#  Virtual homological eigenvalues and
the Weil–Petersson translation length

Yi Liu  Beijing International Center for Mathematical Research, Peking
University
Beijing 100871, China P.R<EMAIL_ADDRESS>
###### Abstract.

For any pseudo-Anosov automorphism on an orientable closed surface, an
inquality is established bounding certain growth of virtual homological
eigenvalues with the Weil–Petersson translation length. The new inquality fits
nicely with other known inequalities due to Kojima and McShane, and due to Lê.

The new quantity to be considered is the square sum of the logarithmic radii
of the homological eigenvalues (with multiplicity) outside the complex unit
circle, called the homological _Jensen square sum_. The main theorem is as
follows. For any cofinal sequence of regular finite covers of a given surface,
together with lifts of a given pseudo-Anosov, the homological Jensen square
sum of the lifts grows at most linearly fast compared to the covering degree,
and the square root of the growth rate is at most $1/\sqrt{4\pi}$ times the
Weil–Petersson translation length of the given pseudo-Anosov.

###### Key words and phrases:

homological eigenvalue, finite cover, Weil–Petersson metric, translation
length

###### 2020 Mathematics Subject Classification:

Primary 57K20; Secondary 57K31

Partially supported by NSFC Grant 11925101, and National Key R&D Program of
China 2020YFA0712800

##  1\. Introduction

Let $S$ be a connected closed orientable surface of genus at least $2$. Let
$f\colon S\to S$ be a pseudo-Anosov automorphism. In [KojM18], S. Kojima and
G. McShane establish an elegant inequality, which we can rearrange as

(1.1)
$\frac{\mathrm{Vol}_{\mathtt{h}}(M_{f})}{6\pi}\leq\frac{\ell_{\mathtt{WP}}(f)}{\sqrt{4\pi}}\times\sqrt{\mathrm{genus}(S)-1}.$

In this inequality, $\mathrm{Vol}_{\mathtt{h}}(M_{f})$ denotes the hyperbolic
volume of the mapping torus $M_{f}$ of $f$, with respect to its isometrically
unique hyperbolic geometry, and $\ell_{\mathtt{WP}}(f)$ denotes the
Weil–Petersson translation length of $f$, as a mapping class acting on the
Teichmüller space of $S$.

The left-hand side of (1.1) is equal to the logarithmic $L^{2}$ torsion of the
orientable closed $3$–manifold $M_{f}$; see [Lüc02, Chapter 3] (with a
different sign convention). This quantity is a known upper bound for the
superior linear growth rate of the virtual homological logarithmic Mahler
measures, with respect to any cofinal sequence of regular finite covers
$(S^{\prime}_{n})_{n\in{\mathbb{N}}}$ of $S$ and lifts
$(f^{\prime}_{n})_{n\in{\mathbb{N}}}$ of $f$. To be precise, if
$\kappa_{n}\colon S^{\prime}_{n}\to S$ are cofinal regular finite covering
maps (that is, the corresponding finite-index normal subgroups of $\pi_{1}(S)$
have trivial common intersection), and if $f^{\prime}_{n}\colon
S^{\prime}_{n}\to S^{\prime}_{n}$ are coverings of $f$ (that is,
$\kappa_{n}\circ f^{\prime}_{n}=f\circ\kappa_{n}$), then the following
inequality holds,

(1.2)
$\varlimsup_{n\to\infty}\frac{\mathbf{m}\left(P_{n}\right)}{[S^{\prime}_{n}:S]}\leq\frac{\mathrm{Vol}_{\mathtt{h}}(M_{f})}{6\pi}.$

In this inequality, $[S^{\prime}_{n}:S]$ denotes the covering degree, and
$\mathbf{m}(P_{n})$ denotes the logarithmic Mahler measure of the
characteristic polynomial $P_{n}$ of the induced linear automorphism
$f^{\prime}_{n*}\colon H_{1}(S^{\prime}_{n};{\mathbb{C}})\to
H_{1}(S^{\prime}_{n};{\mathbb{C}})$. In fact, (1.2) follows easily from an
inequality due to T. T. Q. Lê [Lê18], combined with [Lê14, Theorem 1].

We recall that for any nonzero complex polynomial $Q=Q(z)$, the _logarithmic
Mahler measure_ of $Q$ refers to the non-negative quantity

$\mathbf{m}(Q)=\frac{1}{2\pi}\int_{0}^{2\pi}\log\left|Q\left(e^{\sqrt{-1}\cdot\theta}\right)\right|\,{\mathrm{d}}\theta.$

When $Q$ is monic of degree $d$ with complex roots
$\lambda_{1},\cdots,\lambda_{d}$, the Jensen formula implies that
$\mathbf{m}(Q)$ measures exactly the total logarithmic radii of those roots
outside the complex unit circle, namely,

$\mathbf{m}(Q)=\sum_{j=1}^{d}\log\max(1,\left|\lambda_{j}\right|).$

See [EveW99, Chapter 1].

For any monic polynomial $Q$ of degree $d$ with roots
$\lambda_{1},\cdots,\lambda_{d}$, we introduce another non-negative quantity

(1.3)
$\mathbf{w}(Q)=\sum_{j=1}^{d}\,\log^{2}\max\left(1,\left|\lambda_{j}\right|\right).$

In general, $\mathbf{w}(Q)$ is very different from the average of
$\log^{2}|Q|$ over the unit circle. The latter has been called the second
higher Mahler measure; compare [KurLO08]. Let us call $\mathbf{w}(Q)$ the
_Jensen square sum_ of $Q$.

