# The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and
Reidemeister–Turaev torsion

Léo Bénard, Jan Frahm, Polyxeni Spilioti

###### Abstract

Let $X$ be a compact hyperbolic surface with finite order singularities,
$X_{1}$ its unit tangent bundle. We consider the Ruelle zeta function
$R(s;\rho)$ associated to a representation
$\rho\colon\pi_{1}(X_{1})\to\operatorname{GL}(V_{\rho})$. If $\rho$ does not
factor through $\pi_{1}(X)$, we show that the value at $0$ of the Ruelle zeta
function is the Reidemeister–Turaev torsion of $(X_{1},\rho)$ with respect to
the Euler structure induced by the geodesic flow, thus generalizing Fried’s
conjecture to this setting. We also compute the vanishing order and the
leading coefficient of the Ruelle zeta function at $s=0$ when $\rho$ factors
through $\pi_{1}(X)$.

Keywords — hyperbolic orbisurface, twisted Ruelle zeta function, non-unitary
representation, Reidemeister–Turaev torsion, Selberg trace formula.
Mathematics Subject Classification (2020) — Primary: 11M36; Secondary: 11F72,
37C30, 57Q10.

## Introduction

Given a compact Riemannian manifold $M$, a representation
$\rho\colon\pi_{1}(M)\to\operatorname{GL}(V_{\rho})$ and a vector field
$\mathfrak{X}$ on $M$, the Ruelle zeta function is defined as

$R(s;\rho)=\prod_{\gamma\textup{
prime}}\det\left(\operatorname{Id}-\rho(\gamma)e^{-s\ell(\gamma)}\right),$

where $\gamma$ runs through the _prime_ periodic orbits of $\mathfrak{X}$ and
$\ell(\gamma)$ denote the length of the orbit $\gamma$.

In this work we are interested in the case where $M=X_{1}$ is the unit tangent
bundle of a compact hyperbolic surface $X$ with finitely many singular points
$x_{1},\ldots x_{s}$ of finite order. The geodesic flow $\mathfrak{X}$ acts on
$X_{1}$ and its (prime) periodic orbits are lifts of (prime) closed geodesics
on the surface. For each singular point $x_{j}$ on $X$ there is a class of
loop $c_{j}$ encircling $x_{j}$. It lifts with finite order $\nu_{j}$ to
$\pi_{1}(X_{1})$. We also denote by $t$ the class of the generic fiber. It
represents a loop in $X_{1}$ of a unit vector which makes a full positive
rotation around the base point.

Let $\rho\colon\pi_{1}(X_{1})\to\operatorname{GL}(V_{\rho})$ be a
representation with $\dim V_{\rho}=n$. For any $j=1,\ldots,s$ we denote by
$n_{j}=\dim\operatorname{Fix}\rho(c_{j})$, so that
$\rho(c_{j})=I_{n_{j}}\oplus T_{j}$. Let
$M_{j}(x)=\det(I_{n-n_{j}}-T_{j})(\frac{ix}{\nu_{j}})^{n_{j}}$ and
$M_{0}(x)=(ix)^{n}$ for $x\in\mathbb{C}$.

The main result of this article is the computation of the behavior at zero of
the Ruelle zeta function associated to $\rho$.

######  Theorem A (2.3.1 and 4.3.2).

For any irreducible representation
$\rho\colon\pi_{1}(X_{1})\to\operatorname{GL}(V_{\rho})$, the Ruelle zeta
function $R(s;\rho)$ converges on some right half plane in $\mathbb{C}$ and
extends meromorphically to the whole complex plane. Moreover:

  1. 1.

If $\rho(t)=\operatorname{Id}_{V_{\rho}}$, then $R(s;\rho)$ vanishes at $s=0$
with order and leading coefficient prescribed by:

$R\left(\frac{x}{2\pi},\rho\right)\sim_{x\to
0}\pm\frac{M_{0}(x)^{2g-2+s}}{\prod_{j=1}^{s}M_{j}(x)}.$

  2. 2.

If $\rho(t)\neq\operatorname{Id}_{V_{\rho}}$, then the representation $\rho$
is acyclic. Let $\mathfrak{e}_{geod}$ be the Euler structure induced by the
geodesic flow on $X_{1}$. Then

$R(0;\rho)=\pm\operatorname{tor}(X_{1},V_{\rho},\mathfrak{e}_{geod})$

where $\operatorname{tor}(X_{1},V_{\rho},\mathfrak{e})\in\mathbb{C}^{\times}$
denotes the Reidemeister–Turaev torsion of $X_{1}$ in the representation
$V_{\rho}$ and the Euler structure $\mathfrak{e}_{geod}$ (see below for
details).

When the representation $\rho$ satisfies
$\rho(t)\neq\operatorname{Id}_{V_{\rho}}$ we prove that the twisted homology
groups $H_{*}(X_{1},V_{\rho})$ are trivial (see Section 2) and one can define
a combinatorial invariant, the Reidemeister–Turaev torsion. A theorem of
Chapman (see [3, Theorem 1]) states that it is indeed a topological invariant,
although it depends in general on the choice of an Euler structure (see [27]),
that is, a choice of a lift of each cell of a cell decomposition of $X_{1}$ to
its universal cover $\widetilde{X_{1}}$. For the unit tangent bundle $X_{1}$,
there is a natural choice of an Euler structure $\mathfrak{e}_{geod}$ induced
by the geodesic flow and we compute explicitly in Section 2.3 the
Reidemeister–Turaev torsion
$\operatorname{tor}(X_{1},\rho,\mathfrak{e}_{geod})\in\mathbb{C}$ for this
Euler structure. A (2) says that it coincides with the value at zero of the
Ruelle zeta function, up to sign.

In the case where the representation $\rho$ is unitary, the absolute value of
the Reidemeister torsion is well-defined, independently from an Euler
structure. In this setting A generalizes [10, Theorem 2 and 3], where Fried
proved a similar statement for unitary and _classical_ representation. He
claimed without a proof that his results still hold true for general (non-
classical) unitary representation. Our work fills this gap and extends Fried’s
results to non-unitary representations. Even in the unitary case, we drop the
phase ambiguity for the torsion using the accurate Euler structure and show
that it equals the value of the Ruelle zeta function at zero. The sign
ambiguity for the torsion could also be removed, but not in our computation of
the value $R(0;\rho)$.

Note that the value $s=0$ is not in the convergence domain of the infinite
product $R(s;\rho)$, but as we mentioned above the Ruelle zeta function
extends meromorphically. On the other hand, this function strongly depends on
the metric we fixed on the orbifold $X$ and on the vector field acting on
$X_{1}$. A striking consequence of A (2) is that, for a given hyperbolic
metric on $X$ and for the associated geodesic flow, the value at zero of the
Ruelle zeta function is independent of these choices.

In the case the surface has no elliptic points, A (1) recovers [8, Corollary
C]. Note that A (2) implies that both the Ruelle at $0$ and the torsion are
locally constant on the representation variety, and invariant under the action
of the mapping class group.

##### Relation to previous results.

In a more general setting, metric independence of the value of Ruelle zeta
functions at $0$, and its relation with Reidemeister torsion, has been widely
studied in the past almost 40 years. For $\rho$ unitary, it is known as
Fried’s conjecture and has been settled as a theorem for compact, hyperbolic
odd-dimensional manifolds by Fried in [9], using the Selberg trace formula.
Bunke and Olbrich in [2] unified the treatment for the twisted dynamical zeta
functions with unitary representations, for all locally symmetric spaces of
real rank one, using the Selberg trace formula and tools for harmonic analysis
on locally symmetric spaces. The higher rank case was treated by Moscovici and
Stanton in [16], and Shen in [22]. A similar result, but in variable curvature
for $3$-dimensional manifolds, has been proved by Dang–Guillarmou–Rivière–Shen
in [5], using completely different techniques.

In the case of non-unitary representations $\rho$, fewer is currently known,
despite it is what occurs in many interesting cases, such as holonomy
representations of hyperbolic manifolds. If $\rho$ is obtained as the
restriction of a representation of a Lie group $G$, then Fried’s conjecture
has been established for $G=\operatorname{SL}_{2}(\mathbb{C})$ by Müller [18]
(using the PhD work of Wotzke [29]) and generalized to arbitrary $G$ by Shen
[24]. Moreover, Müller ([19]), Shen ([23]) and the third author ([25]) proved
Fried’s conjecture for more general non-unitary representations, with the
additional assumption that they are close to a unitary and acyclic
representation in the representation variety. In a completely general setting,
Chaubet–Dang established in [4] a variational formula relating the value at
zero of the Ruelle zeta function to the Reidemeister–Turaev torsion. As a
consequence of their work together with our A we have the following corollary:

######  Corollary B.

The statements in A still hold true on an open neighborhood in
$C^{\infty}(X_{1},TX_{1})$ of the geodesic flow $\mathfrak{X}$ of any
hyperbolic metric on X.

In the present paper, the representation $\rho$ is neither required to be
unitary, nor does it have to be close to a unitary acyclic representation. We
even allow representations for which the non-self adjoint Laplacian might have
some generalized $0$-eigenvalues. This is because we do not use the relation
between the Selberg zeta function and the refined analytic torsion. Indeed, we
compute directly the value of the Ruelle zeta function at zero, and identify
it with the Reidemeister torsion.

Recently, Yamaguchi [30] was able to show A (2) in the special case where
$\rho=\rho_{2N}$ is the restriction of the irreducible $2N$-dimensional
representation of $\operatorname{SL}(2,\mathbb{R})$ to $\pi_{1}(X_{1})$. His
proof uses the relation

$R(s;\rho_{2N})=\frac{Z(s-N+\frac{1}{2})}{Z(s+N+\frac{1}{2})}$

between the twisted Ruelle zeta function $R(s;\rho_{2N})$ and the non-twisted
Selberg zeta function $Z(s)$ (associated with the trivial representation of
$\pi_{1}(X_{1})$) and hence does not generalize to arbitrary representations
$\rho$ of $\pi_{1}(X_{1})$.

##### Outline of the proof.

Our proof uses the intimate relation between the twisted Ruelle zeta function
$R(s;\rho)$ and the twisted Selberg zeta function $Z(s;\rho)$ (see Section
4.1):

$R(s;\rho)=\frac{Z(s;\rho)}{Z(s+1;\rho)}.$

The behavior of $Z(s;\rho)$ at $s=0$ and $s=1$ can be studied through its
functional equation relating $Z(s;\rho)$ and $Z(1-s;\rho)$ which we derive in
4.2.1. Both the meromorphic continuation and the functional equation of
$Z(s;\rho)$ are obtained from a twisted Selberg trace formula (see Theorem
3.7.1). This trace formula arises from the trace of the heat operator of
Müller’s twisted Laplacian $\Delta_{\tau,\rho}^{\sharp}$ on $X$ acting on
sections of a certain vector bundle $E_{\tau,\rho}$ over $X$ (see Müller [17,
Section 4] for the definition of the twisted connexion Laplacian operator).
The vector bundle $E_{\tau,\rho}$ over $X$ is associated with the
representation $\rho$ of $\pi_{1}(X_{1})$ and a character $\tau$ of the
universal covering group of $\operatorname{PSO}(2)$. The twist by the
character $\tau$ is necessary in order to obtain a vector bundle over $X$ (see
Section 3.1 for details). We note that $\tau$ can only be trivial in the case
where $\rho$ factors through $\pi_{1}(X)$ and hence this twisted construction
is crucial for A (2).

The technical part of the proof consists of a detailed analysis of the
identity, hyperbolic and elliptic contribution to the geometric side of the
trace formula. This generalizes Hejhal’s trace formula [12, Chapter 9, Theorem
6.2] to the case of non-unitary representations $\rho$ and uses Hoffmann’s
computations of orbital integrals for the universal covering group
$\widetilde{\operatorname{PSL}(2,\mathbb{R})}$ of
$\operatorname{PSL}(2,\mathbb{R})$. The main difficulty is the use of the non-
self adjoint operator $\Delta_{\tau,\rho}^{\sharp}$. This operator is not
necessarily diagonalizable and we have to deal with generalized eigenspaces.
Moreover, the eigenvalues of this operator are not necessarily real and
positive numbers but complex (see Section 3.2 for further details). We remark
that the generalized eigenfunctions of $\Delta_{\tau,\rho}^{\sharp}$ can be
viewed as automorphic forms on $\mathbb{H}^{2}$ of weight $m$ with values in
the non-unitary representation $\rho$ (see [12, Chapter 9], and [13] for the
unitary case).

On the torsion side, the very simple combinatorial nature of the unitary
tangent bundle allows to compute explicitly the Euler structure
$\mathfrak{e}_{geod}$ induced by the geodesic flow, and the
Reidemeister–Turaev torsion in this Euler structure. It relies on Turaev’s
correspondence between _combinatorial_ Euler structures, which yield a choice
of lifts of a cell decomposition of $X_{1}$ to its universal cover necessary
to compute the torsion, and _smooth_ Euler structures, given by non-vanishing
vector fields on $X_{1}$. To our best knowledge, it is the only case this
computation has been performed explicitly, since this correspondence is not
very explicit in the direction we use. Namely, given a non-vanishing vector
field on a 3-manifold $M$, it looks in general like a difficult task to find
the corresponding combinatorial Euler structure.

Finally, the last part of the proof consists of an explicit computation of the
Ruelle zeta function at $s=0$. It follows the ideas of [10, Section 3], where
Fried computed the modulus of each term contributing to $R(0;\rho)$ from the
functional equation of the Selberg zeta function, in the case $\rho$ was
unitary. In this case the Ruelle zeta function is real for $s$ real, hence he
argued that he could forget the arguments of the terms occurring in the
computation. Of course, in our case where $\rho$ is no longer unitary, these
are exactly what we have to compute, and it turns out that after tedious
computations (Section 4.3) they kind of miraculously cancel out with each
other.

##### Organization of the paper.

The paper is organized as follows. In Section 1, we introduce our geometric
setting, recall some facts from the representation theory of
$\widetilde{\operatorname{PSL}(2,\mathbb{R})}$ and fix notation. In Section 2,
we define and compute the Reidemeister–Turaev torsion of the unit tangent
bundle $X_{1}$ of the hyperbolic orbifold $X$ in the Euler structure induced
by the geodesic flow. In Section 3, we establish the Selberg trace formula for
the twisted, non-self adjoint Laplacian on $X$, which will be the key point to
prove A. In Section 4, we introduce the twisted Ruelle and Selberg zeta
functions and prove their meromorphic continuation to the whole complex plane.
Finally, we compute the value of the Ruelle zeta function at zero and prove A.

##### Acknowledgments.

The authors would like to thank Adrien Boulanger, François Costantino, Nguyen
Viet Dang, Pierre Dehornoy, Werner Müller, Joan Porti and Louis-Hadrien Robert
for helpful discussions and comments. The first author is partially funded by
the Research Training Group 2491 "Fourier Analysis and Spectral Theory",
University of Göttingen. The second and third author were supported by a
research grant from the Villum Foundation (Grant No. 00025373).

##  1 Preliminaries

We collect some basic information about compact hyperbolic orbisurfaces $X$
(Section 1.1), their unit tangent bundles $X_{1}$ and their fundamental groups
$\pi_{1}(X_{1})$ (Section 1.2), and recall some representation theory of the
universal cover of $\operatorname{PSL}(2,\mathbb{R})$ (Section 1.3).

###  1.1 Compact hyperbolic orbisurfaces

We consider the upper half plane

$\mathbb{H}^{2}:=\\{z=x+iy:y>0\\}.$

The group $G=\operatorname{PSL}(2,\mathbb{R})$ acts on $\mathbb{H}^{2}$ by
fractional linear transformations. This action is transitive and the
stabilizer of $i\in\mathbb{H}^{2}$ is the maximal compact subgroup
$K=\operatorname{PSO}(2)$, hence

$\mathbb{H}^{2}\cong G/K.$

Let $\mathfrak{g}=~{}\mathfrak{sl}(2,\mathbb{R})$ be the Lie algebra of $G$
and let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be the Cartan
decomposition of $\mathfrak{g}$ with respect to the maximal compact subgroup
$K$. The $G$-invariant metric on $\mathbb{H}^{2}$ induced by the restriction
of the Killing form on $\mathfrak{g}$ to $\mathfrak{p}$ is the Poincaré metric

$ds^{2}=\frac{dx^{2}+dy^{2}}{y^{2}}.$

Let $\Gamma\subseteq G$ be a cocompact Fuchsian group, i.e., a discrete
subgroup of $G$ such that the quotient
$X=\Gamma\backslash\mathbb{H}^{2}=\Gamma\backslash G/K$ is compact. Then, the
Poincaré metric on $\mathbb{H}^{2}$ induces a metric on $X$ that turns it into
a compact hyperbolic orbisurface. Note that in this case, every non-trivial
element in $\Gamma$ is either hyperbolic or elliptic.

