Selberg and Ruelle zeta functions for compact
hyperbolic manifolds
Oberseminar Globale Analysis
Summer Semester 2014
September 26, 2014
The dynamical zeta functions are certain functions on complex numbers. As
their name declares, they are functions that can be used as a tool to count the
periodic orbits of a dynamical system.
These dynamical functions become even
more interesting since one can use a geometrical approach to study them. Namely,
we can consider the periodic orbits of a geodesic ow on a compact hyperbolic
manifold.
But, where are these functions coming from?
The mother (or the grandmother, according to D.Ruelle) of all the zeta functions is the Riemann zeta function, dened by
ζRiem (s) =
∞
X
n−s ,
n=1
for Re(s)
> 1.
The Euler product formula gives another expression of
ζRiem (s)
in
terms of prime numbers:
ζRiem (s) =
Y
(1 − p−s )−1 .
(1)
p prime
The analytic properties of the Riemann zeta function are summarized as follows:
The Riemann zeta function admits a meromorphic continuation to the whole com-
C, with a simple pole at s = 1. It has zeros at sk = −2k, k = 1, 2. . . .,
ζRiem (−2k) = 0. These are the trivial zeros of the zeta function. In addition,
ζRiem (s) satises a functional equation relating ζRiem (s) and ζRiem (1 − s).
plex plane
i.e.
The Riemann hypothesis is still an open problem of mathematics. It can be
1
stated as follows: The non-trivial zeros lie on Re(s) = 2 (the critical line).
1
Pólya and Hilbert approached the Riemann hypothesis from a geometrical and
spectral point of view. The idea was to connect the zeros of the zeta function with
the eigenvalues of certain (self-adjoint) operators.
How are the dynamical zeta functions related to the Riemann zeta function?
There is a correspondence
p prime ↔ prime
periodic orbits,
where the prime periodic orbits correspond to, roughly speaking, the closed geodesics
of minimum length
l(γ).
Regarding the Euler product formula (1), the correspon-
dence above is given by
p prime ⇔ el(γ) .
s ∈ C by
Y
R(s) :=
(1 − e−sl(γ) )−1 ,
The Ruelle zeta function is dened on
[γ] prime
for Re(s)
> 1.
The best way to understand this connection is to consider a compact hyperbolic
surface
X,
dened as
X := Γ\H2 ,
where
H2 ∼
= SL2 (R)/ SO(2)
ds2 = dx2 +dy 2 /y 2 ,
and Γ is a discrete torsion-free cocompact subgroup of SL2 (R). For a given γ ∈ Γ
we denote by [γ] the Γ-conjugacy class of γ . The conjugacy class [γ] is called prime
k
if there exist no k > 1 and γ0 ∈ Γ such that γ = γ0 . If γ 6= e, then there is a
unique closed geodesic cγ associated with [γ] of minimum length. Let l(γ) denote
the length of cγ . We mention here that this minimum length corresponds to the
period of a periodic orbit of a geodesic ow on S(X) rather that on X .
The Selberg zeta function is dened on s ∈ C by
is the uper upper half-plane equipped with the Poincaré mettric
Z(s) :=
∞
Y Y
(1 − e−(s+k) l(γ)),
[γ] prime k=0
for Re(s)
> 1.
Using the Selberg trace formula, it can be shown that the Selberg
zeta function has similar analytic properties to the Riemann zeta function.
particular,
C,
Z(s)
admits a meromorphic continuation to the whole complex plane
it satses functional equations and has zeros at
• s = −m,
In
m = 0, 1, 2, . . .
2
• sk =
λn
where
1
2
±i
q
1
4
− λn ,
n = 1, 2, . . .,
are the discrete positive eigenvalues of the Laplace operator
smooth functions on
X.
Clearly, the zeros
sk
∆ acting on
lie on the critical line and hence,
we can consider these tools from spectral analysis as an approach to prove the
Riemann hypothesis. Moreover, by this approach we have a connection beetween
the classical mechanics (geodesic ow) and quantum mechanics (with the Hamil-
tonian
∆).
This connection has been studied in relation with quantum chaos
(see [Rue02]).
Why is it important to study the Ruelle and Selberg zeta functions?
There are several results concerning the value of the Ruelle zeta function at
s=0
and the analytic torsion.