The polynomials $P_{n}$ in (1.2) are monic of degree
$2\times\mathrm{genus}(S^{\prime}_{n})$. Moreover, they are all reciprocal. We
observe an elementary Cauchy–Schwartz inequality,

$\frac{\mathbf{m}(P_{n})}{[S^{\prime}_{n}:S]}\leq\sqrt{\frac{\mathbf{w}(P_{n})}{[S^{\prime}_{n}:S]}}\times\sqrt{\frac{\mathrm{genus}(S^{\prime}_{n})}{[S^{\prime}_{n}:S]}},$

where the limit of the last factor is clear,

$\lim_{n\to\infty}\sqrt{\frac{\mathrm{genus}(S^{\prime}_{n})}{[S^{\prime}_{n}:S]}}=\sqrt{\mathrm{genus}(S)-1}.$

In light of (1.1) and (1.2), there seems to be some unrevealed inequality
between the growth of $\mathbf{w}(P_{n})$ and ${\ell}_{\mathtt{WP}}(f)$. In
this paper, we establish that inequality, exactly in the predicted form.

This is the following main theorem.

######  Theorem 1.1.

Let $f\colon S\to S$ be a pseudo-Anosov automorphism on a connected closed
orientable surface of genus at least $2$. Then, for any cofinal sequence of
regular finite covers $(S^{\prime}_{n})_{n\in{\mathbb{N}}}$ of $S$ with lifts
$(f^{\prime}_{n})_{n\in{\mathbb{N}}}$ of $f$, the following inequality holds,

$\varlimsup_{n\to\infty}\,\sqrt{\frac{\mathbf{w}(P_{n})}{[S^{\prime}_{n}:S]}}\leq\frac{{\ell}_{\mathtt{WP}}(f)}{\sqrt{4\pi}},$

where $P_{n}$ denotes the characteristic polynomial of $f^{\prime}_{n*}$ on
$H_{1}(S^{\prime}_{n};{\mathbb{C}})$.

With notations of Theorem 1.1, denote by $\mathbf{h}(P_{n})$ the maximum of
$\log\max(1,|\lambda|)$ where $\lambda$ ranges over all the complex roots of
$P_{n}$. Denote by ${\ell}_{\mathtt{T}}(f)$ the Teichmüller translation length
of $f$, which is equal to the entropy of $f$. In [McM13], C. T. McMullen shows
that the well-known inequality

$\varlimsup_{n\to\infty}\,\mathbf{h}(P_{n})\leq{\ell}_{\mathtt{T}}(f),$

must be strict if the invariant foliations of $f$ have a prong singularity of
odd order, and moreover, in that case, there is a uniform gap that depends
only on $f$. It seems reasonable to guess that the inequality in Theorem 1.1
is strict for many, if not all, pseudo-Anosov $f$, and there is a uniform gap
that depends only on $f$.

For any pseudo-Anosov $(S,f)$, one can always find a sequence
$(S^{\prime}_{n},f^{\prime}_{n})_{n\in{\mathbb{N}}}$ as in Theorem 1.1, such
that $\mathbf{h}(P_{n})>0$ holds for each $f^{\prime}_{n}$; see [Liu20].
Hence, $\mathbf{m}(P_{n})>0$ and $\mathbf{w}(P_{n})>0$ also hold.

Note that $\varlimsup_{n}\mathbf{w}(P_{n})/[S^{\prime}_{n}:S]>0$ holds if and
only if $\varlimsup_{n}\mathbf{m}(P_{n})/[S^{\prime}_{n}:S]>0$ holds. Although
it appears heuristically very possible, there are no known examples of a
pseudo-Anosov $(S,f)$ and a sequence
$(S^{\prime}_{n},f^{\prime}_{n})_{n\in{\mathbb{N}}}$ as in Theorem 1.1, such
that $\mathbf{w}(P_{n})$ or $\mathbf{m}(P_{n})$ grows strictly linearly fast
compared to the covering degree.

Below, we outline the proof of Theorem 1.1. Our argument is inspired by
[McM13]. The main difference is that we make use of invariant metrics on the
Teichmüller space related to the Weil–Petersson metric, rather than the
Teichmüller metric.

Using the Bergman metric on Riemann surfaces $X$ marked by $S$, one obtains an
invariant Riemannian metric on the Teichmüller space $\mathrm{Teich}(S)$,
which we call the Habermann–Jost metric. The way is just the same as obtaining
the Weil–Petersson metric on $\mathrm{Teich}(S)$ from the conformal hyperbolic
metric on $X$. Using the natural map $\mathrm{Teich}(S)\to\mathfrak{H}(S)$ of
the Teichmüller space to the Siegel space of Hodge structures on
$H_{1}(S;{\mathbb{C}})$, one obtains an invariant Riemannian pseudometric on
$\mathrm{Teich}(S)$ pulling back the Siegel metric on $\mathfrak{H}(S)$, which
we call the Royden–Siegel metric. Upon suitable normalization, the $L^{2}$
norms of these metrics satisfy the comparison
$2\times\|\xi\|_{\mathtt{HJ}}\geq\|\xi\|_{\mathtt{RS}}$, for any tangent
vector $\xi\in T_{X}\mathrm{Teich}(S)$.

Moreover, virtual versions of the above metrics on $\mathrm{Teich}(S)$ can be
defined using regular finite covering $S^{\prime}\to S$ and the natural
embedding $\mathrm{Teich}(S)\to\mathrm{Teich}(S^{\prime})$. Upon suitable
normalization, the $L^{2}$ norms of the virtual Habermann–Jost metrics satisfy
the convergence
$\lim_{n}\|\xi\|_{\mathtt{HJ}^{\prime}_{n}}=\|\xi\|_{\mathtt{WP}}/\sqrt{4\pi}$,
for any sequence of covers $(S^{\prime}_{n})_{n\in{\mathbb{N}}}$ as assumed.
This is basically because the virtual Bergmann metrics
$g_{\mathtt{B}^{\prime}_{n}}$ on $X$ converges to the rescaled conformal
hyperbolic metric $g_{\mathtt{h}}/4\pi$.