Let $\widetilde{G}$ denote the universal cover of $G$ and write
$\widetilde{H}$ for the preimage of a subgroup $H\subseteq G$ under the
universal covering map. Consider the following one-parameter families in $G$
(modulo $\pm I_{2}$):

$k_{\theta}=\begin{pmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{pmatrix},\qquad
a_{t}=\begin{pmatrix}e^{\frac{t}{2}}&0\\\
0&e^{-\frac{t}{2}}\end{pmatrix},\qquad n_{x}=\begin{pmatrix}1&x\\\
0&1\end{pmatrix},$

where $\theta,t,x\in\mathbb{R}$. Denote by $\widetilde{k}_{\theta}$,
$\widetilde{a}_{t}$ and $\widetilde{n}_{x}$ the unique lifts to the universal
cover $\widetilde{G}$, turning

$\widetilde{K}=\\{\widetilde{k}_{\theta}:\theta\in\mathbb{R}\\},\qquad\widetilde{A}=\\{\widetilde{a}_{t}:t\in\mathbb{R}\\}\qquad\mbox{and}\qquad\widetilde{N}=\\{\widetilde{n}_{x}:x\in\mathbb{R}\\}$

into connected one-parameter subgroups of $\widetilde{G}$. Then the Iwasawa
decomposition

$\widetilde{G}=\widetilde{N}\widetilde{A}\widetilde{K}$

holds. Note that the center $\widetilde{Z}$ of $\widetilde{G}$ is given by

$\widetilde{Z}=\\{\widetilde{k}_{\theta}:\theta\in\mathbb{Z}\pi\\}.$

###  1.2 The relation between $\pi_{1}(X_{1})$ and $\pi_{1}(X)$.

The aim of this section is to shed light on the relations between the
fundamental group of the orbifold surface $X$ and the fundamental group of its
unit tangent bundle $X_{1}$. For this, we follow [10, Section 2].

We first note that the unit tangent bundle of $\mathbb{H}^{2}$ naturally
identifies with $G=\operatorname{PSL}(2,\mathbb{R})$ by the map

$\displaystyle G$ $\displaystyle\to\mathbb{H}^{2}$ $\displaystyle g$
$\displaystyle\mapsto g\cdot\textrm{i}.$

It follows from the preceding discussion that the quotient $\Gamma\backslash
G$ is a compact 3-manifold $X_{1}$, which naturally identifies with the unit
tangent bundle of $X$. This fibration induces an exact sequence

$1\to\mathbb{Z}=\pi_{1}(\operatorname{PSO}(2))\to\pi_{1}(X_{1})\to\Gamma=\pi_{1}(X)\to
1.$

In fact, a compact orbisurface $X$ must have finitely many conical points
$x_{1},\ldots x_{s}$. Removing small disc neighborhoods $D_{1},\ldots D_{s}$
of those points, one gets a compact surface $\overline{X}$ with boundary
$\partial\overline{X}=\\{\partial D_{1},\ldots\partial D_{s}\\}$ a union of
disjoints circles, and with fundamental group

$\pi_{1}(\overline{X})=\langle a_{1},b_{1},\ldots,a_{g},b_{g},c_{1},\ldots
c_{s}\mid\prod_{i}[a_{i},b_{i}]\prod_{j}c_{j}=1\rangle.$

Capping off the boundary components $\partial D_{i}$ by orbidiscs, one gets
the following presentation for the fundamental group of the orbisurface:

$\pi_{1}(X)=\langle a_{1},b_{1},\ldots,a_{g},b_{g},c_{1},\ldots
c_{s}\mid\prod_{i}[a_{i},b_{i}]\prod_{j}c_{j}=1,\,c_{j}^{\nu_{j}}=1\rangle,$

for some non-zero natural integers $\nu_{j},\,i=1,\ldots s$. Those integers
are the order of the elliptic conjugacy classes in $\Gamma$ corresponding to
the singular points $x_{i}$.

To compute the fundamental group of the unitary tangent bundle $X_{1}$ of $X$,
let us denote by $\langle t\rangle$ the fundamental group of the fiber
$\operatorname{PSO}(2)$ which identifies with the center of $\widetilde{G}$.
The generator $t=\widetilde{k}_{\pi}$ identifies with a loop in $X_{1}$
corresponding to a complete clockwise rotation of unit vectors around the base
point in $X$. It is proved in [10, Section 2] that one gets the presentation

(1.1) $\pi_{1}(X_{1})=\langle a_{1},b_{1},\ldots,a_{g},b_{g},c_{1},\ldots
c_{s},t\mid\\\
[t,a_{i}]=1,[t,b_{i}]=1,[t,c_{j}]=1,\,\prod_{i}[a_{i},b_{i}]\prod_{j}c_{j}=t^{2g-2+s},\,c_{j}^{\nu_{j}}=t\rangle.$

Now, let $\rho:\pi_{1}(X_{1})\to\operatorname{GL}(V_{\rho})$ be a finite-
dimensional complex representation of $\pi_{1}(X_{1})$. Since
$t\in\pi_{1}(X_{1})$ is central, $\rho(t)$ commutes with $\rho(\gamma)$ for
all $\gamma\in\pi_{1}(X_{1})$. If $\rho$ is irreducible, $\rho(t)$ must be a
scalar multiple of the identity by Schur’s Lemma, say
$\rho(t)=\lambda\,I_{n}$. We show that $\lambda$ is in fact a root of unity.
For this, let $N=\operatorname{lcm}(1,\nu_{1},\ldots,\nu_{s})$ be the least
common multiple of $1$ and the orders of the elliptic conjugacy classes. The
orbifold Euler characteristic $\chi(X)$ of $X$ is given by

$\chi(X)=2-2g+\sum_{j=1}^{s}\left(\frac{1}{\nu_{j}}-1\right)\in\mathbb{Q}.$

######  Lemma 1.2.1.

Let $\rho\colon\pi_{1}(X_{1})\to\operatorname{GL}(V)$ be an irreducible
representation of dimension $n$. Then, $\rho(t)=\lambda\,I_{n}$ with
$\lambda^{Nn\chi(X)}=1$

###### Proof.

Since $c_{j}^{\nu_{j}}=t$ we find $\rho(c_{j})^{\nu_{j}}=\lambda\,I_{n}$, in
particular

$\prod_{j}\rho(c_{j})^{N}=\prod_{j=1}^{s}\lambda^{N/\nu_{j}}\,I_{n}.$

On the other hand,

$\prod_{j=1}^{s}\rho(c_{j})=\lambda^{2g-2+s}\prod_{i=1}^{g}[\rho(a_{i}),\rho(b_{i})].$

Since $\det(\prod_{i=1}^{g}[\rho(a_{i}),\rho(b_{i})])=1$, we deduce
$\lambda^{Nn\left(\sum_{j=1}^{s}1/\nu_{j}-(2g-2+s)\right)}=1$ and the lemma
follows. ∎

###  1.3 Representation theory of
$\widetilde{\operatorname{PSL}(2,\mathbb{R})}$

We briefly recall the principal series and the (relative) discrete series of
the universal covering group of $\operatorname{PSL}(2,\mathbb{R})$, following
[13, Section 1].

Let $\widetilde{M}=\widetilde{Z}$ denote the center of $\widetilde{G}$, then
$\widetilde{N}\widetilde{A}\widetilde{M}$ is a parabolic subgroup of
$\widetilde{G}$. The unitary dual of $\widetilde{K}\simeq\mathbb{R}$ is
comprised of the unitary characters $\tau_{m}$, $m\in\mathbb{R}$, defined by

(1.2)
$\tau_{m}(\widetilde{k}_{\theta})=e^{im\theta}\qquad(\theta\in\mathbb{R}).$

The restriction $\sigma_{\varepsilon}$ of $\tau_{m}$ to $\widetilde{M}$ only
depends on $\varepsilon=m+2\mathbb{Z}\in\mathbb{R}/2\mathbb{Z}$, and the
unitary dual of $\widetilde{M}$ is given by all $\sigma_{\varepsilon}$,
$\varepsilon\in\mathbb{R}/2\mathbb{Z}$.

For $\sigma=\sigma_{\varepsilon}$, $\varepsilon\in\mathbb{R}/2\mathbb{Z}$, and
$s\in\mathbb{C}$ we form the principal series representation $\pi_{\sigma,s}$
of $\widetilde{\operatorname{PSL}(2,\mathbb{R})}$ on

$I_{\sigma,s}=\\{f\in
C^{\infty}(\widetilde{G}):f(na_{t}mg)=\sigma(m)e^{st}f(g)\\},$

acting by right translation. The $\widetilde{K}$-types in $I_{\sigma,s}$ are
spanned by the functions $\phi_{m}$ with $m+2\mathbb{Z}=\varepsilon$, where

$\phi_{m}(na_{t}k)=e^{st}\tau_{m}(k_{\theta})\qquad(n\in\widetilde{N},t\in\mathbb{R},k\in\widetilde{K}).$

Note that the Casimir element $\Omega=\frac{1}{4}(H^{2}+2EF+2FE)$ with

$H=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},\qquad E=\begin{pmatrix}0&1\\\
0&0\end{pmatrix},\qquad F=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}$

acts in $\pi_{\sigma,s}$ by $s(s-1)\operatorname{Id}$.

##### Unitary principal series.

For $s=\frac{1}{2}+i\lambda$, $\lambda\in\mathbb{R}$, the representation
$\pi_{\sigma,s}$ extends to a unitary representation on the Hilbert space

$\left\\{f:\widetilde{G}\to\mathbb{C}:f(na_{t}mg)=\sigma(m)e^{(i\lambda+\frac{1}{2})t}f(g),\int_{\widetilde{M}\backslash\widetilde{K}}|f(k)|^{2}\,dk<\infty\right\\},$

the _unitary principal series_. We denote the distribution character of this
representation by $\Theta_{\sigma,\lambda}$.

##### (Relative) discrete series.

For $m>0$, the $\widetilde{K}$-types $\phi_{m^{\prime}}$,
$m^{\prime}\in\pm\\{m,m+2,m+4,\ldots\\}$, span an
$\operatorname{\mathfrak{sl}}(2,\mathbb{C})$-invariant subspace of
$I_{\sigma_{\varepsilon},\frac{m}{2}}$ where $\varepsilon=\pm
m+2\mathbb{Z}\in\mathbb{R}/2\mathbb{Z}$, which can be completed to a Hilbert
space of a unitary representation of $\widetilde{G}$, the _(relative) discrete
series_. We write $\Theta_{\pm m}$ for the distribution character of this
representation. Details about the invariant inner product on the (relative)
discrete series can be found in [13, Section 1].

##  2 Reidemeister torsion

In this section, we compute the Reidemeister–Turaev torsion of the unit
tangent bundle $X_{1}$ of a hyperbolic orbisurface $X$ for the Euler structure
given by the geodesic flow. In Section 2.1, we define the Reidemeister–Turaev
torsion. In Section 2.2, we shortly describe the two equivalent notions of
Euler structures introduced by Turaev. We compute explicitly the Reidemeister-
Turaev torsion in the Euler structure given by the geodesic flow in Section
2.3.

###  2.1 Twisted homology, Reidemeister–Turaev torsion and Euler structures

In this section, we define the twisted homology and Reidemeister torsion of a
finite CW-complex $W$ with a representation
$\rho\colon\pi_{1}(W)\to\operatorname{GL}(V_{\rho})$ and an Euler structure
$\mathfrak{e}$. References include [21, 26, 27].

We will denote the fundamental group $\pi_{1}(W)$ by $\pi$. Let
$\widetilde{W}$ be the universal cover of $W$. It has the structure of an
(infinite) CW-complex, whose cells can be listed as follows: if
$\\{c^{i}_{1},\ldots,c^{i}_{k_{i}}\\}$ are the $i$-dimensional cells of $W$,
then
$\\{\pi\cdot\widetilde{c}^{i}_{1},\ldots,\pi\cdot\widetilde{c}^{i}_{k_{i}}\\}$
is the list of the $i$-dimensional cells of $\widetilde{W}$, where
$\pi\cdot\widetilde{c}^{i}_{j}$ denotes the set
$\\{\gamma\cdot\widetilde{c}^{i}_{j}\mid\gamma\in\pi\\}$, see an example in
Fig. 1.

$W$$c^{0}$$c^{1}$$c^{0}$$t^{2}\cdot\widetilde{c}^{0}$$t\cdot\widetilde{c}^{0}$$\widetilde{c}^{0}$$t^{-1}\cdot\widetilde{c}^{0}$$\widetilde{c}^{1}$$t^{-1}\cdot\widetilde{c}^{1}$$t\cdot\widetilde{c}^{1}$$\widetilde{W}$
Figure 1:  In this example, the CW complex $W$ is a circle, with one $0$-cell
$c^{0}$ and one $1$-cell $c^{1}$. Its universal cover $\widetilde{W}$ is the
real line, with the action of the fundamental group $\pi_{1}(W)=\langle
t\rangle$.

It endows $C_{*}(\widetilde{W})$ with the structure of a complex of
$\mathbb{Z}[\pi]$-modules of finite rank, with a basis given by a choice of a
lift $\widetilde{c}^{i}_{j}$ for each cell $c^{i}_{j}\in C_{*}(W)$. Define the
complex

$C_{*}(W,V_{\rho})=V_{\rho}\otimes_{\mathbb{Z}[\pi]}C_{*}(\widetilde{W}).$

Given $\\{v_{1},\ldots,v_{n}\\}$ a basis of $V_{\rho}$, the set
$\\{v_{1}\otimes\widetilde{c}^{i}_{1},v_{2}\otimes\widetilde{c}^{i}_{1}\ldots,v_{n-1}\otimes\widetilde{c}^{i}_{k_{i}},v_{n}\otimes\widetilde{c}^{i}_{k_{i}}\\}$
gives a basis of the vector space $c_{j}(W,V_{\rho})$ for each $i$, and the
boundary operator is given by $\textrm{id}\otimes\partial$.

######  Example 2.1.1.

In the example of Fig. 1, given a representation
$\rho\colon\pi_{1}(\mathbb{S}^{1})\to\operatorname{GL}(V_{\rho})$, the complex
is just

$C_{1}(\mathbb{S}^{1},V_{\rho})\simeq V_{\rho}\to V_{\rho}\simeq
C_{0}(\mathbb{S}^{1},V_{\rho})$

and the boundary map is
$\partial(v\otimes\widetilde{c}^{1})=v\otimes(t-1)\cdot\widetilde{c}^{0}=(\rho(t)-I_{n})v\otimes\widetilde{c}^{0}$.
In the case $\det(\rho(t)-I_{n})\neq 0$, the boundary map $\partial\colon
C_{1}(\mathbb{S}^{1},V_{\rho})\to C_{0}(\mathbb{S}^{1},V_{\rho})$ is an
isomorphism. The twisted homology vector spaces
$H_{*}(\mathbb{S}^{1},V_{\rho})$ are then trivial, and the complex
$C_{*}(\mathbb{S}^{1},V_{\rho})$ is said to be _acyclic_.

######  Remark 2.1.2.

In Example 2.1.1, the identification $C_{j}(\mathbb{S}^{1},V_{\rho})\simeq
V_{\rho}$ is not canonical, it depends on the choice of a lift of the cell
$c^{i}$. Different choices would modify the boundary map $\partial$ by a
factor $\rho(t^{m})$, for some $m\in\mathbb{Z}$. On the other hand, the
acyclicity property does of course not depend on the choices.

In general, given an acyclic based complex $(C_{*},\bf{c^{*}})$ of vector
spaces over a field $\mathbb{K}$, where for each $i$, the set $\bf{c^{i}}$ is
a basis of the vector space $C_{j}$, then the Reidemeister torsion is defined
as follows. The boundary map yields exact sequences

$0\to Z_{i}\to C_{i}\xrightarrow{\partial_{i}}B_{i-1}\to 0.$

Because the complex is acyclic, one has $B_{i}=Z_{i}$ for each $i$. If one
picks arbitrary bases $\bf{b^{i}}$ of the spaces $B_{i}$ and arbitrary lifts
$\bf{\overline{b}^{i}}$ in $C_{i}$, one obtains new bases
$\bf{b^{i}}\sqcup\bf{\overline{b}^{i-1}}$ of $C_{i}$ for each $i$. We denote
by $[\bf{b^{i}}\sqcup\bf{\overline{b}^{i-1}}\colon\bf{c^{i}}]$ the determinant
of the change of basis matrix, the Reidemeister torsion is defined by the
alternating product

$\operatorname{tor}(C_{*},\mathbf{c^{*}})=\prod_{i}[\mathbf{b^{i}}\sqcup\mathbf{\overline{b}^{i-1}}\colon\mathbf{c^{i}}]^{(-1)^{i}}\in\mathbb{K}^{*}/\\{\pm
1\\}.$

It is defined up to sign, it does depend on the basis $\bf{c^{*}}$, but not on
the choices $\bf{b^{*}}$ nor on the lifts $\bf{\overline{b}^{i}}$.

In the case of a CW-complex $W$ with a representation
$\rho\colon\pi_{1}(W)\to\operatorname{GL}(V_{\rho})$, if the complex
$C_{*}(W,V_{\rho})$ is acyclic, one can compute the Reidemeister torsion after
making a choice of lift for each cell of $W$ in $\widetilde{W}$. If we denote
such a choice by $\mathfrak{e}$ (for further details about the definition of
the Euler structure $\mathfrak{e}$, see Section 2.2), then the torsion is
denoted by $\operatorname{tor}(W,\rho,\mathfrak{e})$. Note that the torsion
does not depend on the choice of basis of $V_{\rho}$.

######  Example 2.1.3.

For the circle, with the choice of lifts $\widetilde{c}^{0}$ and
$\widetilde{c}^{1}$ we made in Fig. 1, the torsion equals
$\pm\frac{1}{\det(\rho(t)-I_{n})}$. But if one moves the lift
$\widetilde{c}^{1}$ to $t^{m}\cdot\widetilde{c}^{1}$, then the torsion will be
multiplied by $\det\rho(t^{m})$. Nevertheless, what we computed is a natural
choice of torsion: one chooses _any_ lift $\widetilde{c}^{0}$ of the base
point, and then take $\widetilde{c}^{1}$ as on Fig. 1, taking the edge that
goes out of $\widetilde{c}^{0}$ following the orientation. We denote this
Euler structure on the circle by $\mathfrak{e}_{\circ}$.