We conisder a more abstract geometrical setting. Let
Hd
d-hyperbolic
be the
space
Hd = {(x1 , . . . , xd ) ∈ Rd+1 : x21 − x22 . . . − x2d+1 = 1, x1 > 0},
where
d = 2n + 1, n ∈ N>0 , is
Rd+1
an odd integer, together with the restriction of the
Minkowski metric from
ds2 = dx21 − dx22 . . . − dx2d+1
Hd . Then, SO0 (d, 1) acts transitively on Hd . The stabilizer of the point
(1, 0, . . . , 0) is SO(d), which is a maximal compact subgroup of SO0 (d, 1). We
0
consider the universal coverings G = Spin(d, 1) and K = Spin(d) of SO (d, 1) and
SO(d) respectively. We set
e := G/K.
X
to
Then,
e∼
X
= Hd .
Γ ⊂ G be a discrete torsion-free cocompact subgroup of G. Then X :=
e is a compact hyperbolic manifold of dimension d and constant
Γ\G/K = Γ\X
e . Let Let G = KAN be the
negative curvature −1, with universal covering X
Iwasawa decomposition of G. The subgroup A of G is a mulitplicative torus of
c
dimension 1, i.e. A ∼
= R+ . We set M := CentrK (A). We conisder the set M
of the equivalence classes of the irreducuble unitary representations (σ, Vσ ) of M .
Let also (χ, Vχ ) be a nite-dimensional representation of Γ. We dene the Selberg
zeta function S(s; σ, χ) and the Ruelle zeta function R(s; σ, χ) on s ∈ C associated
with the representations σ and χ of M and Γ respectively.
c. The twisted Ruelle zeta function R(s; σ, χ) is dened
Denition 1. Let σ ∈ M
Let
by the innite product
R(s; σ, χ) :=
Y
det Id −χ(γ) ⊗ σ(mγ )e−sl(γ)
[γ]6=e
[γ] prime
3
(−1)d−1
,
(2)
with Re(s)
> c,
Denition 2.
for some positive constant
Let
c.
σ∈M
c.
The twisted Selberg zeta function
Z(s; σ, χ)
is dened
by the innite product
∞
Y Y
Z(s; σ, χ) :=
det Id −(χ(γ)⊗σ(mγ )⊗S k (Ad(mγ aγ )n ))e−(s+|ρ|)l(γ) ,
(3)
[γ]6=e k=0
[γ] prime
s ∈ C, n = θn is the sum of the negative root spaces of a, S k (Ad(mγ aγ )n )
denotes the k -th symmetric power of the adjoint map Ad(mγ aγ ) restricted to n, and
ρ is a positive number. It converges absolutely and uniformly on compact subsets
of the half-plane Re(s) > r , where r is a positive constant.
where
In Fried ([Fri86]) the zeta functions have been studied explicitly for a closed ori-
X of dimension d. He considers the standard represenM = SO(d−1) on Λj Cd−1 and an orthogonal representation ρ : Γ → O(m)
−t∆j
, induced by the
of Γ. Using the Selberg trace formula for the heat operator e
Hodge Laplacian ∆j on j -forms on X , he managed to prove the meromorphic continuation of the zeta functions to the whole complex plane C, as well as functional
ented hyperbolic manifold
tation of
equations for the Selberg zeta function ([Fri86, p.531-532]).
He proved also the following theorem, in the case of
ρ
d = dim(X)
being odd and
∗
being acyclic, i.e. the vector spaces of the twisted cohomology classes H (X; ρ)
vanish for all
j.
Theorem 1 ([Fri86, Theorem 1]). Let X = Γ\Hd be a compact hyperbolic manifold
of odd dimension. Assume that
ρ : Γ → O(m)
is acyclic, Then, for Re(s)
> d − 1,
the Ruelle zeta function
R(s; ρ) =
Y
det(1 − ρ(γ)e−sl(γ) )
[γ]6=e,
[γ] prime
admits a meromorphic extension to
C
and for
ε = (−1)d−1
|R(0; ρ)ε | = TX (ρ)2 ,
where
TX (ρ)
is the Ray-Singer analytic torsion dened as in [RS71].
This theorem is of great importance, since it connects the Ruelle zeta function
evaluated at zero with the analytic torsion under certain assumptions.