With the above facts, consider any point on the Weil–Petersson axis in
$\mathrm{Teich}(S)$ of a given pseudo-Anosov $f$ on $S$. Then, essentially
speaking, we can obtain comparison of translation lengths (see Definition 2.2)
as follows:

$\frac{{\ell}_{\mathtt{WP}}(f)}{\sqrt{4\pi}}\geq\varlimsup_{n\to\infty}\,{\ell}_{\mathtt{HJ}^{\prime}_{n}}(f)\geq\varlimsup_{n\to\infty}\,\frac{{\ell}_{\mathtt{RS}^{\prime}_{n}}(f)}{2}\geq\varlimsup_{n\to\infty}\,\frac{{\ell}_{\mathtt{S}}(f^{\prime}_{n*})}{2\times\sqrt{[S^{\prime}_{n}:S]}}.$

The nominator ${\ell}_{\mathtt{S}}(f^{\prime}_{n*})$ in the last expression
refers to the Siegel translation length of
$f^{\prime}_{n*}\in\mathrm{Sp}(H_{1}(S^{\prime}_{n};{\mathbb{R}}))$ acting on
$\mathfrak{H}(S^{\prime}_{n})$. On the other hand, we can establish the
formula

${\ell}_{\mathtt{S}}(f^{\prime}_{n*})=2\times\sqrt{\mathbf{w}(P_{n})},$

by studying the Siegel geometry on the generalized upper half plane model.
Then the desired inequality follows.

This paper is organized as follows. In Section 2, we recall structures
associated to the Teichmüller space, and general properties of translation
length. In Section 3, we determine the Siegel translation length of a
symplectic linear transformation. In Section 4, we recall several invariant
Riemannian metrics on the Teichmüller space, as mentioned above, and discuss
their virtual versions. In Section 5, we prove two technical lemmas regarding
comparison and convergence of virtual metrics. In Section 6, we prove our main
result (Theorem 1.1).

##  2\. Preliminaries

In this section, we recall background materials needed for our discussion. For
general reference books, see [Hub06] for Teichmüller theory, and [Sie64] for
the Siegel geometry, and [BriH99, Chapter II.6] about the translation length
of isometries on metric spaces. We only consider closed surfaces of hyperbolic
type, as it suffices for our purpose.

###  2.1. The Teichmüller space

Let $S$ be an oriented closed surface of genus at least $2$. The _Teichmüller
space_ $\mathrm{Teich}(S)$ consists of all the isotopy classes of complex
structures on $S$, compatible with the fixed orientation. Equivalently, we
think of any point of $\mathrm{Teich}(S)$ as represented by a Riemann surface
$X$ together with an orientation-preserving homeomorphism $S\to X$, called the
_marking_ of $X$, and denote the point as $X$ with the marking implicit.

We think of $\mathrm{Teich}(S)$ as a smooth (real) manifold, diffeomorphic to
an open cell of dimension $6\times(\mathrm{genus}(S)-1)$. There are natural
identifications of the tangent and the cotangent spaces (as real vector
spaces)

$\displaystyle T^{*}_{X}\mathrm{Teich}(S)$ $\displaystyle\cong$ $\displaystyle
Q(X),$ $\displaystyle T_{X}\mathrm{Teich}(S)$ $\displaystyle\cong$
$\displaystyle B(X)/Q(X)^{\perp},$

at any point $X\in\mathrm{Teich}(S)$. Here, $Q(X)$ denotes the space of all
the holomorphic quadratic differentials on $X$, and $B(X)$ denotes the space
of all the $L^{\infty}$ Beltrami differentials on $X$. These spaces pair
naturally as

$Q(X)\times B(X)\longrightarrow{\mathbb{R}}\colon(q,\mu)\mapsto\int_{X}q\mu,$

so $Q(X)^{\perp}$ refers to the subspace of $B(X)$ annihilated by $Q(X)$,
namely, $\mu\in B(X)$ lies in $Q(X)^{\perp}$ if and only if $\int_{X}q\mu=0$
holds for all $q\in Q(X)$.

The _mapping class group_ $\mathrm{Mod}(S)$ of $S$, consisting of all the
isotopy classes of orientation-preserving self-homeomorphisms, acts properly
and diffeomorphically on $\mathrm{Teich}(S)$, transforming the markings of
points.

There are many natural differential metrics (or pseudometrics) on
$\mathrm{Teich}(S)$ that are invariant under the action of $\mathrm{Mod}(S)$.
For example, the _Teichmüller metric_ $d_{\mathtt{T}}$ is an invariant Finsler
metric determined uniquely by the formula

$d_{\mathtt{T}}(X,Y)=\frac{\log K(X,Y)}{2},$

for any $X,Y\in\mathrm{Teich}(S)$, where $K(X,Y)>1$ denotes the
quasiconformity constant of the Teichmüller extremal map $X\to Y$ that
commutes homotopically with the markings. The infinitesmal forms of
$d_{\mathtt{T}}$ are the norms

$\|q\|_{\mathtt{T}}=\int_{X}|q|$

for any $q$ in $Q(X)\cong T^{*}_{X}\mathrm{Teich}(S)$, and

$\|\xi\|_{\mathtt{T}}=\sup\left\\{\left|\int_{X}q\mu\right|\colon\int_{X}|q|\leq
1\right\\}$

for any $\xi=\mu+Q(X)^{\perp}$ in $B(X)/Q(X)^{\perp}\cong
T_{X}\mathrm{Teich}(S)$.

We recall the Weil–Petersson metric $d_{\mathtt{WP}}$ and other invariant
Riemannian metrics on $\mathrm{Teich}(S)$ in Section 4 with more discussion.

###  2.2. The Siegel space

Let $V$ be a real vector space of dimension $2p$ furnished with a symplectic
form $\omega$. Then $\omega$ extends complex linearly over the complex vector
space $V\otimes_{\mathbb{R}}{\mathbb{C}}$ as a complex-valued alternating
$2$-form, and determines a Hermitian form of signature $(p,p)$ on
$V\otimes_{\mathbb{R}}{\mathbb{C}}$,

$\langle z,w\rangle=\frac{\sqrt{-1}}{2}\cdot\omega(z,\bar{w}).$

The _Siegel space_ $\mathfrak{H}(V)$ consists of all the Hermitian orthogonal
splittings

$V\otimes_{\mathbb{R}}{\mathbb{C}}=V^{1,0}\oplus V^{0,1},$

such that $\langle\\_,\\_\rangle$ is positive definite on $V^{1,0}$, and the
complex conjugation $z\mapsto\bar{z}$ maps $V^{1,0}$ isomorphically onto
$V^{0,1}$. We call any point of the Siegel space $\mathfrak{H}(V)$ a _Hodge
structure_ on $V\otimes_{\mathbb{R}}{\mathbb{C}}$ with respect to $\omega$.