###  2.2 Combinatorial and smooth Euler structures

In [26], Turaev investigated the concept of Euler structure. The main
achievement is to show that Euler structures can be equivalently defined in
two different ways. Originally Turaev’s description uses simplicial
structures, we slightly reformulate it in the setting of CW complexes.

An Euler chain in a CW complex $W$ with vanishing Euler characteristic is a
choice of a path $c$ in the 1-skeleton $W^{\prime}_{(1)}$ of the barycentric
sub-complex $W^{\prime}$ of $W$, whose boundary is

$\partial c=\sum_{\sigma\in W}(-1)^{\dim\sigma}[\sigma]\in W^{\prime}_{(0)}.$

Two such chains $c$ and $c^{\prime}$ are equivalent, provided their difference
$c-c^{\prime}$ is a boundary. An equivalence class $\mathfrak{e}$ of an Euler
chain is a _combinatorial_ Euler structure.

######  Remark 2.2.1.

If one fixes a $0$-cell $x$ in $W$, an Euler structure defines a path from $x$
to any cell $\sigma$ in $W$. In turn, it defines a lift $\widetilde{\sigma}$
of each cell $\sigma$ to the universal cover $\widetilde{W}$ of $W$, and
allows to define unambiguously the Reidemeister–Turaev torsion as in Section
2.1. Note that two homologous Euler structures will not necessarily define the
same lifts in $\widetilde{W}$, nevertheless the value of the torsion will not
be affected, since it is multiplied by the determinant of an element of the
commutator subgroup $\ker(\pi_{1}(W)\to H_{1}(W))$.

On the other hand, a _smooth_ Euler structure is a homology class of nowhere
vanishing vector fields. Two nowhere vanishing vector fields $\mathfrak{X}$
and $\mathfrak{X}^{\prime}$ on a 3-manifold $X$ are homologous if there exists
a ball $D\subset X$ such that $\mathfrak{X}$ and $\mathfrak{X}^{\prime}$ are
homotopic in $X\setminus D$ as nowhere vanishing vector fields.

The obstruction for two vector fields $\mathfrak{X}$ and
$\mathfrak{X}^{\prime}$ of being homologous is the Chern–Simons class
$cs(\mathfrak{X},\mathfrak{X}^{\prime})\in H_{1}(X)$. It can be defined a
follows: denote by $p\colon X\times[0,1]\to X$ the projection, and choose
$\mathbb{X}$ a smooth section of the bundle $p^{*}TX\to X\times[0,1]$ which
provides a homotopy between $\mathfrak{X}=\mathbb{X}|_{X\times\\{0\\}}$ and
$\mathfrak{X}^{\prime}=\mathbb{X}|_{X\times\\{1\\}}$ and which is transverse
to the zero section. The intersection of $\mathbb{X}$ with the zero section
projects onto a one-dimensional submanifold in $X$, whose homology class is
$cs(\mathfrak{X},\mathfrak{X}^{\prime})$. It can be shown that it only depends
on the homology classes of $\mathfrak{X}$ and $\mathfrak{X}^{\prime}$.

It turns out that both sets of combinatorial and smooth Euler structures are
affine sets on $H_{1}(X)$, and Turaev shows in [26, Section 6] that there is
an $H_{1}(X)$-equivariant isomorphism from one to the other.

###  2.3 The torsion of the unit tangent bundle

Let $X$ be a surface of genus $g$ with $s$ orbifold singularities, such that
$2g-2+s>0$. Let $X_{1}$ be the unit tangent bundle of the orbisurface $X$. We
denote by $c_{1},\ldots c_{s}\in\pi_{1}(X_{1})$ the elliptic elements given by
loops in $X$ around the singularities.

The aim of this section is to prove the following theorem.

######  Theorem 2.3.1.

Let $\rho\colon\pi_{1}(X_{1})\to\operatorname{GL}(V_{\rho})$ be an irreducible
representation of dimension $n$, with $\rho(t)\neq I_{n}$. Then $\rho$ is
acyclic. For any hyperbolic metric on $X$, the geodesic flow on $X_{1}$
induces an Euler structure $\mathfrak{e}_{geod}$ for which

$\operatorname{tor}(X_{1},V_{\rho},\mathfrak{e}_{geod})=\pm\frac{\det(\rho(t)-I_{n})^{2g+s-2}}{\prod_{j=1}^{s}\det(\rho(c_{j})-I_{n})}.$

The proof of this theorem will occupy the rest of this section. It is
subdivided into two parts: first in Section 2.3.1, we decompose the unit
tangent bundle $X_{1}$ into simpler pieces and exhibit a cell decomposition of
each piece together with a natural choice of lifts of the cells in the
universal cover. We prove 2.3.1 for this choice. In the second part Section
2.3.2, we show that this natural choice of lifts corresponds to the Euler
structure induced by the geodesic flow induced by any hyperbolic metric.

####  2.3.1 Computation of the torsion

The 3-manifold $X_{1}$ can be cut along tori into pieces

(2.1) $X_{1}=X_{0}\cup N_{0}\cup N_{1}\cup\ldots\cup N_{s},$

where

  * •

$X_{0}\simeq(\overline{X}\setminus D_{0})\times S^{1}$ is the product of a
surface with $s+1$ boundary components with a circle;

  * •

$N_{0}$ is a solid torus $D_{0}\times S^{1}$;

  * •

$N_{i}$ are solid tori containing the exceptional fibers which correspond to
the elliptic elements $c_{j}$.

We will use the following cell decomposition of $X_{0}$: the surface
$\overline{X}\setminus D_{0}$ retracts on a bouquet of $2g+s$ circles, each of
which can be identified with one of $a_{1},\ldots,b_{g},c_{1},\ldots,c_{s}$.
We denote the circles by $A_{1},B_{1},\ldots A_{g},B_{g},C_{1},\ldots,C_{s}$,
each corresponding to the element of the fundamental group of
$\overline{X}\setminus D_{0}$ denoted by the same tiny letter. Now the product
cell structure on $X_{0}$ is given by

  * •

One $0$-cell $v$;

  * •

$2g+s+1$ one-cells $A_{1},B_{1},\ldots A_{g},B_{g},C_{1},\ldots,C_{s}$ and $T$
(corresponding to the circle factor);

  * •

$2g+s$ two-cells $E_{A_{1}},\ldots,E_{B_{g}},E_{C_{1}},\ldots E_{C_{s}}$, such
that $\partial E_{\gamma}=\gamma T\gamma^{-1}T^{-1}$ for any
$\gamma\in\\{A_{1},\ldots,C_{s}\\}$.

##### Natural lift of the cell structure.

We pick an arbitrary basis $\\{e_{1},\ldots,e_{n}\\}$ of $V_{\rho}$, then the
complex $C_{*}(X_{0},V_{\rho})$ can be computed explicitly, as in [14, Section
4], but we need to fix lifts of the cells of $X_{0}$ in the universal cover
$\widetilde{X}_{0}$. Since $X_{0}$ has the structure of a bouquet of circles
times a circle, its universal cover has the structure of a family of lattices,
a portion of one of them is drawn in Fig. 2.

$\widetilde{v}$
---
$\widetilde{T}$
---
$\widetilde{E}_{A_{1}}$
---
$\widetilde{A}_{1}$
---
$a_{1}\cdot\widetilde{T}$
---
$a_{1}t^{-1}\cdot\widetilde{T}$
---
$t^{-1}\cdot\widetilde{E}_{A_{1}}$
---
$t^{-1}\cdot\widetilde{T}$
---
Figure 2:  A sample of the universal cover of $X_{0}$. Here we draw part of a
lattice corresponding to the universal cover of the two-cell $E_{A_{1}}$. The
Euler structure $\mathfrak{e}_{0}$ is defined by taking the lifts
$\widetilde{v},\widetilde{T},\widetilde{A}_{1}$ and $\widetilde{E}_{A_{1}}$
for each such two-dimensional cell.

We start with an arbitrary lift $\widetilde{v}$ of the unique one-cell $v\in
C_{0}(X_{0})$. Then we choose the lifts $\widetilde{A}_{1},\widetilde{T}$ and
$\widetilde{E}_{A_{1}}$ as in Fig. 2. Similarly, we take the lifts of
$B_{1},A_{2},\ldots,C_{s}$ and of the corresponding two-cells as above.

The set $\\{e_{i}\otimes\widetilde{E}_{\gamma}\mid i=1,\ldots
n,\,\gamma=A_{1},\ldots C_{s}\\}$ is now a basis of $C_{2}(X_{1},V_{\rho})$.
Similarly, $\\{e_{i}\otimes\gamma\mid i=1,\ldots
n,\,\gamma=\widetilde{A}_{1},\ldots\widetilde{C}_{s},\widetilde{T}\\}$ is a
basis of $C_{1}(X_{1},V_{\rho})$ and $\\{e_{i}\otimes\widetilde{v}\mid
i=1,\ldots n\\}$ is a basis of $C_{0}(X_{1},V_{\rho})$.

The boundary operator $\partial_{2}\colon C_{2}(X_{1},V_{\rho})\to
C_{1}(X_{1},V_{\rho})$ reads

$\partial_{2}(e_{i}\otimes
E_{A_{1}})=e_{i}\otimes\left((1-t)\cdot\widetilde{A}_{1}+(a_{1}-1)\cdot\widetilde{T}\right),$

and similarly for other two-cells. Further,
$\partial_{1}(e_{i}\otimes\widetilde{A}_{1})=e_{i}\otimes(a_{1}-1)\widetilde{v}$
and similarly for the other one-cells. Hence, in the given bases, the boundary
maps are given by the following matrices

$\partial_{2}=\begin{pmatrix}I_{n}-\rho(t)&0&\ldots&\ldots&0\\\
0&I_{n}-\rho(t)&0&\ldots&0\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&\ldots&\ldots&I_{n}-\rho(t)\\\
\rho(a_{1})-I_{n}&\ldots&\ldots&\ldots&\rho(c_{s})-I_{n}\end{pmatrix},$
$\partial_{1}=\begin{pmatrix}\rho(a_{1})-I_{n}&\ldots&\ldots&\ldots&\rho(c_{s})-I_{n}&\rho(t)-I_{n}\end{pmatrix}.$

We start with the following lemma:

######  Lemma 2.3.2.

Let $\rho\colon\pi_{1}(X_{1})\to\operatorname{GL}(V_{\rho})$ be an
$n$-dimensional irreducible representation. If $\det(\rho(t)-I_{n})\neq 0$,
then the complex $C_{*}(X_{1},V_{\rho})$ is acyclic.

###### Proof.

For $i=0,\ldots,s$, the solid torus $N_{i}$ has the homotopy type of a circle,
hence the complex $C_{*}(N_{i},V_{\rho})$ is just

$C_{1}(N_{i},V_{\rho})\simeq
V_{\rho}\xrightarrow{\rho(c_{i})-I_{n}}V_{\rho}\simeq C_{0}(N_{i},V_{\rho})$

where $\rho(c_{0})$ is understood as being $\rho(t)$. Since
$\rho(c_{i})^{\nu_{i}}=\rho(t)$, the assumption $\det(\rho(t)-I_{n})\neq 0$
implies $\det(\rho(c_{i})-I_{n})\neq 0$ for any $i$. Hence, we showed that
$C_{*}(N_{i},V_{\rho})$ is acyclic when $\det(\rho(t)-I_{n})\neq 0$. With the
same argument, one shows that $C_{*}(\partial N_{i},V_{\rho})$ is acyclic for
any $i$.

Now we prove that $C_{*}(X_{0},V_{\rho})$ is acyclic. First, the map
$\partial_{1}$ is surjective since $\det(\rho(t)-I_{n})\neq 0$, hence
$H_{0}(X_{0},V_{\rho})=\\{0\\}$. Moreover, the $n(2g+s)\times n(2g+s)$
diagonal submatrix

$\partial^{\prime}_{2}=\begin{pmatrix}I_{n}-\rho(t)&0&\ldots&\ldots&0\\\
0&I_{n}-\rho(t)&0&\ldots&0\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&\ldots&\ldots&I_{n}-\rho(t)\\\ \end{pmatrix}$

of $\partial_{2}$ has determinant equal to $\det(I_{n}-\rho(t))^{2g+s}$, in
particular it is non-zero and $\partial_{2}$ is injective, i.e.,
$H_{2}(X_{0},V_{\rho})=\\{0\\}$. Since the Euler characteristic of $X_{0}$
vanishes, we conclude that $H_{1}(X_{0},V_{\rho})=\\{0\\}$ and the acyclicity
follows. By a Mayer–Vietoris argument one deduces that $C_{*}(X_{1},V_{\rho})$
is acyclic. ∎

###### Proof of 2.3.1.

First, because $\rho(t)\neq I_{n}$ is central, $\det(I_{n}-\rho(t))\neq 0$.
Now we use again a Mayer–Vietoris argument. For any solid torus $N_{i}$ in the
decomposition given in Eq. 2.1, we compute the torsion as in Example 2.1.3 in
the natural Euler structure $\mathfrak{e}_{\circ}$. Denote by $\ell_{i}$ the
core of the solid torus $N_{i}$. Since $c_{i}^{\nu_{i}}t^{-1}$ is a meridian
of $N_{i}$, the curve $\ell_{i}$ is given by $\ell_{i}=c_{i}^{n_{i}}t^{m_{i}}$
with $m_{i}\nu_{i}+n_{i}=1$. Writing $t=c_{i}^{\nu_{i}}$, one sees that
$\ell_{i}=c_{i}$ is the core of the solid torus $N_{i}$. For the torsion we
obtain

$\operatorname{tor}(N_{i},V_{\rho},\mathfrak{e}_{\circ})=\pm\frac{1}{\det(I_{n}-\rho(c_{i}))},$

where $c_{0}$ is understood as being $t$. On the other hand, using [27,
Theorem 2.2], one computes the torsion of $X_{0}$ in the Euler structure
$\mathfrak{e}_{0}$:

$\operatorname{tor}(X_{0},V_{\rho},\mathfrak{e}_{0})=\pm\frac{\det(\partial_{2}^{\prime})}{\det(\rho(t)-I_{n})}$

and it follows

$\operatorname{tor}(X_{0},V_{\rho},\mathfrak{e}_{0})=\pm\det(\rho(t)-I_{n})^{2g+s-1}.$

Finally, we use the well-known multiplicativity of the torsion ([27, Theorem
1.5]). We denote by $\mathfrak{e}$ the natural Euler structure on $X$ induced
by $\mathfrak{e}_{0}$ on $X_{0}$ (which induces an Euler structure on each
boundary torus $\partial N_{i}$ compatible with $\mathfrak{e}_{\circ}$). We
have:

$\operatorname{tor}(X_{1},V_{\rho},\mathfrak{e})=\pm\frac{\operatorname{tor}(X_{0},V_{\rho},\mathfrak{e}_{0})\prod_{i=0}^{s}\operatorname{tor}(N_{i},V_{\rho},\mathfrak{e}_{\circ})}{\prod_{i=0}^{s}\operatorname{tor}(\partial
N_{i},V_{\rho},\mathfrak{e})}.$

For the boundary tori $\partial N_{i}$ an explicit computation yields
$\operatorname{tor}(\partial N_{i},V_{\rho},\mathfrak{e})=1$. We deduce

$\operatorname{tor}(X_{1},V_{\rho},\mathfrak{e})=\pm\frac{\det(\rho(t)-I_{n})^{2g+s-2}}{\prod_{j=1}^{s}\det(\rho(c_{j})-I_{n})}.\qed$

####  2.3.2 Computation of the Euler structure

Now we prove that the Euler structure $\mathfrak{e}$ corresponds to the
geodesic flow through Turaev’s isomorphism between combinatorial and smooth
Euler structure. To do so, we first exhibit an Euler chain realizing
$\mathfrak{e}$, and construct a non-vanishing vector field
$\mathfrak{X}_{\mathfrak{e}}$ realizing the corresponding smooth Euler
structure, following the construction of Turaev in [26, Section 6]. Then, we
compute the Chern–Simons class
$cs(\mathfrak{X}_{geod},\mathfrak{X}_{\mathfrak{e}})$ giving the obstruction
for the geodesic flow $\mathfrak{X}_{geod}$ being homologous to the vector
field $\mathfrak{X}_{\mathfrak{e}}$ and show it is trivial.

##### The Euler chain.

The manifold $X_{0}$ retracts on a CW-complex given by a gluing of tori along
a circle, on each the Euler chain is described as follows (see Fig. 3): we fix
a basepoint $v$ at the intersection of each torus and make a straight path to
the barycenter of each cell, oriented negatively for the one-cells and
positively for the two-cell. The result is a path $c$ with boundary $\partial
c=[v]-[T]-[\gamma]+[E_{\gamma}]$, hence an Euler chain. It is clear that it
induces the Euler structure $\mathfrak{e}_{0}$ on $X_{0}$.

$v$
---
$\gamma$
---
$T$
---
$E_{\gamma}$
---
Figure 3: For any one-cell $\gamma$ in $\\{A_{1},\ldots,C_{s}\\}$, we draw the
corresponding two-cell $E_{\gamma}$ with an Euler chain joining the vertex $v$
with each cell, oriented accordingly to the dimensions as in Section 2.2.