How can one generalize this result for a non-unitary representation of
case of a compact hyperbolic odd-dimensional manifold?
4
Γ
in the
Wotzke dealt with this conjecture in his thesis ([Wot08]).
he considered a nite-dimensional complex representation
and its restrictions
τ |K
and
τ |Γ
to
K
and
Γ
More specically,
τ : G → GL(V )
of
G
respectively. By [MM63, Proposition
3.1] there exists an isomorphism between the locally homogenous vector bundle
Eτ
over
with
X
τ |Γ
associated with
:
τ |K
and the at vector bundle
Cartan involution of
R(s; τ |Γ )
over
X
associated
Γ\(G/K × V ) ∼
= (Γ\G × V )/K.
τ 6= τθ ,
With the additional assumption that
Theorem 2
Ef l
G,
τθ = τ ◦ θ
and
θ
denotes the
he proved the following theorem.
.
([Wot08, Theorem 8.13])
is regular at
where
(4)
s=0
Let
τ 6= τθ .
Then the Ruelle zeta function
and
R(0; τ |Γ ) = TX (τ |Γ )4 .
For the applications (see [Mül12]), it is important to have results available for
general nite-dimensional representations.
How can one generalize these results for an arbitrary non-unitary representation
of
Γ?
The answer to this question gives rise to further research.
arbitrary nite-dimensional representation
χ : Γ → GL(Vχ )
of
We consider an
Γ.
Our approach
to the problem of proving the meromorphic continuation and functional equations
for both the Selberg and Ruelle zeta functions is dierent from the method of
Wotzke, since we consider an arbitrary representation of
Γ
and we can not apply
the isomorphism (4).
Then, we dene the Selberg and the Ruelle zeta functions associated with
χ
and prove that they converge in some half-plane Re(s)
> c.
σ and
We prove also that
they admit a meromorphic continuation to the whole complex plane and describe
the singularities of the Selberg zeta function in terms of the discrete spectrum of
certain twisted dierential operators on
equations relating their values at
s
X.
Furthermore, we provide functional
with those at
−s.
The main tool that we use
is the Selberg trace formula for integral trace-class operators induced by the at
]
Laplacian ∆ , acting on smooth sections of the twisted vector bundle E ⊗ Eχ , rst
introduced in [Mül11].
τ : K → GL(Vτ ) be a complex nite-dimensional unitary representation of
eτ := G ×τ Vτ → X
e be the associated homogenous vector bundle. Let
K . Let E
Eτ := Γ\(G ×τ Vτ ) → X be the locally homogenous vector bundle. Let ∆τ be the
]
Bochner-Laplace operator associated with τ . We dene the operator ∆τ,χ acting
∞
on C (X, Eτ ⊗ Eχ ). Locally it can be described as
Let
e] = ∆
e τ ⊗ IdVχ ,
∆
τ,χ
5
(5)
where
e]
∆
τ,χ
and
eτ
∆
are the lifts to
e
X
of
∆]τ,χ
and
∆τ ,
respectively.
Contrary to the setting of Wotzke, our operator is not self-adjoint. However, it
]
still has nice spectral properties, i.e. the spectrum of ∆τ,χ is a discrete subset of a
−t∆]τ,χ
positive cone in C. We consider the corresponding heat semi-group e
acting
on the space of smooth sections of the vector bundle
Eτ ⊗ Eχ .
It is an integral
operator with smooth kernel, given by
Hτ,χ (x, y) =
X
Htτ (e
x, γe
y ) ⊗ χ(γ) IdVχ .
γ∈Γ
The kernel function belongs to the Harish-Chandra
Lq -Schwartz
space. We have
Htτ ∈ (C q (G) ⊗ End(Vτ ))K×K
for
t>0
and for every
q > 0.
Hence, we can consider the trace of the operator
]
e−t∆τ,χ
and derive a corresponding
trace formula.
Proposition 1 (Selberg trace formula for non unitary representations).
Let Eχ be
e
a at vector bundle over X = Γ\X , associated with a nite-dimensional complex
]
representation χ : Γ → GL(Vχ ) of Γ. Let ∆τ,χ be the twisted Bochner-Laplace
∞
operator acting on C (X, Eτ ⊗ Eχ ). Then,
Z
X
−t∆]τ,χ
tr Htτ (g −1 γg)dg.