The group $\mathrm{Sp}(V)\cong\mathrm{Sp}(2p,{\mathbb{R}})$ of symplectic
linear transformations acts transitively on $\mathfrak{H}(V)$, such that any
$\varphi\in\mathrm{Sp}(V)$ extends complex linearly over
$V\otimes_{\mathbb{R}}{\mathbb{C}}$, and takes any Hodge structure
$V^{1,0}\oplus V^{0,1}$ to $\varphi(V^{1,0})\oplus\varphi(V^{0,1})$. The
isotropy group at $V^{1,0}\oplus V^{0,1}$ is isomorphic to the unitary group
$\mathrm{U}(V^{1,0})\cong\mathrm{U}(p)$, acting simultaneously on the summands
$V^{1,0}$ and $V^{0,1}$ as canonical unitary transformations and their complex
conjugates, respectively. In particular, $\mathfrak{H}(V)$ is the homogenous
space associated to $\mathrm{Sp}(V)$, diffeomorphic to an open cell of
dimension $p^{2}+p$.

The most important example to our interest is the Siegel space of Hodge
structures on a surface. If $S$ is an oriented closed surface of genus at
least $2$, then the first real cohomology $H^{1}(S;{\mathbb{R}})$ is furnished
with a natural symplectic form, evaluating the cup product of any pair of
$1$-classes on the fundamental class. In this case, we simply denote the
Siegel space of Hodge structures on $H^{1}(S;{\mathbb{C}})$ as
$\mathfrak{H}(S)$. For any Teichmüller point $X\in\mathrm{Teich}(S)$, we
obtain a unique Hermitian orthogonal splitting

$H^{1}(S;{\mathbb{C}})\cong H^{1}(X;{\mathbb{C}})=H^{1,0}(X)\oplus
H^{0,1}(X)\cong\Omega^{1}(X)\oplus\overline{\Omega^{1}(X)},$

where the complex linear subspaces $H^{1,0}(X)$ and $H^{0,1}(X)$ of
$H^{1}(X;{\mathbb{C}})$ are naturally identified as the spaces $\Omega^{1}(X)$
and $\overline{\Omega^{1}(X)}$ of the holomorphic and the anti-holomorphic
differentials on $X$, respectively, and where $H^{1}(S;{\mathbb{C}})$ is
identified with $H^{1}(X;{\mathbb{C}})$ via the marking. Therefore, the
construction determines a natural map

(2.1) $J\colon\mathrm{Teich}(S)\to\mathfrak{H}(S),$

which is equivariant with respect to the natural group homomorphism
$\mathrm{Mod}(S)\to\mathrm{Sp}(H^{1}(S;{\mathbb{R}}))$. It is known that $J$
is surjective and smooth, and the tangent map of $J$ is injective except along
the hyperelliptic locus; see [Roy74, Section 3] for more information.

Back to the general setting with $(V,\omega)$. There is a unique invariant
Riemannian metric on $\mathfrak{H}(V)$ with respect to $\mathrm{Sp}(V)$, up to
normalization. In fact, $(\mathfrak{H}(V),\mathrm{Sp}(V))$ forms a simple
Riemannian symmetric space of noncompact type, (and morover, a Hermitian
symmetric domain). This metric has been studied by C. L. Siegel systematically
in his expository book [Sie64]. Fixing a symplectic basis of $V$, we can
identify $V$ as the real linear space spanned by
$\vec{x}_{1},\cdots,\vec{x}_{p},\vec{y}_{1},\cdots,\vec{y}_{p}$ and furnished
with the standard symplectic form
$\vec{x}^{*}_{1}\wedge\vec{y}^{*}_{1}+\cdots+\vec{x}^{*}_{p}\wedge\vec{y}^{*}_{p}$.
Then the transformational geometry of the Siegel space
$(\mathfrak{H}(V),\mathrm{Sp}(V))$ can be identified with Siegel’s generalized
upper half plane model, on which the invariant metric can be described
explicitly. We recall the explicit description below, following Siegel’s
normalization in [Sie64].

###  2.3. The generalized upper half plane

For any natural number $p$, the _generalized upper half plane_
$\mathfrak{H}_{p}$ of rank $p$ refers to the open subset of symmetric $p\times
p$–matrices of complex entries, such that the imaginary parts are positive
definite. Namely,

(2.2) $\mathfrak{H}_{p}=\left\\{Z\in\mathrm{Sym}_{p\times
p}({\mathbb{C}})\colon\Im Z>0\right\\},$

where we denote $\mathrm{Sym}_{p\times
p}({\mathbb{C}})=\\{Z\in\mathrm{Mat}_{p\times p}({\mathbb{C}})\colon
Z^{\dagger}=Z\\}$ ($\dagger$ meaning transpose) and $\Im
Z=(Z-\bar{Z})/2\sqrt{-1}$ [Sie64, Chapter I, §2].

The Riemannian metric tensor of the the Siegel metric at any point
$Z\in\mathfrak{H}_{p}$ is defined as

(2.3) $g_{\mathtt{S}}=\mathrm{tr}\left(Y^{-1}\cdot{\mathrm{d}}Z\cdot
Y^{-1}\cdot\overline{{\mathrm{d}}Z}\right)$

where we denote $Y=\Im Z$ and treat the matrix entries of the real and the
imaginary parts as coordinates [Sie64, Chapter III, §11]. The Siegel metric
$g_{\mathtt{S}}$ has nonpositive sectional curvature everywhere [Sie64,
Chapter III, §17]. In other words, $(\mathfrak{H}_{p},g_{\mathtt{S}})$ forms a
Hadamard manifold, (by definition, simply connected, metric complete, and
nonpositively curved).