On the solid tori we just take the natural Euler chain denoted
$\mathfrak{e}_{\circ}$ in Example 2.1.3.

##### A vector field realizing $\mathfrak{e}$.

We construct a non-vanishing vector field on the CW-complex for $X_{0}$. On
each two-cell as in Fig. 3, we start with a Stiefel vector field: the
barycenter of each cell $\sigma$ is a zero of Hopf index $(-1)^{\dim\sigma}$,
and the vector field flows from the lower dimensional cells to the higher
dimensional cells (see [21, Section 3.2]). Then we modify it as depicted in
[21, Figure 3.5]: around each barycenter we modify it so that the index in the
cone neighborhood of an edge joining two barycenters is the coefficient of
this one-cell in the Euler chain $c$. In particular it remains untouched
except for the 3 one-cells involved in Fig. 3.

It follows from the fact that $c$ is an Euler chain that the index of the new
vector field is zero around each vertex, hence the new vector field extends to
a non-vanishing vector field $\mathfrak{X}_{\mathfrak{e}_{0}}$ on the whole
2-dimensional CW complex whose homology class does not depend on the choices
we made.

On each solid torus $N_{i}$ in the decomposition given in Eq. 2.1, the natural
Euler structure $\mathfrak{e}_{\circ}$ yields a vector field
$\mathfrak{X}_{\circ}$ tangent to the core of the solid torus. We extend it to
the whole solid torus by a radial homotopy
$r\mathfrak{X}_{e_{0}}+(1-r)\mathfrak{X}_{\circ}$, on any disc section of the
torus. The vector field we obtain equals $\mathfrak{X}_{\mathfrak{e}_{0}}$ on
the boundary.

Gluing all together, it defines an Euler structure $\mathfrak{e}$ on the
manifold $X_{1}$.

##### The Chern-Simons class
$cs(\mathfrak{X}_{geod},\mathfrak{X}_{\mathfrak{e}})$.

Now we prove that the vector field we computed is homologous to the geodesic
flow $\mathfrak{X}_{geod}$. We denote by $\mathfrak{X}_{\theta}$ the vector
field directed everywhere positively along the fiber of the unit bundle
$X_{1}$. Observe that the geodesic flow $\mathfrak{X}_{geod}$ and
$\mathfrak{X}_{\theta}$ are homotopic as nowhere vanishing vector fields: it
is clear on each solid torus $N_{i}$, and a nowhere vanishing homotopy can be
realized as $t\mathfrak{X}_{\theta}+(1-t)\mathfrak{X}_{geod}$ on $X_{0}$.

$\gamma$
---
$E_{\gamma}$
---
$t$
---
$\partial D$
---
$y_{2}<0$
---
$y_{2}>0$
---
$D_{\gamma}$
---
$\mathfrak{X}_{\theta}$
---
$\mathfrak{X}_{\mathfrak{e}}$
---
Figure 4:  We draw the vector fields $\mathfrak{X}_{\theta}$ and
$\mathfrak{X}_{\mathfrak{e}}$ in the 2-cell $E_{\gamma}$ for
$\gamma\in\\{a_{1},\ldots,a_{g},b_{1},\ldots,b_{g},c_{1},\ldots,c_{s}\\}$.
More precisely, we describe them on a spherical neighborhood of the vertex
corresponding to $\gamma$. The red segment represents the homology class of
$t$. After a homotopy of the yellow vector field
$\mathfrak{X}_{\mathfrak{e}}$, it coincides with the green vector field
$\mathfrak{X}_{\theta}$ on the two extremal points $\partial D_{\gamma}$ at
the intersection between the black segment $\gamma$ and the blue circle
$\partial D$. The new vector field obtained by gluing these two has index 0 on
the circle $\partial D$, hence the homology class $[t]$ does not contribute to
$cs(\mathfrak{X}_{\theta},\mathfrak{X}_{\mathfrak{e}})$.

Hence, we are led to compute
$cs(\mathfrak{X}_{\mathfrak{e}},\mathfrak{X}_{\theta})$. We follow the
procedure given by Turaev in [26, Section 5.2]. We compute the Chern–Simons
class on each two-cell. In Fig. 4 we calculate the contribution of the
homology class of $t$ in one of the 2-cells, and show it is trivial. The same
computation for the homology class of any $\gamma$ shows

$cs(\mathfrak{X}_{\theta},\mathfrak{X}_{\mathfrak{e}})=0$

hence

$cs(\mathfrak{X}_{geod},\mathfrak{X}_{\mathfrak{e}})=0.$

It concludes the proof of 2.3.1, since it shows that the Euler structure
$\mathfrak{e}$ used in Section 2.3.1 is induced by the geodesic flow.

######  Remark 2.3.3.

Unlike the manifold $X_{1}$, the pieces $X_{0}$ and $N_{i}$ are manifolds with
boundary. In this case, Turaev introduces a thickening of the boundary so that
the vector fields representing Euler structure are transverse to the boundary.
Here, we deal with vector fields tangent everywhere to the boundary. It makes
the description much simpler, and we can recover the setting of Turaev by
taking a thickening of the boundaries of $X_{0}$ and of each $N_{0}$ and by
extending the vector fields so that

  * •

Each vector field is tangent to the slice $\partial\Theta\times\\{0\\}$ for
$\Theta\in\\{X_{0},N_{0},\ldots,N_{s}\\}$.

  * •

The vector field is pointing outward on $\partial X_{0}\times\\{1\\}$.

  * •

The vector field are pointing inward on $\partial
N_{i}\times\\{1\\},i=0,\ldots s$.

##  3 A Selberg trace formula for non-unitary twists

We introduce the twisted Laplacian on sections of vector bundles over $X$ and
use it to derive a trace formula for the corresponding heat operator. In
Section 3.1, we define the vector bundles we will make use of, and in Section
3.2, the corresponding twisted Laplace operators. Then, in Section 3.3, we
write the pre-trace formula for the corresponding heat operator, and we
compute the contribution of the identity (Section 3.4), the hyperbolic
(Section 3.5) and elliptic (Section 3.6) contributions. We gather all this in
Section 3.7 to write down the Selberg trace formula for (a shift of) the
twisted Laplacian. Finally, in Section 3.8, we prove a technical result on
multiplicities of the eigenvalues of the Laplacians that will be used in
Section 4 to prove the meromorphicity of the twisted Ruelle zeta function.

###  3.1 Vector bundles over $X$

Consider a finite-dimensional, complex representation
$\rho:\widetilde{\Gamma}\to\operatorname{GL}(V_{\rho})$. Let

$E_{\rho}=V_{\rho}\times_{\widetilde{\Gamma}}\widetilde{G}\to\widetilde{\Gamma}\backslash\widetilde{G}=X_{1}$

denote the associated flat vector bundle over $X_{1}$ and equip it with a flat
connection $\nabla^{E_{\chi}}$. In general, this vector bundle does not define
a vector bundle over $X=X_{1}/\widetilde{K}$, because both
$\widetilde{\Gamma}$ and $\widetilde{K}$ contain the center $\widetilde{Z}$ of
$\widetilde{G}$ and $\rho$ is not necessarily trivial on $\widetilde{Z}$. In
order to obtain a vector bundle on $X$, we have to twist this construction by
a character of $\widetilde{K}$ which is compatible with $\rho$ on
$\widetilde{Z}$. We therefore assume that $\rho(t)=e^{-im\pi}$ for some
$m\in\mathbb{R}$. Note that this is automatically satisfied if $\rho$ is
irreducible by 1.2.1.

Let $\tau=\tau_{m}:\widetilde{K}\to\operatorname{U}(V_{\tau})$ denote the
character of $\widetilde{K}$ defined in Eq. 1.2, then $\tau(t)=\rho(t)^{-1}$.
We consider the homogeneous vector bundle

$E_{\tau}=\widetilde{G}\times_{\widetilde{K}}V_{\tau}\to\mathbb{H}=\widetilde{G}/\widetilde{K}.$

The invariant inner product on $V_{\tau}$ induces a Hermitian metric and a
metric connection $\nabla^{E_{\tau}}$ on $E_{\tau}$. By the same reason as
above, the bundle $E_{\tau}$ does not factor through
$X=\widetilde{\Gamma}\backslash\widetilde{G}/\widetilde{K}$.

In order to obtain a vector bundle on $X$ associated with $\rho$, we consider
the tensor product representation $V_{\tau}\otimes V_{\rho}$ of
$\widetilde{K}\times\widetilde{\Gamma}$ and let

$E_{\tau,\rho}=\widetilde{G}\times_{\widetilde{K}\times\widetilde{\Gamma}}(V_{\tau}\otimes
V_{\rho})\to\widetilde{\Gamma}\backslash\widetilde{G}/\widetilde{K},$

where $\widetilde{K}\times\widetilde{\Gamma}$ acts on $\widetilde{G}$ by
$g\cdot(k,\gamma)=\gamma^{-1}gk$. Note that this action is not effective, but
the elements in $\widetilde{K}\times\widetilde{\Gamma}$ acting trivially on
$\widetilde{G}$ are of the form $(z,z^{-1})$, $z\in\widetilde{Z}$, and hence
also act trivially on $V_{\tau}\otimes V_{\rho}$ by construction. Smooth
sections of the bundle $E_{\tau,\rho}$ can be identified with smooth functions
$f:\widetilde{G}\to V_{\tau}\otimes V_{\rho}$, such that

$f(\gamma
gk)=\left[\tau(k)^{-1}\otimes\rho(\gamma)\right]f(g)\qquad(g\in\widetilde{G},k\in\widetilde{K},\gamma\in\widetilde{\Gamma}).$

We now define a connection $\nabla^{E_{\tau,\rho}}$ on $E_{\tau,\rho}$ in
terms of its associated covariant derivative. For this we identify vector
fields $\mathfrak{X}\in C^{\infty}(X,TX)$ on $X$ with smooth functions
$\mathfrak{X}:\widetilde{G}\to\mathfrak{p}$ such that $\mathfrak{X}(\gamma
gk)=\operatorname{Ad}(k)|_{\mathfrak{p}}^{-1}\mathfrak{X}(g)$. Then,

$\nabla^{E_{\tau,\rho}}_{\mathfrak{X}}f(g)=\left.\frac{d}{dt}\right|_{t=0}f(g\exp
t\mathfrak{X}(g))\qquad(f\in C^{\infty}(X,E_{\tau,\rho}),\mathfrak{X}\in
C^{\infty}(X,TX))$

defines a covariant derivative and hence a connection on $E_{\tau,\rho}$. Note
that in general there is no Hermitian metric on $E_{\tau,\rho}$ compatible
with the connection, since the representation $\rho$ is not necessarily
unitary.

###  3.2 The twisted Laplacian

For a complex vector bundle $E\to X$ with covariant derivative $\nabla^{E}$,
the second covariant derivative $(\nabla^{E})^{2}$ is defined by

$(\nabla^{E})^{2}_{\mathfrak{X},\mathfrak{X}^{\prime}}=\nabla_{\mathfrak{X}}^{E}\nabla_{\mathfrak{X}^{\prime}}^{E}-\nabla^{E}_{\nabla_{\mathfrak{X}}^{\operatorname{LC}}\mathfrak{X}^{\prime}}\qquad(\mathfrak{X},\mathfrak{X}^{\prime}\in
C^{\infty}(X,TX)),$

where $\nabla^{\operatorname{LC}}$ is the Levi–Civita connection on $TX$. The
negative of the trace of the second covariant derivative is the corresponding
connection Laplacian:

$\Delta_{E}=-\operatorname{tr}\left((\nabla^{E})^{2}\right).$

For $\rho$ and $\tau$ as in the previous section, the twisted Laplacian
$\Delta_{\tau,\rho}^{\sharp}$ is defined as the connection Laplacian

$\Delta_{\tau,\rho}^{\sharp}=\Delta_{E_{\tau,\rho}}=-\operatorname{tr}\left((\nabla^{E_{\tau,\rho}})^{2}\right).$

Note that the orbisurface $X$ has a finite cover, which is a manifold.
Therefore, the spectral theory of $\Delta_{\tau,\rho}^{\sharp}$ on $X$ is the
same as the spectral theory of its lift to the finite cover, acting on
sections invariants under the action of the finite group of Deck
transformations. Hence, by [15, Theorem 4.3], we have the following properties
(see also for the manifold case the work of Müller [17], and for the orbifold
case the work of Fedosova [6] and Shen [23, Section 7]):

  1. (1)

If we choose a Hermitian metric on $E_{\tau,\rho}$, then
$\Delta_{\tau,\rho}^{\sharp}$ acts in $L^{2}(X,E_{\tau,\rho})$ with domain
$C^{\infty}(X,E_{\tau,\rho})$. However, it is not a formally self-adjoint
operator in general. Note that while the inner product on
$L^{2}(X,E_{\tau,\rho})$ depends on the chosen inner product on $V_{\rho}$,
the space $L^{2}(X,E_{\tau,\rho})$ as a topological vector space does not,
thanks to the compactness of $X$.

  2. (2)

$\Delta_{\tau,\rho}^{\sharp}$ is an elliptic second order differential
operator with purely discrete spectrum
$\operatorname{spec}(\Delta_{\tau,\rho}^{\sharp})\subseteq\mathbb{C}$,
consisting of generalized eigenvalues. The spectrum is contained in a
translate of a positive cone in $\mathbb{C}$. The generalized eigenspaces

$\\{f\in
L^{2}(X,E_{\tau,\rho}):(\Delta_{\tau,\rho}^{\sharp}-\mu\operatorname{Id})^{N}f=0\mbox{
for some }N\\}$

are finite-dimensional, contained in $C^{\infty}(X,E_{\tau,\rho})$, and their
direct sum, as $\mu$ runs through
$\operatorname{spec}(\Delta_{\tau,\rho}^{\sharp})$, is dense in
$L^{2}(X,E_{\tau,\rho})$.

  3. (3)

If $\widetilde{E}_{\tau,\rho}\simeq E_{\tau}\times V_{\rho}\to\mathbb{H}$
denotes the pullback of $E_{\tau,\rho}$ to $\mathbb{H}$, the lift
$\widetilde{\Delta}_{\tau,\rho}^{\sharp}$ of $\Delta_{\tau,\rho}^{\sharp}$ to
$\mathbb{H}$ takes the form

(3.1)
$\widetilde{\Delta}_{\tau,\rho}^{\sharp}=\Delta_{\tau}\otimes\operatorname{Id}_{V_{\rho}},$

where $\Delta_{\tau}$ denotes the Laplace–Beltrami operator on
$E_{\tau}\to\mathbb{H}$.

###  3.3 The pre-trace formula

The heat operator $e^{-t\Delta_{\tau_{m},\rho}^{\sharp}}$ corresponding to the
twisted Laplacian $\Delta_{\tau_{m},\rho}^{\sharp}$ is of trace class and by
Eq. 3.1 its integral kernel is the smooth function

$H_{t}^{\tau_{m},\rho}(g,h)=\sum_{[\gamma]\subseteq\widetilde{\Gamma}/\widetilde{Z}}\rho(\gamma)\otimes
H_{t}^{\tau_{m}}(h^{-1}\gamma g)\qquad(g,h\in\widetilde{G}),$

where the summation is over the conjugacy classes of
$\widetilde{\Gamma}/\widetilde{Z}\simeq\Gamma$ and

$H_{t}^{\tau_{m}}\in(C^{\infty}(\widetilde{G})\otimes\operatorname{End}(V_{\tau_{m}}))^{\widetilde{K}\times\widetilde{K}}$

denotes the heat kernel of the Laplace–Beltrami operator on
$E_{\tau_{m}}\to\mathbb{H}$. Note that the center $\widetilde{Z}$ of
$\widetilde{G}$ lies in $\widetilde{K}$, hence each summand is independent on
the choice of $\gamma$ relatively to $\widetilde{Z}$. The trace of
$e^{-t\Delta_{\tau_{m},\rho}^{\sharp}}$ can be computed in two different ways,
by summing over the generalized eigenvalues of
$\Delta_{\tau_{m},\rho}^{\sharp}$ (the spectral side) and by integrating the
heat kernel along the diagonal in $\widetilde{\Gamma}\backslash\widetilde{G}$
(the geometric side). For the spectral side we denote for an eigenvalue
$\mu\in\operatorname{spec}(\Delta_{\tau_{m},\rho}^{\sharp})$ its algebraic
multiplicity by

$\operatorname{mult}(\mu;\Delta_{\tau_{m},\rho}^{\sharp})=\dim\\{f\in
L^{2}(X,E_{\tau_{m},\rho}):(\Delta_{\tau_{m},\rho}^{\sharp}-\mu\operatorname{Id})^{N}f=0\mbox{
for some }N\\}.$

Note that $\operatorname{mult}(\mu;\Delta_{\tau_{m},\rho}^{\sharp})$ is
independent of the chosen metric on $E_{\rho}$. For the geometric side, we use
the standard arguments grouping conjugacy classes in
$\widetilde{\Gamma}/\widetilde{Z}$ (see e.g., [28, Section 2]). This leads to
the following pre-trace formula which was made rigorous by Müller [17,
Proposition 5.1] for manifolds and generalized to orbifolds by Shen [23,
Theorem 7.1]:

(3.2)
$\sum_{\mu\in\operatorname{spec}(\Delta_{\tau_{m},\rho}^{\sharp})}\operatorname{mult}(\mu;\Delta_{\tau_{m},\rho}^{\sharp})e^{-t\mu}\\\
=\sum_{[\gamma]\subseteq\widetilde{\Gamma}/\widetilde{Z}}\operatorname{Vol}(\widetilde{\Gamma}_{\gamma}\backslash\widetilde{G}_{\gamma})\operatorname{tr}{\rho(\gamma)}\int_{\widetilde{G}_{\gamma}\backslash\widetilde{G}}\operatorname{tr}H_{t}^{\tau_{m}}(g^{-1}\gamma
g)\,d\dot{g}.$

Since $\widetilde{\Gamma}/\widetilde{Z}\simeq\Gamma$ consists of the identity
element $e$, hyperbolic and elliptic elements, we can compute the three
contributions to the right-hand side in (3.2) separately.