Tr(e
)=
tr χ(γ)
Γ\G
γ∈Γ
]
In fact, we use specic twisted Bochner-Laplace-type operators Aτ,χ (σ), in]
duced by ∆τ,χ . These operators are dened as follows. We want to associate the
Selberg and Ruelle zeta functions with irreducible representations σ of M . We
∗
obtain these representations by the pullback i : R(K) → R(M ) of the embedding
i : M ,→ K , where R(K), R(M ) denote the representation
M , respectively. There are always two distinct cases:
•
case (a): σ
•
case (b): σ
rings over
Z
of
K
is invariant under the action of the restricted Weyl group
and
WA .
is not invariant under the action of the restricted Weyl group
WA .
In both cases we construct a graded vector bundle of the form
E(σ) = ⊕ τ ∈Kb Eτ ,
mτ (σ)
6
mτ (σ) ∈ {−1, 0, 1}. We
sections of E(σ) ⊗ Eχ , given by
where
consider the operator
M
A]τ,χ (σ) :=
A]τ,χ (σ)
acting on smooth
A]τ,χ + c(σ),
mτ (σ)6=0
c(σ)
The operator Aτ,χ
]
is dened in a similar way as the twisted Bochner-Laplace operator ∆τ,χ in (5).
where
is a number dened by the highest weight of
Namely, if we consider the operator
Ω,
σ.
Aτ := −R(Ω) induced by the Casimir element
then
eτ ⊗ IdVχ ,
e]τ,χ = A
A
where
e] , A
eτ
A
τ,χ
denotes the lift to
e
X
of
A]τ,χ , Aτ
respectively. We describe here the
case (a), since the case (b) can be treated similarly. We prove the trace formula
for the corresponding heat semi-group
]
e−tAτ,χ .
Theorem 3 (trace formula for the operator e−tAτ,χ (σ) ).
]
−tA]τ,χ (σ)
Trs (e
Z
) = dim(Vχ ) Vol(X)
For every
c we
σ∈M
have
2
e−tλ Pσ (iλ)dλ
R
2
X l(γ)
e−l(γ) /4t
+
Lsym (γ; σ)
.
nΓ (γ)
(4πt)1/2
[γ]6=e
Let
N ∈ N.
We use the generalized resolvent identity:
N
Y
R(s2i )
=
i=1
N Y
N
X
i=1
1
R(s2i ),
2
2
s
−
s
i
j=1 j
(6)
j6=i
where
R(s2i ) := (A]τ,χ (σ) + s2i )−1 , si ∈ C − spec(A]τ,χ (σ)).
The trace formula
together with this identity are the main tools to prove our results. The proofs of
the meromorphic continuation of the zeta functions are based on the fact that if
−tA]τ,χ (σ)
we insert the trace formula for the operator e
in the integral
N Z
X
i=1
0
∞
Y
N
]
1
2
e−tsi Tr e−tAτ,χ (σ) dt,
2
2
s − si
j=1 j
j6=i
then the integral that includes the term of the hyperbolic contribution is related
to the logarithmic derivative of the Selberg zeta function:
L(s) :=
d
log Z(s; σ, χ).
ds
7
Dχ] (σ) acting on the space C ∞ (X, Eτs (σ) ⊗
Eχ ). Here Eτs (σ) denotes the locally homogenous vector bundle over X , associated
b.
with τs (σ) := s ⊗ τ (σ), where s is the spin representation of K , and τ (σ) ∈ K
We dene the twisted Dirac operator
Locally it can be described as follows
e
e χ] (σ) = D(σ)
⊗ IdVχ ,
D
where
e χ] (σ), D(σ)
e
e
D
are the lifts to X
Dirac operator associated with the
Dχ] (σ), D(σ), respectively, and D(σ) is the
representation τs (σ) of K . We state our main
of
results.
Theorem 4.
Z(s; σ, χ)
admits a meromorphic contin±
uation to the whole complex plane C. The set of the singularities equals {sk =
±iλk : λk ∈ spec(Dχ] (σ)), k ∈ N}. The orders of the singularities are equal to
m(λk ), where m(λk ) ∈ N denotes the algebraic multiplicity of the eigenvalue λk .
For
λ0 = 0,
The Selberg zeta function
the order of the singularity is equal to
Theorem 5.