For any pair of points $Z,W\in\mathfrak{H}_{p}$, the generalized cross-ratio
matrix

$R=R(Z,W)=(Z-W)\cdot(Z-\bar{W})^{-1}\cdot(\bar{Z}-\bar{W})\cdot(\bar{Z}-W)^{-1}$

is defined in $\mathrm{Mat}_{p\times p}({\mathbb{C}})$, with all
characteristic roots contained in $[0,1)$. Hence the series expression

$\log^{2}\left(\frac{1+R^{1/2}}{1-R^{1/2}}\right)=4R\cdot\left(\sum_{m=0}^{\infty}\frac{R^{m}}{2m+1}\right)^{2}$

converges absolutely to a matrix in $\mathrm{Mat}_{p\times p}({\mathbb{C}})$.
The Siegel distance between the pair of points $Z,W\in\mathfrak{H}_{p}$ can be
calculated as

(2.4)
$d_{\mathtt{S}}(Z,W)=\sqrt{\mathrm{tr}\left(\log^{2}\left(\frac{1+R^{1/2}}{1-R^{1/2}}\right)\right)}.$

See [Sie64, Chapter III, §13].

The symplectic group $\mathrm{Sp}(2p,{\mathbb{R}})$ acts on $\mathfrak{H}_{p}$
by generalized fractional linear transformations, as follows. Any matrix
$\varphi\in\mathrm{Sp}(2p,{\mathbb{R}})$ can be divided into four blocks
$A,B,C,D\in\mathrm{Mat}_{p\times p}({\mathbb{R}})$, such that the basis
transforms as

$\varphi_{*}(\begin{array}[]{cccccc}\vec{x}_{1}&\cdots&\vec{x}_{p}&\vec{y}_{1}&\cdots&\vec{y}_{p}\end{array})=(\begin{array}[]{cccccc}\vec{x}_{1}&\cdots&\vec{x}_{p}&\vec{y}_{1}&\cdots&\vec{y}_{p}\end{array})\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right).$

Then $\varphi$ being symplectic is equivalent to the conditions
$A^{\dagger}C=C^{\dagger}A$, and $D^{\dagger}B=B^{\dagger}D$ and
$A^{\dagger}D-C^{\dagger}B=I$ altogether ($I$ denoting the identity matrix).
The action of $\mathrm{Sp}(2p,{\mathbb{R}})$ on $\mathfrak{H}_{p}$ is
explicitly

$\varphi\colon Z\mapsto(AZ+B)\cdot(CZ+D)^{-1}$

for any $\varphi\in\mathrm{Sp}(2p,{\mathbb{R}})$ and $Z\in\mathfrak{H}_{p}$.
See [Sie64, Chapter II, §§4–5].

######  Remark 2.1.

For any real vector space $V$ of dimension $2p$ furnished with a symplectic
form $\omega$, an identification
$(\mathfrak{H}(V),\mathrm{Sp}(V))\cong(\mathfrak{H}_{p},\mathrm{Sp}(2p,{\mathbb{R}}))$
is canonically determined by fixing a symplectic basis of $V$. Since the
canonical identifications $V\cong{\mathbb{R}}^{2p}$ and
$\mathrm{Sp}(V)\cong\mathrm{Sp}(2p,{\mathbb{R}})$ are obvious, it remains to
identify any point $Z\in\mathfrak{H}_{p}$ with a Hodge structure on
$V\otimes_{\mathbb{R}}{\mathbb{C}}\cong{\mathbb{C}}^{2p}$. We first require
the distinguished point $Z=\sqrt{-1}\cdot I$ to be identified with the Hodge
structure $V^{1,0}\oplus V^{0,1}$ where $V^{1,0}$ is the real linear subspace
spanned by the complex vectors:

$\begin{array}[]{ccc}\vec{z}_{1}=\vec{x}_{1}+\sqrt{-1}\cdot\vec{y}_{1},&\cdots,&\vec{z}_{p}=\vec{x}_{p}+\sqrt{-1}\cdot\vec{y}_{p}\end{array}$

Hence, $V^{0,1}=\overline{V^{1,0}}$. In general, any point $X+\sqrt{-1}\cdot
Y$ in $\mathfrak{H}_{p}$ is the transformation image of $\sqrt{-1}\cdot I$
under the symplectic matrix

$M=\left(\begin{array}[]{cc}I&X\\\
0&I\end{array}\right)\cdot\left(\begin{array}[]{cc}Y^{1/2}&0\\\
0&Y^{-1/2}\end{array}\right)=\left(\begin{array}[]{cc}Y^{1/2}&X\cdot
Y^{-1/2}\\\ 0&Y^{-1/2}\end{array}\right),$

so the Hodge structure corresponding to $Z=X+\sqrt{-1}\cdot Y$ has $V^{1,0}$
spanned by $M\vec{z}_{1},\cdots,M\vec{z}_{p}$.

###  2.4. Translation length

######  Definition 2.2.

For any metric space $({\mathcal{X}},d_{\mathcal{X}})$, the _translation
length_ of an isometry $\sigma\colon{\mathcal{X}}\to{\mathcal{X}}$ is defined
as

${\ell}_{\mathcal{X}}(\sigma)=\inf_{x\in{\mathcal{X}}}d_{\mathcal{X}}(x,\sigma.x).$

The translation length can be thought of as a function on the isometry group
of ${\mathcal{X}}$,

${\ell}_{\mathcal{X}}\colon\mathrm{Isom}({\mathcal{X}})\to[0,+\infty).$

This function is clearly invariant under conjugations and inversion. Moreover,
it satisfies the following upper semi-continuity with respect to sequences of
pointwise convergence.