######  Remark 3.3.1.

Since $m\in\mathbb{Q}$ we could just as well formulate everything in terms of
a finite covering group $G_{1}$ of $G$ which is a connected, semisimple group
with finite center. This formulation allows us to use the Harish-Chandra
$L^{q}$-Schwartz space $\mathcal{C}^{q}(G_{1})$ (see e.g. [1, p. 161–162] for
its definition). By [1, Lemma 2.3 and Proposition 2.4], the heat kernel
$H_{t}^{\tau_{m}}$ is contained in
$\mathcal{C}^{q}(G_{1})\otimes\operatorname{End}(V_{\tau_{m}})$ for any $q>0$,
and hence it is an admissible function in the sense of Gangolli [11, p. 407].
We can therefore apply the distribution character of a unitary representation
of $G_{1}$ to $H_{t}^{\tau_{m}}$ in Section 3.4, Section 3.5 and Section 3.6.

###  3.4 The identity contribution

The contribution of $\gamma=e$ to the right-hand side of (3.2) is

$\operatorname{Vol}(X)\dim(V_{\rho})H_{t}^{\tau_{m}}(e).$

We can employ the Plancherel formula for $L^{2}(\mathbb{H},E_{\tau_{m}})$ from
[13, Lemma 5] to find that

$H_{t}^{\tau_{m}}(e)=\frac{1}{4\pi}\Bigg{[}\int_{\mathbb{R}}\Theta_{\sigma,\lambda}(H_{t}^{\tau_{m}})\frac{\lambda\sinh
2\pi\lambda}{\cosh 2\pi\lambda+\cos\pi m}\,d\lambda\\\ +\sum_{n\equiv m+1\mod
2}|n|\Theta_{n+\operatorname{sign}(n)}(H_{t}^{\tau_{m}})\Bigg{]},$

where $\sigma$ is the restriction of $\tau_{m}$ to $\widetilde{M}$. By [20, p.
2634] we find that

(3.3)
$\Theta_{\sigma,\lambda}(H_{t}^{\tau_{m}})=e^{-t(\lambda^{2}+\frac{1}{4})}$

and

(3.4)
$\Theta_{m^{\prime}}(H_{t}^{\tau_{m}})=\begin{cases}e^{-\frac{|m^{\prime}|}{2}(1-\frac{|m^{\prime}|}{2})t}&\mbox{for
$m=m^{\prime}+2\operatorname{sign}(m^{\prime})\ell$, $\ell\in\mathbb{Z}_{\geq
0}$,}\\\ 0&\mbox{else.}\end{cases}$

Note that, in contrast to the case of for instance higher-dimensional
hyperbolic manifolds, the expression for
$\Theta_{\sigma,\lambda}(H_{t}^{\tau_{m}})$ is independent of $\sigma$ since
$\widetilde{M}$ is discrete. Write
$n+\operatorname{sign}(n)=\operatorname{sign}(m)(|m|-(\ell-1))$,
$1\leq\ell<|m|$ odd, then the identity contribution becomes

$\frac{\operatorname{Vol}(X)\dim(V_{\rho})}{4\pi}\Bigg{[}\int_{\mathbb{R}}e^{-t(\lambda^{2}+\frac{1}{4})}\frac{\lambda\sinh
2\pi\lambda}{\cosh 2\pi\lambda+\cos\pi m}\,d\lambda\\\
+\sum_{\begin{subarray}{c}1\leq\ell<|m|\\\ \ell\textup{
odd}\end{subarray}}(|m|-\ell)e^{\frac{(|m|-\ell+1)(|m|-\ell-1)}{4}t}\Bigg{]}.$

###  3.5 The hyperbolic contribution

For a conjugacy class $[\gamma]\subseteq\widetilde{\Gamma}/\widetilde{Z}$ of
hyperbolic elements we choose a representative $\gamma\in\widetilde{\Gamma}$
which is conjugate in $\widetilde{G}$ to $\widetilde{a}_{\ell(\gamma)}$,
instead of $z\widetilde{a}_{\ell(\gamma)}$, $z\in\widetilde{Z}$. Here,
$\ell(\gamma)>0$ is the length of the geodesic corresponding to $[\gamma]$.
Then, by [13, Lemma 2] and [28, Section 6], the contribution of $[\gamma]$ to
the right-hand side of (3.2) is given by

$\operatorname{tr}\rho(\gamma)\frac{l(\gamma)}{n_{\Gamma}(\gamma)D(\gamma)}\frac{1}{2\pi}\int_{\mathbb{R}}\Theta_{\sigma,\lambda}(H_{t}^{\tau_{m}})e^{-il(\gamma)\lambda}\,d\lambda,$

where $\sigma$ is the restriction of $\tau_{m}$ to $\widetilde{M}$,
$n_{\Gamma}(\gamma)$ is the multiplicity of the (not necessarily prime)
geodesic $[\gamma]$ and $D(\gamma)=2\sinh\frac{l(\gamma)}{2}$ is the Weyl
denominator. By (3.3), the total hyperbolic contribution becomes

$\sum_{[\gamma]\textup{
hyp.}}\operatorname{tr}\rho(\gamma)\frac{l(\gamma)}{n_{\Gamma}(\gamma)D(\gamma)}\frac{1}{2\pi}\int_{\mathbb{R}}e^{-t(\lambda^{2}+\frac{1}{4})}e^{-il(\gamma)\lambda}\,d\lambda.$

Computing the integral, we obtain

$\frac{e^{-\frac{t}{4}}}{2\sqrt{4\pi t}}\sum_{[\gamma]\textup{
hyp.}}\frac{l(\gamma)\operatorname{tr}\rho(\gamma)}{n_{\Gamma}(\gamma)\sinh\frac{l(\gamma)}{2}}e^{-\frac{l(\gamma)^{2}}{4t}}.$

We remark that this sum converges absolutely by the same arguments as in [8,
Proposition 4.1.1].

###  3.6 The elliptic contribution

For a conjugacy class $[\gamma]\subseteq\widetilde{\Gamma}/\widetilde{Z}$ of
elliptic elements, we choose a representative $\gamma\in\widetilde{\Gamma}$
which is conjugate in $\widetilde{G}$ to $\widetilde{k}_{\theta(\gamma)}$ with
$\theta(\gamma)\in(0,\pi)$. Let $M(\gamma)$ denote the order of the stabilizer
$\widetilde{\Gamma}_{\gamma}/\widetilde{Z}$ of $\gamma$ in
$\widetilde{\Gamma}/\widetilde{Z}$, then

$\operatorname{Vol}(\widetilde{\Gamma}_{\gamma}\backslash\widetilde{G}_{\gamma})=\operatorname{Vol}(\Gamma_{\gamma}\backslash
G_{\gamma})=\operatorname{Vol}(\langle
k_{\theta(\gamma)}\rangle\backslash\operatorname{PSO}(2))=\frac{1}{M(\gamma)}$

and hence, by [13, Lemma 4] the contribution of $[\gamma]$ to the pre-trace
formula (3.2) is given by

$\frac{\operatorname{tr}\rho(\gamma)}{M(\gamma)}\cdot\frac{1}{2i\sin\theta(\gamma)}\Bigg{[}\frac{i}{2}\int_{\mathbb{R}}\Theta_{\sigma,\lambda}(H_{t}^{\tau_{m}})\frac{\cosh
2(\pi-\theta(\gamma))\lambda+e^{i\pi m}\cosh 2\theta(\gamma)\lambda}{\cosh
2\pi\lambda+\cos\pi m}\,d\lambda\\\ -\delta_{(|m|-1)\\!\\!\\!\\!\\!\mod
2}\frac{\Theta_{1}(H_{t}^{\tau_{m}})-\Theta_{-1}(H_{t}^{\tau_{m}})}{2}\\\
-\sum_{\begin{subarray}{c}n\equiv m+1\\!\\!\\!\\!\mod 2\\\ n\neq
0\end{subarray}}\operatorname{sign}(n)e^{in\theta(\gamma)}\Theta_{n+\operatorname{sign}(n)}(H_{t}^{\tau_{m}})\Bigg{]},$

where $\delta_{(|m|-1)\\!\\!\mod 2}$ is meant to be $1$ for $|m|-1\equiv 0\mod
2$ and $0$ otherwise. Using (3.3) and (3.4), the elliptic contribution becomes

$\sum_{[\gamma]\textup{
ell.}}\frac{\operatorname{tr}\rho(\gamma)}{4M(\gamma)\sin\theta(\gamma)}\Bigg{[}\int_{\mathbb{R}}e^{-t(\lambda^{2}+\frac{1}{4})}\frac{\cosh
2(\pi-\theta(\gamma))\lambda+e^{i\pi m}\cosh 2\theta(\gamma)\lambda}{\cosh
2\pi\lambda+\cos\pi m}\,d\lambda\\\
+2i\operatorname{sign}(m)\sum_{\begin{subarray}{c}1\leq\ell<|m|\\\
\ell\textup{
odd}\end{subarray}}e^{i\operatorname{sign}(m)(|m|-\ell)\theta(\gamma)}e^{\frac{(|m|-\ell+1)(|m|-\ell-1)}{4}t}\Bigg{]}.$

Note that this sum is always finite since $\Gamma$ only contains finitely many
elliptic elements.

###  3.7 The trace formula

Adding the identity, the hyperbolic and the elliptic contribution finally
shows the trace formula. For the statement let

$A_{\tau_{m},\rho}^{\sharp}=\Delta_{\tau_{m},\rho}^{\sharp}-\frac{1}{4}.$

######  Theorem 3.7.1 (Selberg trace formula with non-unitary twist).

Let $\rho:\widetilde{\Gamma}\to\operatorname{GL}(V_{\rho})$ be a finite-
dimensional complex representation with $\rho(t)=e^{-im\pi}$,
$m\in\mathbb{R}$. Then, for every $t>0$ we have

$\displaystyle\operatorname{Tr}(e^{-tA_{\tau_{m},\rho}^{\sharp}})$
$\displaystyle\quad=\frac{\operatorname{Vol}(X)\dim(V_{\rho})}{4\pi}\Bigg{[}\int_{\mathbb{R}}e^{-t\lambda^{2}}\frac{\lambda\sinh
2\pi\lambda}{\cosh 2\pi\lambda+\cos\pi m}\,d\lambda$ $\displaystyle\hskip
199.16928pt+\sum_{\begin{subarray}{c}1\leq\ell<|m|\\\ \ell\textup{
odd}\end{subarray}}(|m|-\ell)e^{(\frac{|m|-\ell}{2})^{2}t}\Bigg{]}.$
$\displaystyle\qquad+\frac{1}{2\sqrt{4\pi t}}\sum_{[\gamma]\textup{
hyp.}}\frac{l(\gamma)\operatorname{tr}\rho(\gamma)}{n_{\Gamma}(\gamma)\sinh\frac{l(\gamma)}{2}}e^{-\frac{l(\gamma)^{2}}{4t}}$
$\displaystyle\qquad+\sum_{[\gamma]\textup{
ell.}}\frac{\operatorname{tr}\rho(\gamma)}{4M(\gamma)\sin\theta(\gamma)}\Bigg{[}\int_{\mathbb{R}}e^{-t\lambda^{2}}\frac{\cosh
2(\pi-\theta(\gamma))\lambda+e^{i\pi m}\cosh 2\theta(\gamma)\lambda}{\cosh
2\pi\lambda+\cos\pi m}\,d\lambda$ $\displaystyle\hskip
113.81102pt+2i\operatorname{sign}(m)\sum_{\begin{subarray}{c}1\leq\ell<|m|\\\
\ell\textup{
odd}\end{subarray}}e^{i\operatorname{sign}(m)(|m|-\ell)\theta(\gamma)}e^{(\frac{|m|-\ell}{2})^{2}t}\Bigg{]}.$

Here, the summation is over the conjugacy classes $[\gamma]$ of hyperbolic,
resp. elliptic, elements in $\widetilde{\Gamma}/\widetilde{Z}\simeq\Gamma$ and
the representative $\gamma\in\widetilde{\Gamma}$ is chosen such that it is
conjugate in $\widetilde{G}$ to $\widetilde{a}_{\ell(\gamma)}$
($\ell(\gamma)>0$), resp. $\widetilde{k}_{\theta(\gamma)}$
($\theta(\gamma)\in(0,\pi)$).

###  3.8 An application to the determination of multiplicities

In order to conclude that the residues of the logarithmic derivative of the
Selberg zeta function are integers, we now deduce from the trace formula an
expression for the difference of the multiplicities
$\operatorname{mult}(\mu;\Delta_{\tau_{m},\rho}^{\sharp})$ for $m$ and $m+2$.
For this, it is more convenient to work with $A_{\tau_{m},\rho}^{\sharp}$. For
$\mu\in\mathbb{C}$ we let

$L^{2}(X,E_{\tau_{m},\rho})_{\mu}=\\{f\in
L^{2}(X,E_{\tau_{m},\rho}):(A_{\tau_{m},\rho}^{\sharp}-\mu\operatorname{Id})^{N}f=0\mbox{
for some }N\\}$

denote the corresponding generalized eigenspace of
$A_{\tau_{m},\rho}^{\sharp}$ and

$\operatorname{mult}(\mu;A_{\tau_{m},\rho}^{\sharp})=\dim
L^{2}(X,E_{\tau_{m},\rho})_{\mu}$

its multiplicity.

Define $\kappa,x_{\pm}\in\operatorname{\mathfrak{sl}}(2,\mathbb{C})$ by

$\kappa=E-F,\qquad x_{\pm}=\frac{1}{2}(H\pm i(E+F)),$

then

$[\kappa,x_{\pm}]=\pm 2ix_{\pm},\qquad[x_{+},x_{-}]=-i\kappa.$

We remark that the complexification
$\operatorname{\mathfrak{sl}}(2,\mathbb{C})$ of the Lie algebra
$\operatorname{\mathfrak{sl}}(2,\mathbb{R})$ of $G$ has to be used here since
$\operatorname{ad}(\kappa)$ does not act diagonalizably on
$\operatorname{\mathfrak{sl}}(2,\mathbb{R})$. The Casimir element $\Omega$ for
$\operatorname{\mathfrak{sl}}(2,\mathbb{C})$ can be expressed in terms of the
two bases $H,E,F$ and $\kappa,x_{+},x_{-}$ as

$\Omega=\frac{1}{4}(H^{2}+2EF+2FE)=-\frac{1}{4}(\kappa^{2}-2x_{+}x_{-}-2x_{-}x_{+}).$

Moreover, we can write

(3.5)
$\Omega=-\frac{1}{4}\kappa^{2}-\frac{1}{2}i\kappa+x_{-}x_{+}=-\frac{1}{4}\kappa^{2}+\frac{1}{2}i\kappa+x_{+}x_{-}.$

For $u\in C^{\infty}(\mathbb{H},E_{\tau_{m}})=(C^{\infty}(G)\otimes
V_{\tau_{m}})^{K}$ we define $\partial_{\pm}u(g)=d\pi_{R}(x_{\pm})u(g)$, where
$d\pi_{R}$ denotes the Lie algebra representation corresponding to the right
regular representation $\pi_{R}$ of $G$ on $C^{\infty}(G)$. Then
$[\kappa,x_{\pm}]=\pm 2ix_{\pm}$ implies

$\partial_{\pm}:C^{\infty}(\mathbb{H},E_{\tau_{m}})\to
C^{\infty}(\mathbb{H},E_{\tau_{m\mp 2}}).$

Further, by (3.5) we find that for $u\in C^{\infty}(\mathbb{H},E_{\tau_{m}})$:

$\displaystyle\partial_{+}\partial_{-}u$
$\displaystyle=d\pi_{R}(\Omega)+\frac{1}{4}d\pi_{R}(\kappa)^{2}-\frac{1}{2}id\pi_{R}(\kappa)=d\pi_{L}(\Omega)-\frac{m^{2}}{4}-\frac{m}{2},$
$\displaystyle\partial_{-}\partial_{+}u$
$\displaystyle=d\pi_{R}(\Omega)+\frac{1}{4}d\pi_{R}(\kappa)^{2}+\frac{1}{2}id\pi_{R}(\kappa)=d\pi_{L}(\Omega)-\frac{m^{2}}{4}+\frac{m}{2},$

where $d\pi_{L}$ is the Lie algebra representation corresponding to the left
regular representation $\pi_{L}$. Note that
$d\pi_{L}(\Omega)=-\widetilde{\Delta}_{\tau_{m},\rho}^{\sharp}=-\widetilde{A}_{\tau_{m},\rho}^{\sharp}-\frac{1}{4}$,
where $\widetilde{\Delta}_{\tau_{m},\rho}^{\sharp}$ resp.
$\widetilde{A}_{\tau_{m},\rho}^{\sharp}$ denotes the lift of
$\Delta_{\tau_{m},\rho}^{\sharp}$ resp. $A_{\tau_{m},\rho}^{\sharp}$ to
$\mathbb{H}$. Since $\partial_{+}$ and $\partial_{-}$ are left-invariant, they
induce operators on $C^{\infty}(X,E_{\tau_{m},\rho})$ which, by the previous
considerations, satisfy

