvariantfoliationofφ. Bythedefinitionofsuchafoliation,φ( )=
φ φ
F F
λ , where λ >1 is the dilatation of φ.
φ φ φ
FConsider the Jacobian Jac ( ) = m of the measured foliation . Since
φ φ φ
F F
is aninvariantfoliationofthe pseudo-Anosovautomorphismφ, onegetsthe
φ
F
following equality of the Z-modules:
m =λ m , λ = 1. (1)
φ φ φ φ
6 ±
Let v(1),...,v(n) be a basis in the module m , such that v(i) >0. In view of
φ
{ }
(1), one obtains the following system of linear equations:
λ v(1) = a v(1)+a v(2)+...+a v(n)
φ 11 12 1n
λφv(2) = a21v(1)+a22v(2)+...+a2nv(n)
... (2)
λ v(n) = a v(1)+a v(2)+...+a v(n),
φ n1 n2 nn
where a Z. The matrix A = (a ) is invertible. Indeed, since the foliation
ij ij
∈
is minimal, the real numbers v(1),...,v(n) are linearly independent over Q.
φ
F
So do the numbers λ v(1),...,λ v(n), which therefore can be taken for a basis
φ φ
of the module m . Thus, there exists an integer matrix B = (b ), such that
φ ij
v(j) = w(i), where w(i) =λ v(i). Clearly, B is an inverse to the matrix A.
i,j φ
TherefoPre, A GLn(Z).
∈
Moreover,withoutlossofthegeneralityonecanassumethata 0. Indeed,
ij
≥
if it is not yet the case, consider the conjugacy class [A] of the matrix A. It is
known that there exists a matrix A+ [A], whose entries are the non-negative
integers. One has to replace the basis∈v =(v(1),...,v(n)) in the module m by
φ
the basis Tv, where A+ =TAT 1. It will be further assumed that A=A+.
−
Lemma 5 The vector (v(1),...,v(n)) is the limit of a periodic Jacobi-Perron
continued fraction.
Proof. It follows from the discussion above, that there exists a non-negative
integer matrix A, such that Av = λ v. In view of the Proposition 3 of [1], the
φ
matrix A admits a unique factorization:
0 1 0 1
A= ... , (3)
(cid:18)I b1(cid:19) (cid:18)I bk(cid:19)
where b = (b(i),...,b(i))T are the vectors of the non-negative integers. Let us
i 1 n
consider a periodic Jacobi-Perroncontinued fraction:
0 1 0 1 0
Per ... . (4)
(cid:18)I b1(cid:19) (cid:18)I bk(cid:19)(cid:18)I(cid:19)
9
According to Satz XII of [12], the above periodic fraction converges to the
vectorw =(w(1),...,w(n)),suchthatwsatisfiestheequation(B B ...B )w =
1 2 k
Aw = λ w. In view of the equation Av = λ v, we conclude that the vectors v
φ φ
andwarethecollinearvectors. Therefore,theJacobi-Perroncontinuedfractions
of v and w must coincide. (cid:3)
Itisnowstraightforwardtoprove,thattheAF-algebraattachedtothefoliation
is a stationary AF-algebra. Indeed, by the lemma 5, the vector of periods
φ
F
v(i) = ω unfolds into a periodic Jacobi-Perron continued fraction. By the
γi
definitioRn, the Bratteli diagram of the AF-algebra Aφ is periodic as well. In
other words, the AF-algebra A is a stationary AF-algebra. (cid:3)
φ
4.2 Proof of Theorem 2
(i) Let us prove the first statement. For the sake of completeness, let us give a
proof of the following (well-known) lemma.
Lemma 6 Let φ and φ be the conjugate pseudo-Anosov automorphisms of a
′
surface X. Then their invariant foliations φ and φ′ are the equivalent mea-
F F
sured foliations.
Proof. Let φ,φ Mod (X) be conjugate, i.e φ = h φ h 1 for an auto-
′ ′ −
∈ ◦ ◦
morphism h Mod (X). Since φ is the pseudo-Anosov automorphism, there
∈
exists a measured foliation , such that φ( ) = λ . Let us evaluate the
φ φ φ φ
F F F
automorphism φ on the foliation h( ):
′ φ
F
φ′(h( φ)) =hφh−1(h( φ)) =hφ( φ)=
F F F
=hλ =λ (h( )). (5)
φ φ φ φ
F F
Thus, φ′ = h( φ) is the invariant foliation for the pseudo-Anosov automor-
F F
phismφ′and φ, φ′ areequivalentfoliations. Notealsothatthepseudo-Anosov
automorphismFφFhas the same dilatation as the automorphism φ. (cid:3)
′
Suppose that φ,φ Mod (X) are the conjugate pseudo-Anosov automor-
′
∈
phisms. The functor F acts by the formulas φ Aφ and φ′ Aφ′, where
7→ 7→
Aφ,Aφ′ are the AF-algebras of the invariant foliations φ, φ′. In view of
F F
lemma 6, φ and φ′ are the equivalent measured foliations. Then, by lemma
F F
4, the AF-algebras Aφ and Aφ′ are the stably isomorphic AF-algebras. The
item (i) follows.
(ii) Let us prove the second statement. We start with an elementary ob-
servation. Let φ Mod (X) be a pseudo-Anosov automorphism. Then there
∈
exists a unique measured foliation, , such that φ( )=λ , where λ >1
φ φ φ φ φ
F F F
10