######  Lemma 2.3.

If $(\sigma_{m})_{m\in{\mathbb{N}}}$ is a sequence of isometries on a metric
space $({\mathcal{X}},d_{\mathcal{X}})$ that converges to an isometry
$\varphi_{\infty}$ everywhere in ${\mathcal{X}}$, then

$\varlimsup_{m\to\infty}{\ell}_{\mathcal{X}}(\sigma_{m})\leq{\ell}_{\mathcal{X}}(\sigma_{\infty}).$

###### Proof.

For any $x\in{\mathcal{X}}$ and $\epsilon>0$, we obtain
$d_{\mathcal{X}}(x,\sigma_{m}.x)\leq
d_{\mathcal{X}}(x,\sigma_{\infty}.x)+\epsilon$ for all sufficiently large $m$.
Then ${\ell}_{\mathcal{X}}(\sigma_{m})\leq
d_{\mathcal{X}}(x,\sigma_{\infty}.x)+\epsilon$, and then
$\varlimsup_{m\to\infty}{\ell}_{\mathcal{X}}(\sigma_{m})\leq
d_{\mathcal{X}}(x,\sigma_{\infty}.x)+\epsilon$. Taking the infimum over all
$x\in{\mathcal{X}}$, we obtain
$\varlimsup_{m\to\infty}{\ell}_{\mathcal{X}}(\sigma_{m})\leq{\ell}_{\mathcal{X}}(\sigma_{\infty})+\epsilon$.
Let $\epsilon$ tend to $0+$, the asserted inequality follows. ∎

##  3\. The Siegel translation length of symplectic linear transformations

In this section, we figure out the translation length of a symplectic matrix
with respect to its fractional linear transformation on the generalized upper
half plane and the Siegel metric. This is not a difficult task, the most
importantly because symplectic linear transformations have been well
understood. In fact, conjugacy classes in $\mathrm{Sp}(2p,{\mathbb{R}})$ have
been classified with normal forms all listed. The theory is similar to the
well-known Jordan normal form theory for $\mathrm{SL}(n,{\mathbb{R}})$,
although notationally more involved. For details of that theory, we refer to
J. Gutt [Gu14]. However, we need a few ingredients from [Gu14] in our proof of
Theorem 3.1, and we elaborate when we use them.

######  Theorem 3.1.

Let $(V,\omega)$ be a finite-dimensional real symplectic vector space. Then,
for any symplectic linear transformation $\varphi\in\mathrm{Sp}(V)$, the
Siegel translation length of $\varphi$ satisfies the formula

${\ell}_{\mathtt{S}}(\varphi)=2\times\sqrt{\mathbf{w}(P_{\varphi})}$

where $P_{\varphi}$ denotes the characteristic polynomial of $\varphi$, and
where $\mathbf{w}$ denotes the Jensen square sum as defined in (1.3).

The rest of this section is devoted to the proof of Theorem 3.1.

######  Lemma 3.2.

If $V=V_{1}\oplus\cdots\oplus V_{k}$ is a symplectic orthogonal decomposition
of $V$ into symplectic summands $V_{i}$ that are invariant under $\varphi$.
Then

${\ell}_{\mathtt{S}}(\varphi)=\sqrt{{\ell}_{\mathtt{S}}(\varphi_{1})^{2}+\cdots+{\ell}_{\mathtt{S}}(\varphi_{k})^{2}}$

where $\varphi_{i}\in\mathrm{Sp}(V_{i})$ denotes the restricted transformation
of $\varphi$.

###### Proof.

The decomposition induces a natural inclusion
$\mathfrak{H}(V_{1})\times\cdots\times\mathfrak{H}(V_{k})$ into
$\mathfrak{H}(V)$. With respect to the cartesian product of Siegel metrics on
$\mathfrak{H}(V_{1})\times\cdots\times\mathfrak{H}(V_{k})$ and the Siegel
metric on $\mathfrak{H}(V)$, the inclusion is an isometric embedding with
totally geodesic image, by (2.3) and simple observation. Since the Siegel
space $\mathfrak{H}(V)$ is Hadamard with respect to the Siegel metric, the
closest point projection
$\mathfrak{H}(V)\to\mathfrak{H}(V_{1})\times\cdots\times\mathfrak{H}(V_{k})$
is well-defined and distance non-increasing; see [BriH99, Chapter II,
Corollary 2.5]. It follows that ${\ell}_{\mathtt{S}}(\varphi)$ is witnessed by
some sequence of points in
$\mathfrak{H}(V_{1})\times\cdots\times\mathfrak{H}(V_{k})$, so it agrees with
the translation distance of $\varphi$ acting on
$\mathfrak{H}(V_{1})\times\cdots\times\mathfrak{H}(V_{k})$. The latter is
evidently equal to
$({\ell}_{\mathtt{S}}(\varphi_{1})^{2}+\cdots+{\ell}_{\mathtt{S}}(\varphi_{k})^{2})^{1/2}$.
∎

Let $(V,\omega)$ be a finite-dimensional real symplectic vector space. For any
element $\varphi\in\mathrm{Sp}(V)$, it follows from the Jordan–Chevalley
decomposition that there is a unique, commutative factorization of $\varphi$
in $\mathrm{Sp}(V)$,

$\varphi=\varphi_{\mathtt{ss}}\varphi_{\mathtt{u}}=\varphi_{\mathtt{u}}\varphi_{\mathtt{ss}},$

such that the eigenvectors of $\varphi_{\mathtt{ss}}$ span
$V\otimes_{\mathbb{R}}{\mathbb{C}}$, and the only eigenvalue of
$\varphi_{\mathtt{u}}$ is $1$. In fact, the Jordan–Chevalley decomposition in
the semisimple real Lie algebra $\mathfrak{sp}(V)$ is the same as inherited
from $\mathfrak{gl}(V)$, and it detemines the factorization in the linear
group $\mathrm{Sp}(V)$; see [Hum72, Chapter II, §6.4].

We refer to $\varphi_{\mathtt{ss}}$ and $\varphi_{\mathtt{u}}$ as the
_semisimple_ factor and the _unipotent_ factor of $\varphi$, respectively.