(3.6)
$\partial_{+}\partial_{-}u=-A_{\tau_{m},\rho}^{\sharp}-\frac{(m+1)^{2}}{4},\qquad\partial_{-}\partial_{+}u=-A_{\tau_{m},\rho}^{\sharp}-\frac{(m-1)^{2}}{4}.$

######  Lemma 3.8.1.

For $\mu\neq-\frac{(m+1)^{2}}{4}$:

$\operatorname{mult}(\mu;A_{\tau_{m},\rho}^{\sharp})=\operatorname{mult}(\mu;A_{\tau_{m+2},\rho}^{\sharp}).$

###### Proof.

Since $\partial_{\pm}$ is given by the right regular representation and
$A_{\tau_{m},\rho}^{\sharp}$ by the left regular representation, we have

$\partial_{\pm}\circ A_{\tau_{m},\rho}^{\sharp}=A_{\tau_{m\mp
2},\rho}^{\sharp}\circ\partial_{\pm}.$

This implies

$\partial_{\pm}:L^{2}(X,E_{\tau_{m},\rho})_{\mu}\to L^{2}(X,E_{\tau_{m\mp
2},\rho})_{\mu}.$

By (3.6), the map

$\partial_{+}:L^{2}(X,E_{\tau_{m}}\otimes E_{\rho})_{\mu}\to
L^{2}(X,E_{\tau_{m-2},\rho})_{\mu}$

is an isomorphism whenever $\mu\neq-\frac{(m-1)^{2}}{4}$, and the map

$\partial_{-}:L^{2}(X,E_{\tau_{m}}\otimes E_{\rho})_{\mu}\to
L^{2}(X,E_{\tau_{m+2},\rho})_{\mu}$

is an isomorphism whenever $\mu\neq-\frac{(m+1)^{2}}{4}$. ∎

######  Theorem 3.8.2.

For all $m\in\mathbb{R}$, we have

$\operatorname{mult}(-\tfrac{(m+1)^{2}}{4};A_{\tau_{m+2},\rho}^{\sharp})-\operatorname{mult}(-\tfrac{(m+1)^{2}}{4};A_{\tau_{m},\rho}^{\sharp})\\\
=\frac{\operatorname{Vol}(X)\dim(V_{\rho})}{4\pi}(m+1)+\sum_{[\gamma]\textup{
ell.}}\frac{i\operatorname{tr}\rho(\gamma)}{2M(\gamma)\sin\theta(\gamma)}e^{i(m+1)\theta(\gamma)}.$

In particular, the right-hand side is an integer.

###### Proof.

We subtract the trace formula in Theorem 3.7.1 for $m$ from the one for $m+2$.
By 3.8.1, the left-hand side becomes

$e^{\frac{(m+1)^{2}}{4}t}\big{(}\operatorname{mult}(-\tfrac{(m+1)^{2}}{4};A_{\tau_{m+2},\rho}^{\sharp})-\operatorname{mult}(-\tfrac{(m+1)^{2}}{4};A_{\tau_{m},\rho}^{\sharp})\big{)}.$

Carefully comparing terms on the right-hand side for $m$ and $m+2$, we find

$\frac{\operatorname{Vol}(X)\dim(V_{\rho})}{4\pi}(m+1)e^{\frac{(m+1)^{2}}{4}t}+\sum_{[\gamma]\textup{
ell.}}\frac{i\operatorname{tr}\rho(\gamma)}{2M(\gamma)\sin\theta(\gamma)}e^{i(m+1)\theta(\gamma)}e^{\frac{(m+1)^{2}}{4}t}.\qed$

##  4 Twisted Ruelle and Selberg zeta functions

We establish the meromorphic continuation for the Selberg zeta function in
Section 4.1, and a functional equation in Section 4.2. Then, we compute the
behavior of the Ruelle zeta function at zero in Section 4.3, and prove A.

###  4.1 Meromorphic continuation

Every closed oriented geodesic $\gamma$ on $X$ lifts canonically to a homotopy
class in $\widetilde{\Gamma}=\pi_{1}(X_{1})$, still denoted by $\gamma$. The
conjugacy class $[\gamma]$ in
$\widetilde{\Gamma}/\widetilde{Z}\simeq\Gamma=\pi_{1}(X)$ consists of
hyperbolic elements and conversely every such conjugacy class of hyperbolic
elements contains exactly one representative of a closed geodesic.

A closed geodesic $\gamma$ on $X$ is called _prime_ if it cannot be written as
a multiple of a shorter geodesic.

Let $\rho:\widetilde{\Gamma}=\pi_{1}(X_{1})\to\operatorname{GL}(V_{\rho})$ be
a finite-dimensional, complex representation. For $s\in\mathbb{C}$, we define
the twisted Selberg zeta function

$Z(s;\rho)=\prod_{\gamma\text{
prime}}\prod_{k=0}^{\infty}\det\left(\operatorname{Id}-\rho(\gamma)e^{-(s+k)\ell(\gamma)}\right)$

and the twisted Ruelle zeta function

$R(s;\rho)=\prod_{\gamma\text{
prime}}\det\left(\operatorname{Id}-\rho(\gamma)e^{-s\ell(\gamma)}\right),$

where the products run over the prime closed geodesics in $X$.

It is shown in [7, Theorem 3.1] that $Z(s;\rho)$, and hence $R(s;\rho)$,
converges for $s$ in some right half plane of $\mathbb{C}$ and defines a
holomorphic function on this half plane (see also [29, Section 1.2] for the
torsion-free case). We note that the logarithmic derivative $L(s;\rho)$ of
$Z(s;\rho)$ is given by

$L(s;\rho):=\frac{d}{ds}\log
Z(s;\rho)=\sum_{\gamma}\frac{\ell(\gamma)\operatorname{tr}(\rho(\gamma))}{2n_{\Gamma}(\gamma)\sinh(\frac{\ell(\gamma)}{2})}e^{-(s-\frac{1}{2})\ell(\gamma)},$

where the summation is over _all_ closed geodesics.

Moreover, we have

(4.1) $R(s;\rho)=\frac{Z(s;\rho)}{Z(s+1;\rho)}.$

To show meromorphic continuation of $Z(s;\rho)$ and $R(s;\rho)$ to
$s\in\mathbb{C}$, we let $s_{1}=s\in\mathbb{C}$ and $s_{2}\in\mathbb{C}$ with
$\textup{Re}(s_{1}-\frac{1}{2})^{2},\textup{Re}(s_{2}-\frac{1}{2})^{2}$
sufficiently large and insert the trace formula from Theorem 3.7.1 into the
resolvent identity

$\operatorname{tr}\left(\left(A_{\tau_{m},\rho}^{\sharp}+(s_{1}-\tfrac{1}{2})^{2}\right)^{-1}-\left(A_{\tau_{m},\rho}^{\sharp}+(s_{2}-\tfrac{1}{2})^{2}\right)^{-1}\right)\\\
=\int_{0}^{\infty}\left(e^{-t(s_{1}-\frac{1}{2})^{2}}-e^{-t(s_{2}-\frac{1}{2})^{2}}\right)\operatorname{tr}(e^{-tA_{\tau_{m},\rho}^{\sharp}})\,dt,$

which holds by the same arguments as in [8, Section 4.2] (especially the proof
of [8, Lemma 4.2.2] is identical), to obtain

(4.2)
$\operatorname{tr}\left(\left(A_{\tau_{m},\rho}^{\sharp}+(s_{1}-\tfrac{1}{2})^{2}\right)^{-1}-\left(A_{\tau_{m},\rho}^{\sharp}+(s_{2}-\tfrac{1}{2})^{2}\right)^{-1}\right)\\\
=I(s_{1},s_{2};\rho)+H(s_{1},s_{2};\rho)+E(s_{1},s_{2};\rho),$

with the obvious meaning of the identity contribution $I(s_{1},s_{2};\rho)$,
the hyperbolic contribution $H(s_{1},s_{2};\rho)$ and the elliptic
contribution $E(s_{1},s_{2};\rho)$. For the computation of these three terms
we choose $|m|\leq 1$ which is possible since $m$ is determined by
$\rho(t)=e^{-im\pi}$ and hence only determined by $\rho$ modulo $2$. Then, all
contributions from the (relative) discrete series vanish.

We first compute the identity contribution

(4.3)
$I(s_{1},s_{2};\rho)=\frac{\operatorname{Vol}(X)\dim(V_{\rho})}{4\pi}\int_{0}^{\infty}\left(e^{-t(s_{1}-\frac{1}{2})^{2}}-e^{-t(s_{2}-\frac{1}{2})^{2}}\right)\\\
\times\int_{\mathbb{R}}e^{-t\lambda^{2}}\frac{\lambda\sinh 2\pi\lambda}{\cosh
2\pi\lambda+\cos\pi m}\,d\lambda\,dt.$

Interchanging the integrals shows that

$=\frac{\operatorname{Vol}(X)\dim(V_{\rho})}{4\pi}\int_{\mathbb{R}}\frac{\lambda\sinh
2\pi\lambda}{\cosh 2\pi\lambda+\cos\pi
m}\left(\frac{1}{\lambda^{2}+(s_{1}-\frac{1}{2})^{2}}-\frac{1}{\lambda^{2}+(s_{2}-\frac{1}{2})^{2}}\right)\,d\lambda.$

This integral is computed in [12, Chapter 10, Lemma 2.4]:

(4.4)
$=\frac{\operatorname{Vol}(X)\dim(V_{\rho})}{4\pi}\sum_{n=0}^{\infty}\Bigg{[}\frac{1}{n+\frac{m}{2}+s_{1}}+\frac{1}{n-\frac{m}{2}+s_{1}}-\frac{1}{n+\frac{m}{2}+s_{2}}-\frac{1}{n-\frac{m}{2}+s_{2}}\Bigg{]}.$

The hyperbolic contribution

$\displaystyle H(s_{1},s_{2};\rho)={}$
$\displaystyle\int_{0}^{\infty}\left(e^{-t(s_{1}-\frac{1}{2})^{2}}-e^{-t(s_{2}-\frac{1}{2})^{2}}\right)$
$\displaystyle\hskip 42.67912pt\times\frac{1}{2\sqrt{4\pi
t}}\sum_{[\gamma]\textup{
hyp.}}\frac{l(\gamma)\operatorname{tr}(\rho(\gamma))}{n_{\Gamma}(\gamma)\sinh\frac{l(\gamma)}{2}}e^{-\frac{l(\gamma)^{2}}{4t}}\,dt$

was calculated in [8, Proposition 4.2.4 & 4.2.5] to be

$=\frac{L(s_{1};\rho)}{2(s_{1}-\frac{1}{2})}-\frac{L(s_{2};\rho)}{2(s_{2}-\frac{1}{2})}.$

Finally, the elliptic contribution is given by

$E(s_{1},s_{2};\rho)=\sum_{[\gamma]\textup{
ell.}}\frac{\operatorname{tr}\rho(\gamma)}{4M(\gamma)\sin\theta(\gamma)}\int_{0}^{\infty}\left(e^{-t(s_{1}-\frac{1}{2})^{2}}-e^{-t(s_{2}-\frac{1}{2})^{2}}\right)\\\
\times\int_{\mathbb{R}}e^{-t\lambda^{2}}\frac{\cosh
2(\pi-\theta(\gamma))\lambda+e^{i\pi m}\cosh 2\theta(\gamma)\lambda}{\cosh
2\pi\lambda+\cos\pi m}\,d\lambda\,dt.$

Note that summation and integration can be interchanged since the sum is
finite. Interchanging the two integrals and computing the integral over $t$
gives

$=\sum_{[\gamma]\textup{
ell.}}\frac{\operatorname{tr}\rho(\gamma)}{4M(\gamma)\sin\theta(\gamma)}\int_{\mathbb{R}}\left(\frac{1}{\lambda^{2}+(s_{1}-\frac{1}{2})^{2}}-\frac{1}{\lambda^{2}+(s_{2}-\frac{1}{2})^{2}}\right)\\\
\times\frac{\cosh 2(\pi-\theta(\gamma))\lambda+e^{i\pi m}\cosh
2\theta(\gamma)\lambda}{\cosh 2\pi\lambda+\cos\pi m}\,d\lambda.$

The integral over $\lambda$ is computed in [12, Chapter 10, Lemma 2.7]:

(4.5) $=\sum_{[\gamma]\textup{
ell.}}\frac{\operatorname{tr}\rho(\gamma)}{4M(\gamma)\sin\theta(\gamma)}\Bigg{[}\frac{i}{s_{1}-\frac{1}{2}}\sum_{n=0}^{\infty}\Bigg{(}\frac{e^{-2i\theta(\gamma)(n-\frac{m}{2}+\frac{1}{2})}}{n-\frac{m}{2}+s_{1}}-\frac{e^{2i\theta(\gamma)(n+\frac{m}{2}+\frac{1}{2})}}{n+\frac{m}{2}+s_{1}}\Bigg{)}\\\
-\frac{i}{s_{2}-\frac{1}{2}}\sum_{n=0}^{\infty}\Bigg{(}\frac{e^{-2i\theta(\gamma)(n-\frac{m}{2}+\frac{1}{2})}}{n-\frac{m}{2}+s_{2}}-\frac{e^{2i\theta(\gamma)(n+\frac{m}{2}+\frac{1}{2})}}{n+\frac{m}{2}+s_{2}}\Bigg{)}\Bigg{]}.$

######  Proposition 4.1.1.

The logarithmic derivative $L(s;\rho)$ of the twisted Selberg zeta function
$Z(s;\rho)$ has a meromorphic extension to the whole complex plane
$\mathbb{C}$. Its singularities are given by the following formal expression:

$\sum_{j=0}^{\infty}\left[\frac{1}{s-\frac{1}{2}-i\mu_{j}}+\frac{1}{s-\frac{1}{2}+i\mu_{j}}\right]\\\
+\frac{\operatorname{Vol}(X)\dim(V_{\rho})}{4\pi}\sum_{n=0}^{\infty}\left[\frac{1-2s}{s+\frac{m}{2}+n}+\frac{1-2s}{s-\frac{m}{2}+n}\right]\\\
+i\sum_{[\gamma]\textup{
ell.}}\frac{\operatorname{tr}\rho(\gamma)}{2M(\gamma)\sin\theta(\gamma)}\sum_{n=0}^{\infty}\left[\frac{e^{2i\theta(\gamma)(n+\frac{m}{2}+\frac{1}{2})}}{s+\frac{m}{2}+n}-\frac{e^{-2i\theta(\gamma)(n-\frac{m}{2}+\frac{1}{2})}}{s-\frac{m}{2}+n}\right]$

where $(\frac{1}{4}+\mu_{j}^{2})_{j\in\mathbb{Z}_{\geq 0}}\subseteq\mathbb{C}$
are the eigenvalues of $\Delta_{\tau_{m},\rho}^{\sharp}$, counted with
algebraic multiplicity.

###### Proof.

Fix $s_{2}\in\mathbb{C}$ with $\textup{Re}(s_{2}-\frac{1}{2})^{2}$
sufficiently large and let $s=s_{1}\in\mathbb{C}$. Multiplying (4.2) by
$2(s_{1}-\frac{1}{2})$ gives

$L(s;\rho)=2(s-\tfrac{1}{2})\operatorname{tr}\left(\left(A_{\rho}^{\sharp}+(s-\tfrac{1}{2})^{2}\right)^{-1}-\left(A_{\rho}^{\sharp}+(s_{2}-\tfrac{1}{2})^{2}\right)^{-1}\right)\\\
-2(s-\tfrac{1}{2})I(s,s_{2};\rho)-2(s-\tfrac{1}{2})E(s,s_{2};\rho)+\frac{s-\frac{1}{2}}{s_{2}-\frac{1}{2}}L(s_{2};\rho).$

Together with (4.4) and (4.5), this shows that $L(s;\rho)$ extends
meromorphically to all $s\in\mathbb{C}$. The contribution of the first two
terms to the poles of $L(s;\rho)$ was obtained in [8, Proposition 4.2.5]. The
last term is holomorphic in $s$, and for the term $E(s,s_{2};\rho)$ one can
use (4.5). ∎

######  Theorem 4.1.2 (Meromorphic continuation of the Selberg zeta function).

The twisted Selberg zeta function $Z(s;\rho)$ has a meromorphic extension to
the whole complex plane $\mathbb{C}$ with poles and zeros contained in the set
$\\{\frac{1}{2}\pm i\mu_{j}:j\in\mathbb{Z}_{\geq
0}\\}\cup\\{\pm\frac{m}{2}-n:n\in\mathbb{Z}_{\geq 0}\\}$, where
$(\frac{1}{4}+\mu_{j}^{2})_{j\in\mathbb{Z}_{\geq 0}}\subseteq\mathbb{C}$ are
the eigenvalues of $\Delta_{\tau_{m},\rho}^{\sharp}$.