The Selberg zeta function
Z(s; σ, χ)
= exp
Z(−s; σ, χ)
where
Pσ
Z(s; σ, χ)
2m(0).
satises the functional equation
Z
− 4π dim(Vχ ) Vol(X)
s
Pσ (r)dr ,
(7)
0
denotes the Plancherel polynomial associated with
c.
σ∈M
The proof of this theorem is based again on the generalised resolvent identity
(6) and the connection of the hyperbolic contribution in the trace formula to the
logarithmic derivative
L(s)
of the Selberg zeta function. Then, one can show that
the logarithmic derivative satses the following equation
L(s) + L(−s) = −4π dim(Vχ ) Vol(X)Pσ (s),
and hence obtain (7).
Furthermore, we have the following theorem.
Theorem 6. Let det(A]τ,χ (σ) + s2 ) be the regularized determinant associated to the
operator
A]τ,χ (σ) + s2 .
Z(s; σ, χ) =
Then, the Selberg zeta function has the representation
det(A]τ,χ (σ)
2
+ s ) exp
Z
− 2π dim(Vχ ) Vol(X)
s
Pσ (t)dt .
(8)
0
This determinant formula is crucial for connecting the Ruelle zeta function and
a renement of the analytic torsion, as it is dened in [BK05]. In particular, one
can derive a product formula that provides a representation of the Ruelle zeta
function as a product of Selberg zeta functions with shifted arguments.
8
Theorem 7.
Let
c.
σ∈M
Then the Ruelle zeta function has the representation
R(s; σ, χ) =
d−1
Y
p
Zp (s; σ, χ)(−1) ,
(9)
p=0
where
Zp (s; σ, χ) :=
Y
Z(s + ρ − λ; ψp ⊗ σ, χ).
(ψp ,λ)∈Jp
The above representation of the Ruelle zeta function, given by equation (9),
leads to the meromorphic comtinuation of the Ruelle zeta function.
Theorem 8.
For every
c,
σ∈M
the Ruelle zeta function
morphic continuation to the whole complex plane
R(s; σ, χ)
admits a mero-
C.
By equations (8) and (9), it can be proved that there exists also a determinant
formula for
R(s; σ, χ).
Proposition 2.
The Ruelle zeta function has the representation
R(s; σ, χ) =
d
Y
(−1)p
det(A]τ,χ (σp ⊗ σ) + (s + ρ − λ)2 )
p=0
exp
− 2π(d + 1) dim(Vχ ) dim(Vσ ) Vol(X)s .
(10)
Hence, we are motivated to consider the Ruelle zeta function evaluated at zero
as a candidate for the rened analytic torsion.
References
[BK05]
Maxim Braverman and Thomas Kappeler, A renement of the Ray-Singer
torsion, C. R. Math. Acad. Sci. Paris
[Fri86]
341 (2005), no. 8, 497502.
David Fried, Analytic torsion and closed geodesics on hyperbolic mani-
folds, Invent. Math.
84 (1986), no. 3, 523540.
[MM63] Yozô Matsushima and Shingo Murakami, On vector bundle valued har-
monic forms and automorphic forms on symmetric riemannian manifolds,
Ann. of Math. (2)
78 (1963), 365416.
[Mül11] Werner Müller, A Selberg trace formula for non-unitary twists, Int. Math.
Res. Not. IMRN (2011), no. 9, 20682109.
9
[Mül12]
, The asymptotics of the Ray-Singer analytic torsion of hyperbolic
3-manifolds., Metric and dierential geometry. The Je Cheeger anniversary volume. Selected papers based on the presentations at the international conference on metric and dierential geometry, Tianjin and Beijing,
China, May 1115, 2009, Berlin: Springer, 2012, pp. 317352 (English).
[RS71]
D. B. Ray and I. M. Singer,
R-torsion
manifolds, Advances in Math.
and the Laplacian on Riemannian
7 (1971), 145210.
[Rue02] David Ruelle, Dynamical zeta functions and transfer operators, Notices
Amer. Math. Soc.
49 (2002), no. 8, 887895.
[Wot08] A. Wotzke, Die Ruellesche Zetafunktion und die analytische Torsion hy-
perbolischer Mannigfaltigkeiten, Ph.d thesis, Bonn, Bonner Mathematische Schriften, 2008.
10