######  Lemma 3.3.

Theorem 3.1 holds if $\varphi=\varphi_{\mathtt{ss}}$.

###### Proof.

We say that $(V,\varphi)$ is _symplectic simple_ if
$\varphi=\varphi_{\mathtt{ss}}$, and if $V$ contains no nontrivial
$\varphi$–invariant symplectic subspaces. In this case, $(V,\varphi)$
decomposes as a direct sum of symplectic simple transformations
$(V_{i},\varphi_{i})$. In view of the Pythagorean formula (Lemma 3.2) for
${\ell}_{\mathtt{S}}(\varphi)$ and the obvious similar formula for
$2\times\sqrt{\mathbf{w}(P_{\varphi})}$, it suffices to prove the lemma for
any symplectic simple transformation.

Any symplectic simple transformation $(V,\varphi)$ is either $2$–dimensional
central (with a unique eigenvalue $\lambda=\pm 1$ of multiplicity $2$), or
$2$–dimensional hyperbolic (with distinct real eigenvalues
$\\{\lambda,1/\lambda\\}$), or $2$–dimensional elliptic (with distinct
unimodular eigenvalues $\\{\lambda,\bar{\lambda}\\}$), or otherwise,
$4$–dimensional ‘loxodromic’ (with distinct complex eigenvalues
$\\{\lambda,\bar{\lambda},1/\lambda,1/\bar{\lambda}\\}$ off the unit circle
and the real axis). See [Gu14, Theorem 1.2] for general normal forms of
matrices in $\mathrm{Sp}(2p,{\mathbb{R}})$.

When $V$ is $2$–dimensional, the formula for ${\ell}_{\mathtt{S}}(\varphi)$ is
actually well-known. In this case, the Siegel metric on Siegel space
$\mathfrak{H}(V)$ agrees with the hyperbolic metric on the upper-half complex
plane (of constant curvature $-1$), and
$\mathrm{Sp}(V)\cong\mathrm{SL}(2,{\mathbb{R}})$ acts the same way as the
fractional linear transformations, so

${\ell}_{\mathtt{S}}(\varphi)=\begin{cases}0&\varphi\mbox{
central/elliptic}\\\ |2\log|\lambda||&\varphi\mbox{ hyperbolic}\end{cases}.$

In any of the subcases, ${\ell}_{\mathtt{S}}(\varphi)$ is equal to
$2\times\sqrt{\mathbf{w}(P_{\varphi})}$, as asserted.

When $V$ is $4$–dimensional, the Siegel space $\mathfrak{H}(V)$ can be
identified with the generalized upper half plane $\mathfrak{H}_{2}$,
consisting of all the symmetric complex $2\times 2$–matrices with positive
definite imaginary part. Any symplectic transformation with distinct
eigenvalues $\\{\lambda,\bar{\lambda},1/\lambda,1/\bar{\lambda}\\}$ can be
conjugated in $\mathrm{Sp}(4,{\mathbb{R}})$ into the normal form

$\varphi=\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)=\left(\begin{array}[]{cc}\rho\cdot\left({\begin{array}[]{cc}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{array}}\right)&\\\
&\rho^{-1}\cdot\left({\begin{array}[]{cc}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{array}}\right)\end{array}\right)$

denoting $\rho=|\lambda|$ and $\theta=\arg\lambda$. Recall that $\varphi$ acts
isometrically on $\mathfrak{H}_{2}$ as $Z\mapsto(AZ+B)(CZ+D)^{-1}$. The purely
positive imaginary scalar $2\times 2$–matrices form a $\varphi$–invariant
geodesic $l$ in the Hadamard manifold $\mathfrak{H}_{2}$. As the closest point
projection $\mathfrak{H}_{2}\to l$ is distance non-increasing [BriH99, Chapter
II, Corollary 2.5], the translation length of $\varphi$ is realized at every
point on $l$. On $l$ we can apply (2.4) and compute directly

${\ell}_{\mathtt{S}}(\varphi)=\sqrt{2}\cdot\left|2\log|\lambda|\right|,$

which is also $2\times\sqrt{\mathbf{w}(P_{\varphi})}$, as asserted. ∎

######  Lemma 3.4.

${\ell}_{\mathtt{S}}(\varphi)={\ell}_{\mathtt{S}}(\varphi_{\mathtt{ss}})$

###### Proof.

For any $r\in{\mathbb{R}}$, the element
$\varphi_{\mathtt{u}}^{r}\in\mathrm{Sp}(V)$ is well defined via the series
expansion $(1+x)^{r}=\sum_{j=0}^{\infty}{r\choose j}x^{j}$, because
$\varphi_{\mathtt{u}}-\mathbf{1}$ is nilpotent in
$\mathrm{End}_{\mathbb{R}}(V)$. Moreover, for any $r>0$, the element
$\varphi_{\mathtt{ss}}\varphi_{\mathtt{u}}^{r}$ is conjugate to
$\varphi=\varphi_{\mathtt{ss}}\varphi_{\mathtt{u}}$ in $\mathrm{Sp}(V)$. For
example, this follows from Gutt’s characterization of conjugacy classes; the
characterization only uses eigenvalues, and dimensions of generalized
eigenspaces, and the ranks and signatures of certain symmetric forms on the
generalized eigenspaces, all of which are obviously preserved as $r>0$ varies;
see [Gu14, Theorem 1.2]. Observe ${\ell}_{\mathtt{S}}$ is constant on any
conjugacy class of $\mathrm{Sp}(V)$. By Lemma 2.3, we obtain

${\ell}_{\mathtt{S}}(\varphi)=\lim_{m\to\infty}{\ell}_{\mathtt{S}}\left(\varphi_{\mathtt{ss}}\varphi_{\mathtt{u}}^{1/m}\right)\leq{\ell}_{\mathtt{S}}(\varphi_{\mathtt{ss}}).$