###### Proof.

In view of 4.1.1 it suffices to show that the residues of $L(s;\rho)$ are
integers. For the poles at $s=\frac{1}{2}\pm i\mu_{j}$ the residue is the
algebraic multiplicity of $\mu_{j}$. For $s=\mp\frac{m}{2}-n$ the residue is

$\frac{\operatorname{Vol}(X)\dim(V_{\rho})}{4\pi}(\pm m+2n+1)\pm
i\sum_{[\gamma]\\\ \textup{
ell.}}\frac{\operatorname{tr}\rho(\gamma)}{2M(\gamma)\sin\theta(\gamma)}e^{\pm
2i\theta(\gamma)(n\pm\frac{m}{2}+\frac{1}{2})}$

which is an integer by Theorem 3.8.2. ∎

######  Remark 4.1.3.

Since $\Delta_{\tau_{m},\rho}^{\sharp}$ is in general not self-adjoint, its
eigenvalues might be complex, although contained in a positive cone in
$\mathbb{C}$. It may therefore happen that finitely many of the singularities
$\frac{1}{2}\pm i\mu_{j}$ of $L(s;\rho)$ coincide with finitely many of the
singularities $\pm\frac{m}{2}-n$. In particular, we are not able to decide
more precisely what are the poles and the zeros of the Selberg zeta function
in 4.1.1.

###  4.2 The functional equation

From the trace formula we further conclude:

######  Theorem 4.2.1 (Functional equation).

The twisted Selberg zeta function $Z(s;\rho)$ satisfies the following
functional equation:

$\eta(s;\rho)=\frac{Z(s;\rho)}{Z(1-s;\rho)}=\exp\Bigg{(}\int_{0}^{s-\frac{1}{2}}\dim(V_{\rho})\operatorname{Vol}(X)\frac{\xi\sin
2\pi\xi}{\cos 2\pi\xi+\cos\pi m}\\\ -\sum_{[\gamma]\textup{
ell.}}\frac{\pi\operatorname{tr}\rho(\gamma)}{M(\gamma)\sin\theta(\gamma)}\frac{\cos
2(\pi-\theta(\gamma))\xi+e^{i\pi m}\cos 2\theta(\gamma)\xi}{\cos
2\pi\xi+\cos\pi m}\,d\xi\Bigg{)},$

where the integral is along any contour from $0$ to $s-\frac{1}{2}$ in the
complex plane avoiding the zeros of the denominator.

###### Proof.

Fix $s_{2}\in\mathbb{C}$ with $\textup{Re}(s_{2}-\frac{1}{2})^{2}$
sufficiently large and subtract from (4.2) with $s_{1}=s$ the same identity
with $s_{1}=1-s$ instead. This yields

$0=I(s,1-s;\rho)+E(s,1-s;\rho)+H(s,1-s;\rho).$

The identity contribution equals

$\displaystyle I(s,1-s;\rho)$
$\displaystyle=-\frac{\dim(V_{\rho})\operatorname{Vol}(X)}{4}\big{(}\tan\pi(s-\tfrac{1}{2}+\tfrac{m}{2})+\tan\pi(s-\tfrac{1}{2}-\tfrac{m}{2})\big{)},$

by [8, proof of Theorem 4.2.8]. Using

$\tan(x+y)+\tan(x-y)=\frac{2\sin(2x)}{\cos(2x)+\cos(2y)},$

this becomes

$I(s,1-s;\rho)=-\frac{\dim(V_{\rho})\operatorname{Vol}(X)}{2}\frac{\sin
2\pi(s-\frac{1}{2})}{\cos 2\pi(s-\frac{1}{2})+\cos\pi m}.$

The hyperbolic contribution becomes

$H(s,1-s;\rho)=\frac{L(s;\rho)+L(1-s;\rho)}{2(s-\frac{1}{2})}$

and the elliptic contribution is

$\displaystyle E(s,1-s;\rho)={}$ $\displaystyle\sum_{[\gamma]\textup{
ell.}}\frac{\operatorname{tr}\rho(\gamma)}{4M(\gamma)\sin\theta(\gamma)}\Bigg{[}\frac{i}{s-\frac{1}{2}}\sum_{n\in\mathbb{Z}}\Bigg{(}\frac{e^{2i\theta(\gamma)(n+\frac{m}{2}+\frac{1}{2})}}{s-\frac{m}{2}-n-1}-\frac{e^{2i\theta(\gamma)(n+\frac{m}{2}+\frac{1}{2})}}{s+\frac{m}{2}+n}\Bigg{)}\Bigg{]}$
$\displaystyle={}$ $\displaystyle\sum_{[\gamma]\textup{
ell.}}\frac{\pi\operatorname{tr}\rho(\gamma)}{2M(\gamma)(s-\frac{1}{2})\sin\theta(\gamma)}\frac{\cos
2(\pi-\theta(\gamma))(s-\frac{1}{2})+e^{i\pi m}\cos
2\theta(\gamma)(s-\frac{1}{2})}{\cos 2\pi(s-\frac{1}{2})+\cos\pi m}$

by [12, equation (**) on page 444]. This implies

(4.6) $\frac{d}{ds}\log\frac{Z(s;\rho)}{Z(1-s;\rho)}=L(s;\rho)+L(1-s;\rho)\\\
=\dim(V_{\rho})\operatorname{Vol}(X)(s-\tfrac{1}{2})\frac{(s-\frac{1}{2})\sin
2\pi(s-\frac{1}{2})}{\cos 2\pi(s-\frac{1}{2})+\cos\pi m}\\\
-\sum_{[\gamma]\textup{
ell.}}\frac{\pi\operatorname{tr}\rho(\gamma)}{M(\gamma)\sin\theta(\gamma)}\frac{\cos
2(\pi-\theta(\gamma))(s-\frac{1}{2})+e^{i\pi m}\cos
2\theta(\gamma)(s-\frac{1}{2})}{\cos 2\pi(s-\frac{1}{2})+\cos\pi m}$

and the claim follows by integration and exponentiation. ∎

###  4.3 Fried’s conjecture

We compute the behavior of the Ruelle zeta function at $s=0$, and we prove A.

We start with the following proposition:

######  Proposition 4.3.1.

The Ruelle zeta function extends meromorphically to the whole complex plane.
Moreover,

$R(s;\rho)\sim_{s\to 0}\pm\eta(s+1;\rho)^{-1}$

with $\eta(s;\rho)$ as in 4.2.1.

###### Proof.

Using Eq. 4.1 and 4.1.2 we directly deduce that $R(s;\rho)$ extends
meromorphically. Together with the functional equation for the Selberg zeta
function established in 4.2.1, we get

$R(s;\rho)=\frac{Z(s;\rho)}{Z(1-s;\rho)}=\frac{Z(s;\rho)}{\eta(1+s;\rho)Z(-s;\rho)}$

and the result follows. ∎

Now, we are led to compute the value of $\eta(s+1;\rho)$ at $s=0$. The rest of
this section is devoted to this computation. We summarize the result in the
following theorem. Recall that each singular point in the orbisurface $X$ is
represented by an elliptic loop $c_{j}\in\pi_{1}(X_{1})$ of finite order
$\nu_{j}$. For any $j=1,\ldots,s$ we denote by
$n_{j}=\dim\operatorname{Fix}\rho(c_{j})$, so that
$\rho(c_{j})=I_{n_{j}}\oplus T_{j}$. Let
$M_{j}(x)=\det(I_{n-n_{j}}-T_{j})(\frac{ix}{\nu_{j}})^{n_{j}}$ and
$M_{0}(x)=(ix)^{n}$ for $x\in\mathbb{C}$.

######  Theorem 4.3.2.

For $\rho\colon\pi_{1}(X_{1})\to\operatorname{GL}(V)$ an irreducible
representation:

  1. 1.

If $m=0$, then

$R\left(\frac{x}{2\pi},\rho\right)\sim_{x\to
0}\pm\frac{M_{0}(x)^{2g-2+s}}{\prod_{j=1}^{s}M_{j}(x)}.$

  2. 2.

If $m\neq 0$, then

$R(0;\rho)=\pm\frac{\det(\rho(t)-I_{n})^{2g+s-2}}{\prod_{j=1}^{s}\det(\rho(c_{j})-I_{n})}.$

Now we prove 4.3.2 by computing the behavior of $\eta(1+s;\rho)$ near $s=0$.
We start with the following observation:

######  Lemma 4.3.3.

(4.7)
$\eta(1+s;\rho)=\exp\int_{0}^{s+\frac{1}{2}}\frac{\eta^{\prime}(\xi+\frac{1}{2};\rho)}{\eta(\xi+\frac{1}{2};\rho)}d\xi$

######  Remark 4.3.4.

Note that the previous expression makes sense independently on the chosen path
between $0$ and $s+\frac{1}{2}$, since the residues of the logarithmic
derivative of the meromorphic function $\eta$ are in $2i\pi\mathbb{Z}$.

###### Proof.

Note that the equality holds true at $s=-\frac{1}{2}$, where indeed both side
are equal to 1. Let $s\in\mathbb{C}$ such that $\eta(1+s;\rho)$ is neither $0$
nor $\infty$. Then, taking a determination of the logarithm and
differentiating both sides, we find that both sides of the equation locally
coincide, up to a multiplicative constant. Since both sides define a
meromorphic function, the constant should be global. Considering
$s=-\frac{1}{2}$, we conclude that it is equal to 1. ∎

In order to compute the integral in Eq. 4.7, we regroup the terms as in Eq.
4.6:

(4.8)
$\frac{\eta^{\prime}(\xi+\frac{1}{2};\rho)}{\eta(\xi+\frac{1}{2};\rho)}=\dim(V_{\rho})\operatorname{Vol}(X)\frac{-2\pi\xi\sin
2\pi\xi}{\cos 2\pi\xi+\cos\pi m}+\sum_{j}E_{j}(\xi),$

where

(4.9)
$E_{j}(\xi)=-\sum_{k=1}^{\nu_{j}-1}\frac{\pi\operatorname{tr}\rho(c_{j}^{k})}{\nu_{j}\sin(k\pi/\nu_{j})}\frac{\cos\left(2(\pi-k\pi/\nu_{j})\xi\right)+e^{i\pi
m}\cos(2k\pi\xi/\nu_{j})}{\cos 2\pi\xi+\cos\pi m}$

regroups all the summands corresponding to the elliptic element $c_{j}$ and
its powers.

In view of 4.3.4, one needs to make a choice of an integration path avoiding
the poles of the integrand in Eq. 4.7. The integrand has a unique pole in
$[0,\frac{1}{2}]$, which we denote by $\xi_{0}$ (its value is
$\frac{1-m}{2}\mod\mathbb{Z}$). We choose a path from $0$ to $\frac{1}{2}$
which avoids $\xi_{0}$ along a small half-circle in the upper-half plane and
we denote this path by $\mathscr{C}_{+}$, oriented following the increasing
real part. Let $\mathscr{C}_{-}=\overline{\mathscr{C}_{+}}$, then
$\mathscr{C}=\mathscr{C}_{+}-\mathscr{C}_{-}$ is a small circle encircling
$\xi_{0}$.

First, we compute the contribution of the identity, namely the integral of the
first summand in the right-hand side of (4.8).

######  Lemma 4.3.5.

(4.10) $\displaystyle\exp\left(\int_{0}^{1/2}\frac{-2\pi\xi\sin 2\pi\xi
d\xi}{\cos 2\pi\xi+\cos\pi m}\right)$ $\displaystyle=\pm(1-e^{i\pi m})$
$\displaystyle\text{for }m\neq 0,$
$\displaystyle\exp\left(\int_{0}^{1/2+s}\frac{-2\pi\xi\sin 2\pi\xi d\xi}{\cos
2\pi\xi+1}\right)$ $\displaystyle\sim_{s=0}-2\pi s+o(s)$
$\displaystyle\text{for }m=0.$

###### Proof.

The asymptotic case (for $m=0$) goes through exactly as in [10, Lemma 3], so
we just prove the Eq. 4.10 (for $m\neq 0$). By [10, Lemma 2], we know that

$\textup{Re}\left(\int_{0}^{1/2}\frac{-2\pi\xi\sin 2\pi\xi d\xi}{\cos
2\pi\xi+\cos\pi m}\right)=\log|1-e^{i\pi m}|.$

Hence, we only need to show that

$\textup{Im}\left(\int_{0}^{1/2}\frac{-2\pi\xi\sin 2\pi\xi d\xi}{\cos
2\pi\xi+\cos\pi m}\right)=\arg(1-e^{i\pi m})\mod\pi,$

where for the integral we mean $\int_{\mathscr{C}_{+}}\frac{-2\pi\xi\sin
2\pi\xi d\xi}{\cos 2\pi\xi+\cos\pi m}$.

Let $f(\xi)=\cos 2\pi\xi+\cos\pi m$. We want to compute the imaginary part of
the integral

$\int_{\mathscr{C}_{+}}\xi\frac{f^{\prime}(\xi)}{f(\xi)}d\xi.$

Note that the function $\xi\mapsto\xi\frac{f^{\prime}(\xi)}{f(\xi)}$ sends the
real axis to itself, hence we have
$\int_{\mathscr{C}_{-}}\xi\frac{f^{\prime}(\xi)}{f(\xi)}d\xi=\overline{\int_{\mathscr{C}_{+}}\xi\frac{f^{\prime}(\xi)}{f(\xi)}d\xi}$
and
$\textup{Im}\left(\int_{\mathscr{C}_{-}}\xi\frac{f^{\prime}(\xi)}{f(\xi)}d\xi\right)=-\textup{Im}\left(\int_{\mathscr{C}_{+}}\xi\frac{f^{\prime}(\xi)}{f(\xi)}d\xi\right)$.
We deduce that

$\textup{Im}\left(\int_{\mathscr{C}_{+}}\xi\frac{f^{\prime}(\xi)}{f(\xi)}d\xi\right)=\frac{1}{2}\textup{Im}\left(\int_{\mathscr{C}}\xi\frac{f^{\prime}(\xi)}{f(\xi)}d\xi\right)=\pi\operatorname{Res}\left(\xi\frac{f^{\prime}(\xi)}{f(\xi)},\xi_{0}\right)=\pi\xi_{0}.$

A computation shows $\arg((1-e^{i\pi m})^{2})=\pi(1-m)\mod 2\pi$, hence
$\arg(1-e^{i\pi m})=\pi\xi_{0}\mod\pi$ and Eq. 4.10 follows. ∎

Now, we compute the terms coming from the elliptic elements. We first treat
the case $m\neq 0$, which generalizes [10, Lemma 5]:

######  Lemma 4.3.6.

For $m\neq 0$ we have

(4.11) $\displaystyle\exp\int_{0}^{1/2}E_{j}$
$\displaystyle=\pm\frac{\det(I_{n}-\rho(c_{j}))}{\det(I_{n}-\rho(t))^{1/\nu_{j}}}.$

###### Proof.

In [10, Lemma 5], it is shown that the equality Eq. 4.11 holds true in moduli.
Following the lines of [10, Proof of Lemma 5], we compute the arguments of the
terms involved. For each $i$, recall that the contribution $E_{j}$ of the
elliptic element $c_{j}$ and its powers is given in Eq. 4.9. The term
$\operatorname{Tr}\rho(c_{j})^{k}$ is $\sum_{\alpha}\alpha^{k}$, for $\alpha$
running through the eigenvalues of $\rho(c_{j})$. In particular
$\alpha^{M}=e^{-i\pi m}$, where $M=\nu_{j}$ is the order of $c_{j}$ (recall
that $\rho(t)=e^{-i\pi m}$). We compute the contribution of each $\alpha$ to
$\int_{\mathscr{C}_{+}}E_{j}$ as follows: we set $z_{0}=e^{i\pi/M}$ and
$z=e^{2i\pi\xi/M}$ and get that $\alpha$ contributes
$\int_{\mathcal{C}_{+}}\frac{P(z)}{Q(z)}dz$, where

$\displaystyle P(z)$
$\displaystyle=-\frac{1}{2iz}\sum_{k=1}^{M-1}\frac{\alpha^{k}}{\sin(k\pi/M)}(z^{M-k}+z^{k-M}+e^{i\pi
m}(z^{k}+z^{-k}))z^{M},$ $\displaystyle Q(z)$ $\displaystyle=(z^{M}+e^{i\pi
m})(z^{M}+e^{-i\pi m})$

are polynomials in $z$, and $\mathcal{C}_{+}$ is the image of the path
$\mathscr{C}_{+}$, by the map $\xi\mapsto e^{2i\pi\xi/M}$. It is a small
deformation of an arc of the unit circle, avoiding the pole occurring at
$e^{2i\pi\xi_{0}/M}$.

Indeed, one can rearrange $P$ by grouping the terms corresponding to $k$ and
$M-k$ in the sum, factorizing $(z^{k}+z^{-k})$ and re-expanding:

$P(z)=-\frac{1}{2iz}\sum_{k=1}^{M-1}\frac{\alpha^{k}e^{i\pi
m}+\alpha^{-k}e^{-i\pi m}}{\sin(k\pi/M)}(z^{k}+z^{-k})z^{M}.$

Let us assume that $m\neq\pm 1$, so that the polynomial $Q$ splits as the
product $Q(z)=\prod_{l=1}^{2M}(z-z_{l})$ with $z_{l}$ simple roots. Hence, we
can decompose the rational fraction $P/Q$ as

$\frac{P(z)}{Q(z)}=\sum_{l=1}^{2M}\frac{P(z_{l})/Q^{\prime}(z_{l})}{z-z_{l}}.$

Since $m\notin\mathbb{Z}$, one can integrate to obtain

(4.12)
$\int_{\mathcal{C}_{+}}\frac{P(z)}{Q(z)}dz=\sum_{l=1}^{2M}\frac{P(z_{l})}{Q^{\prime}(z_{l})}\log\left(\frac{z_{0}-z_{l}}{1-z_{l}}\right),$

keeping in mind that the final result will not depend on the chosen
determination of the logarithm.

Now

$\frac{P(z_{l})}{Q^{\prime}(z_{l})}=\sum_{k=1}^{M-1}\frac{(\alpha^{k}e^{i\pi
m}+\alpha^{-k}e^{-i\pi m})(z_{l}^{k}+z_{l}^{-k})}{2iM\varepsilon(e^{i\pi
m}-e^{-i\pi m})\sin(k\pi/M)},$

where $\varepsilon=\pm 1$ depending on $z_{l}^{M}=e^{\varepsilon i\pi m}$. In
particular, this term is real, and is the same for both $z_{l}$ and
$z_{l}^{-1}$. Hence, the contribution of $z_{l}$ and $z_{l}^{-1}$ together to
Eq. 4.12 equals $P(z_{l})/Q^{\prime}(z_{l})$ times

$\displaystyle\log\left(\frac{z_{0}-z_{l}}{1-z_{l}}\right)-\log\left(\frac{z_{0}-z_{l}^{-1}}{1-z_{l}^{-1}}\right)$
$\displaystyle=\log(z_{0})+\log(1-\frac{z_{l}}{z_{0}})-\log(1-z_{l}z_{0})$
$\displaystyle=\frac{i\pi}{M}+\log(1-\frac{z_{l}}{z_{0}})-\log(1-z_{l}z_{0}).$

All together, setting $\omega=\frac{z_{l}}{z_{0}}$ for $z_{l}$ with
$\varepsilon=1$, and $\omega=z_{l}z_{0}$ for the corresponding $z_{l}^{-1}$,
we obtain:

$\displaystyle\int_{\mathcal{C}_{+}}\frac{P(z)}{Q(z)}dz={}$ $\displaystyle
i\pi+\sum_{\omega^{M}=e^{-i\pi
m}}\log(1-\omega)\sum_{k=1}^{M-1}\frac{(\alpha^{k}e^{i\pi
m}+\alpha^{-k}e^{-i\pi m})}{2iM(e^{i\pi m}-e^{-i\pi m})\sin(k\pi/M)}$
$\displaystyle\hskip
108.12054pt\times(z_{0}^{k}\omega^{k}+z_{0}^{-k}\omega^{-k}-z_{0}^{-k}\omega^{k}-z_{0}^{k}\omega^{-k})$
$\displaystyle={}$ $\displaystyle i\pi+\sum_{\omega^{M}=e^{-i\pi
m}}\log(1-\omega)\sum_{k=1}^{M-1}\frac{(\alpha^{k}e^{i\pi
m}+\alpha^{-k}e^{-i\pi m})}{M(e^{i\pi m}-e^{-i\pi
m})}(\omega^{k}-\omega^{-k}).$

Note that $\omega=\zeta\alpha$, with $\zeta^{M}=1$. In the sum above, the term
for $k$ simplifies with the term for $M-k$ and one gets

$\frac{1}{M(e^{i\pi m}-e^{-i\pi m})}\sum_{k=1}^{M-1}e^{-i\pi
m}\zeta^{k}-e^{i\pi m}\zeta^{-k}=\begin{cases}-1/M\text{ if }\zeta\neq 1\\\
(M-1)/M\text{ if }\zeta=1\end{cases}.$

Hence, we obtain

$\int_{\mathcal{C}_{+}}\frac{P}{Q}=i\pi+\log(1-\alpha)-\frac{1}{M}\sum_{\omega^{M}=e^{-i\pi
m}}\log(1-\omega).$

The last sum above is just $\log(1-e^{-i\pi m})$. Adding all this for every
$\alpha$ and every index $i$, and taking the exponential, we obtain Eq. 4.11.

For $m=\pm 1$, one needs to be a bit more careful when decomposing $Q(z)$.
However, the computation goes through, combining what we have just done and
[10, Section 3]. Hence, we omit this case. ∎

Now we generalize [10, Lemma 6], which computes the elliptic contribution in
the case $m=0$:

######  Lemma 4.3.7.

For $m=0$ we have

(4.13) $\displaystyle\exp\int_{0}^{1/2-\varepsilon}E_{j}$
$\displaystyle\sim_{\varepsilon\to 0}\pm(\det
T_{j})^{-\frac{1}{2}}\det(I_{n-n_{j}}-T_{j})(2\pi\varepsilon)^{-\frac{n}{\nu_{j}}+n_{j}}\,i^{n-n_{j}}\nu_{j}^{-n_{j}}.$

###### Proof.

We follow [10, Proof of Lemma 6] and compute in each step the phase factors
involved. Let $\alpha$, $z_{0}$, $P$ and $Q$ be as in the proof of 4.3.6. By
the same arguments as in [10, proof of Lemma 6], we can write

$\frac{P(z)}{Q(z)}=\sum_{z_{1}^{M}=-1}\frac{\mu(z_{1})}{z-z_{1}},$

with $z=e^{2\pi i\xi/M}$. The residues $\mu(z_{1})$ are computed in [10, Lemma
7] and are given by

$\mu(z_{0}^{2k+1})=v(k)-\frac{2k+1}{M}\qquad(0\leq k\leq M-1),$

where $v(k)=\\#\\{x\in\mathbb{Z}:|x|\leq k,x\equiv p\mod M\\}$ with $p$ chosen
such that $\alpha=z_{0}^{2p}$. For $z_{1}\neq z_{0}$ we find

$\int_{0}^{\frac{1}{2}}\frac{\mu(z_{1})}{z-z_{1}}d\xi=\int_{1}^{z_{0}}\frac{\mu(z_{1})}{z-z_{1}}dz=\mu(z_{1})\log\left(\frac{z_{0}-z_{1}}{1-z_{1}}\right),$

and for $z_{1}=z_{0}$ we let $z_{\varepsilon}=z_{0}e^{-2\pi i\varepsilon/M}$
and get

$\int_{0}^{\frac{1}{2}-\varepsilon}\frac{\mu(z_{0})}{z-z_{0}}d\xi=\int_{1}^{z_{\varepsilon}}\frac{\mu(z_{0})}{z-z_{0}}dz=\mu(z_{0})\log\left(\frac{z_{\varepsilon}-z_{0}}{1-z_{0}}\right).$

######  Remark 4.3.8.

Note once again that the final result does not depend on the chosen
determination of the logarithm. In particular the residues $\mu(z_{0}^{k})$
are rational numbers in $\frac{1}{M}\mathbb{Z}$, hence the values of the
complex numbers computed in the sequel such as
$\left(\frac{z_{\varepsilon}-z_{0}}{1-z_{0}}\right)^{\mu(z_{0})}$ or
$\left(\frac{z_{0}-z_{1}}{1-z_{1}}\right)^{\mu(z_{1})}$ are only well-defined
up to a (fixed) $M$-th root of unity. Since at the end, we take the product of
$M$ such factors, this ambiguity cancels out.

Hence, the total contribution of the eigenvalue $\alpha$ to
$\exp\int_{0}^{\frac{1}{2}-\varepsilon}E_{j}(\xi)\,d\xi$ amounts to

$\displaystyle\exp\int_{0}^{\frac{1}{2}-\varepsilon}\frac{P(z)}{Q(z)}d\xi$
$\displaystyle=\left(\frac{z_{\varepsilon}-z_{0}}{1-z_{0}}\right)^{\mu(z_{0})}\prod_{\begin{subarray}{c}z_{1}^{M}=-1\\\
z_{1}\neq
z_{0}\end{subarray}}\left(\frac{z_{0}-z_{1}}{1-z_{1}}\right)^{\mu(z_{1})}$
(4.14) $\displaystyle\sim\left(\frac{-2\pi
i\varepsilon}{M(1-z_{0})}\right)^{\mu(z_{0})}\prod_{\begin{subarray}{c}z_{1}^{M}=-1\\\
z_{1}\neq
z_{0}\end{subarray}}\left(\frac{1-z_{1}/z_{0}}{1-z_{1}}\right)^{\mu(z_{1})},$

where we have used $e^{-2\pi i\varepsilon/M}-1\sim-2\pi i\varepsilon/M$ and
the fact that $\sum_{z_{1}}\mu(z_{1})=0$ by the Residue Theorem. To compute
the product, we generalize [10, Lemma 8 and Lemma 9].

We first claim that the following generalization of [10, Lemma 8] holds:

######  Claim.

(4.15)
$\frac{1}{M(1-z_{0})}\prod_{k=1}^{M-1}\left(\frac{1-z_{0}^{2k}}{1-z_{0}^{2k+1}}\right)^{2k+1}=(-i)^{M-1}\left(\frac{M}{2}\right)^{M}.$

###### Proof of the claim.

In the numerator we can combine the factor for $k$ with the one for $M-k$ to
find

$(1-z_{0}^{2k})^{2k+1}(1-z_{0}^{2M-2k})^{2M-2k+1}=(-1)^{M}z_{0}^{(2k)^{2}}(1-z_{0}^{2k})^{M+1}(1-z_{0}^{2M-2k})^{M+1},$

where we have used that $(1-z_{0}^{2k})=-z_{0}^{2k}(1-z_{0}^{2M-2k})$. This
results in

(4.16)
$\prod_{k=1}^{M-1}(1-z_{0}^{2k})^{2k+1}=\prod_{k=1}^{\lfloor\frac{M-1}{2}\rfloor}(-1)^{M}z_{0}^{(2k)^{2}}\prod_{k=1}^{M-1}(1-z_{0}^{2k})^{M+1}.$

Similarly, in the denominator we rewrite

$(1-z_{0}^{2k+1})^{2k+1}(1-z_{0}^{2M-2k-1})^{2M-2k-1}\\\
=(-1)^{M}z_{0}^{(2k+1)^{2}}(1-z_{0}^{2k+1})^{M}(1-z_{0}^{2M-2k-1})^{M}$

to find

(4.17)
$\prod_{k=0}^{M-1}(1-z_{0}^{2k+1})^{2k+1}=\prod_{k=0}^{\lfloor\frac{M-2}{2}\rfloor}(-1)^{M}z_{0}^{(2k+1)^{2}}\prod_{k=0}^{M-1}(1-z_{0}^{2k+1})^{M}.$

Note that the sign $(-1)^{M}$ in (4.16) and (4.17) is either $+1$ for $M$ even
or occurs the same number of times for $M$ odd. Therefore, combining (4.16)
and (4.17) yields

$\frac{\prod_{k=1}^{M-1}(1-z_{0}^{2k})^{2k+1}}{\prod_{k=0}^{M-1}(1-z_{0}^{2k+1})^{2k+1}}=z_{0}^{K}\frac{\prod_{k=1}^{M-1}(1-z_{0}^{2k})^{M+1}}{\prod_{k=0}^{M-1}(1-z_{0}^{2k+1})^{M}},$

where

$K=\sum_{k=1}^{\lfloor\frac{M-1}{2}\rfloor}(2k)^{2}-\sum_{k=1}^{\lfloor\frac{M}{2}\rfloor}(2k-1)^{2}=\frac{M(M-1)}{2}\times\begin{cases}+1&\mbox{for
$M$ odd,}\\\ -1&\mbox{for $M$ even.}\end{cases}$

Hence,

$z_{0}^{K}=e^{\pm\frac{i\pi(M-1)}{2}}=(\pm i)^{M-1}=(-i)^{M-1},$

where we have used that $i^{M-1}=(-i)^{M-1}$ for $M$ odd. Finally, (4.15)
follows after evaluating the remaining products using the fact that
$\prod_{i=1}^{q-1}1-y^{i}=q$ for any primitive $q$-th root of unity. ∎

Next, we claim that the following generalization of [10, Lemma 9] holds:

######  Claim.

(4.18)
$\frac{\prod_{k=1}^{M-1}(1-z_{0}^{2k})^{v(k)}}{\prod_{k=0}^{M-1}(1-z_{0}^{2k+1})^{v(k)}}=\frac{M}{2}\times\begin{cases}1&\mbox{for
$\alpha=1$,}\\\ \pm\alpha^{-\frac{1}{2}}(1-\alpha)&\mbox{for $\alpha\neq
1$.}\end{cases}$

###### Proof of the claim.

To show this, first observe that for $|p|\leq\frac{M}{2}$:

$v(k)=\begin{cases}0&\mbox{for $k<|p|$,}\\\ 1&\mbox{for $|p|\leq k<M-|p|$,}\\\
2&\mbox{for $M-|p|\leq k<M$.}\end{cases}$

Then one can again combine terms in the products as above to find that the
left hand side in (4.18) equals

$\frac{\prod_{k=1}^{M-1}(1-z_{0}^{2k})}{\prod_{k=0}^{M-1}(1-z_{0}^{2k+1})}\times\begin{cases}1&\mbox{for
$p=0$,}\\\ (-z_{0}^{|p|})(1-z_{0}^{-2|p|})&\mbox{for $p\neq 0$.}\end{cases}$

Evaluating the products as before and rewriting

$(-z_{0}^{|p|})(1-z_{0}^{-2|p|})=\pm z_{0}^{-p}(1-z_{0}^{2p})$

shows (4.18), using $z_{0}^{2p}=\alpha$. ∎

Now we are back to the proof of 4.3.7. We insert (4.15) and (4.18) into
(4.14), but we need to distinguish the case $\alpha=1$ from the case
$\alpha\neq 1$. Indeed, if $\alpha=1$, one finds $v(0)=1$, while $v(0)=0$ if
$\alpha\neq 1$.

Now for each $\alpha$, the contribution to
$\int_{0}^{\frac{1}{2}-\varepsilon}E_{j}$ equals

$E_{j}(\alpha)=\begin{cases}\pm\alpha^{-1/2}(1-\alpha)(2i\pi\varepsilon)^{-\frac{1}{M}}\,i^{1-\frac{1}{M}}&\text{for
}\alpha\neq 1,\\\
\pm(2i\pi\varepsilon)^{1-\frac{1}{M}}\,i^{1-\frac{1}{M}}M^{-1}&\text{for
}\alpha=1.\end{cases}$

Thanks to 4.3.8 we can deal with the $M$-th roots of $i$ involved in (4.14),
in particular they cancel out in the case $\alpha=1$. We obtain

$\int_{0}^{\frac{1}{2}-\varepsilon}E_{j}=E_{j}(1)^{n_{j}}\prod_{\alpha\neq
1}E_{j}(\alpha)\\\
=\pm\left(\frac{(2\pi\varepsilon)^{1-\frac{1}{M}}}{M}\right)^{n_{j}}\prod_{\alpha\neq
1}\alpha^{-\frac{1}{2}}(1-\alpha)(2i\pi\varepsilon)^{-\frac{1}{M}}\,i^{1-\frac{1}{M}}\\\
=\det(T_{j})^{-\frac{1}{2}}\det(I_{n-n_{j}}-T_{j})(2\pi\varepsilon)^{n_{j}-\frac{n}{M}}\,i^{n-n_{j}}M^{-n_{j}}$

Replacing $M$ by $\nu_{j}$ finishes the proof of 4.3.7. ∎

###### Proof of 4.3.2.

First we assume $m=0$. Then $\rho(t)=I_{n}$, and 4.3.2 (1) reads

$R\left(\frac{x}{2\pi};\rho\right)\sim_{x\to
0}\frac{(ix)^{n(2g-2+s)-\sum_{j=1}^{s}n_{j}}}{\prod_{j=1}^{s}\det(I_{n-n_{j}}-T_{j})\nu_{j}^{n_{j}}}+o(x).$

Now combining the second statement of 4.3.5 with 4.3.7 yields

$\eta\left(1+\frac{x}{2\pi};\rho\right)\sim_{x\to 0}\pm
x^{\dim(V_{\rho})\chi(X)}\\\ \times\prod_{j=1}^{s}(\det
T_{j})^{-\frac{1}{2}}\det(I_{n-n_{j}}-T_{j})x^{-\dim(V_{\rho})/\nu_{j}+n_{j}}\,i^{n-n_{j}}\nu_{j}^{-n_{j}}+o(x).$

Recall that $n=\dim(V_{\rho})$, $-\chi(X)=2g-2+s-\sum_{i}\frac{1}{\nu_{j}}$
and that

$\prod_{j=1}^{s}\det
T_{j}=\prod_{j=1}^{s}\det\rho(c_{j})=\det\rho(t)^{2g-2+s}=1$

by (1.1). Moreover, observe that

$\prod_{j=1}^{s}i^{n-n_{j}}=i^{ns-\sum_{j}n_{j}}=\pm
i^{n(2g-2+s)-\sum_{j=1}^{s}n_{j}}$

and the result follows.

Now if $m\neq 0$, we directly deduce 4.3.2 (2) from the first statement in
4.3.5 and 4.3.6, since $\rho(t)=e^{-i\pi m}I_{n}$. ∎

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L. Bénard: Mathematisches Institut, Georg–August Universität Göttingen,
Bunsenstraße 3–5, 37073 Göttingen, Germany

E-mail address<EMAIL_ADDRESS>

J. Frahm: Department of Mathematics, Aarhus University, Ny Munkegade 118,
8000, Aarhus C, Denmark

E-mail address<EMAIL_ADDRESS>

P. Spilioti: Department of Mathematics, Aarhus University, Ny Munkegade 118,
8000, Aarhus C, Denmark

E-mail address<EMAIL_ADDRESS>