To see the other direction, we observe that symplectic linear transformations
with no redundant characteristic roots form a dense open subset of
$\mathrm{Sp}(V)$. (In fact, the discriminant of the characteristic polynomial
being zero determines an algebraic, proper subset of the real algebraic group
$\mathrm{Sp}(V)$, whose interior has to be empty.) Therefore, we can take a
sequence $(\psi_{m})_{m\in{\mathbb{N}}}$ in $\mathrm{Sp}(V)$, converging to
$\varphi$, such that each $\psi_{m}$ has no redudant characteristic roots. In
particular, $\psi_{m}$ are all semisimple. It follows from Lemma 3.3 that the
formula in Theorem 3.1 holds for all ${\ell}_{\mathtt{S}}(\psi_{m})$ and for
${\ell}_{\mathtt{S}}(\varphi_{\mathtt{ss}})$. Note that
$\varphi_{\mathtt{ss}}$ has the same characteristic polynomial as that of
$\varphi$, so the characteristic roots of $\psi_{m}$ converge to those of
$\varphi_{\mathtt{ss}}$. By Lemma 2.3, we obtain

${\ell}_{\mathtt{S}}(\varphi_{\mathtt{ss}})=\lim_{m\to\infty}{\ell}_{\mathtt{S}}(\psi_{m})\leq{\ell}_{\mathtt{S}}(\varphi).$

We conclude that ${\ell}_{\mathtt{S}}(\varphi)$ must be equal to
${\ell}_{\mathtt{S}}(\varphi_{\mathtt{ss}})$. ∎

Theorem 3.1 follows from Lemmas 3.3 and 3.4.

##  4\. Invariant Riemannian metrics on the Teichmüller space

In this section, we recall some invariant Riemannian metrics or pseudometrics
on the Teichmüller space, and also consider similar constructions associated
to regular finite covers of the surface. As some of the metrics have name
collision in the literature, we rename a few in this paper, not necessarily
suggesting an accurate attribution.

###  4.1. Ordinary versions

####  4.1.1. The Weil–Petersson metric

By the uniformization theorem, there is a unique conformal hyperbolic metric
$g_{\mathtt{h}}$ on $X$, (that is, a Riemannian metric of constant curvature
$-1$ conformal to the complex charts). The well-known Gauss–Bonnet theorem
implies

$\mathrm{Area}_{\mathtt{h}}(X)=-2\pi\chi(S)=4\pi\times(\mathrm{genus}(S)-1).$

With respect to $g_{\mathtt{h}}$, any tangent vector $\xi\in
T_{X}\mathrm{Teich}(S)$ is represented by a unique harmonic Beltrami
differential $\mu\in B(X)$, that is, $\mu=q/g_{\mathtt{h}}$ for some
holomorphic quadratic differential $q\in Q(X)$. The _Weil–Petersson metric_ on
$\mathrm{Teich}(S)$ is the Riemannian metric determined by

(4.1)
$\|\xi\|_{\mathtt{WP}}^{2}=\int_{X}|\mu|^{2}\,{\mathrm{d}}\mathrm{Area}_{\mathtt{h}}=\int_{X}\frac{q\bar{q}}{g_{\mathtt{h}}}$

for any $\xi\in T_{X}\mathrm{Teich}(S)=B(X)/Q(X)^{\perp}$ and the hyperbolic
harmonic representative $\mu=\overline{q/g_{\mathtt{h}}}$.

The Weil–Petersson metric on $\mathrm{Teich}(S)$ has nonpositive sectional
curvature. It is not complete, but any pair of points in $\mathrm{Teich}(S)$
can be connected with a unique geodesic segment. The mapping class group
$\mathrm{Mod}(S)$ acting on $\mathrm{Teich}(S)$ preserves the Weil–Petersson
metric. Moreover, any pseudo-Anosov mapping class $f\in\mathrm{Mod}(S)$ has an
axis, namely, a unique invariant complete geodesic, along which the
Weil–Petersson translation length ${\ell}_{\mathtt{WP}}(f)$ is realized. See
the survey [Wol09] of S. A. Wolpert for more information.

####  4.1.2. The Habermann–Jost metric

Using the Bergmann metric instead of the hyperbolic metric, we obtain what we
call the Habermann–Jost metric on $\mathrm{Teich}(S)$. This metric has been
studied by L. Habermann and J. Jost [HabJ96] in detail.

For any $X\in\mathrm{Teich}(S)$, the _Bergmann metric_ $g_{\mathtt{B}}$ on $X$
refers to the pullback of the canonical flat metric on
$\mathrm{Jac}(X)=\Omega^{1}(X)^{*}/H_{1}(X;{\mathbb{Z}})$ via the Abel–Jacobi
map $X\to\mathrm{Jac}(X)$, which is independent of the auxiliary base point.
To avoid confusion about the normalization, one makes sure

$\mathrm{Area}_{\mathtt{B}}(X)=\mathrm{genus}(S).$

The _Haberman–Jost metric_ on $\mathrm{Teich}(S)$ is the Riemannian metric
determined by

(4.2)
$\|\xi\|_{\mathtt{HJ}}^{2}=\int_{X}|\mu|^{2}\,{\mathrm{d}}\mathrm{Area}_{\mathtt{B}}=\int_{X}\frac{q\bar{q}}{g_{\mathtt{B}}},$

for any $\xi\in T_{X}\mathrm{Teich}(S)=B(X)/Q(X)^{\perp}$ and its Bergman
harmonic representative $\mu=\overline{q/g_{\mathtt{B}}}\in B(X)$, for some
unique $q\in Q(X)$.

######  Lemma 4.1.

Let $\theta_{1},\cdots,\theta_{p}$ be a Hermitian orthonormal basis of
$\Omega^{1}(X)$, where $p=\dim_{\mathbb{C}}\Omega^{1}(X)$ is equal to the
genus of $S$. Then, the following formulas hold,

$g_{\mathtt{B}}=\sum_{j=1}^{p}\theta_{j}\bar{\theta}_{j},$

and for any $\xi\in T_{X}\mathrm{Teich}(S)$ represented uniquely as
$\mu=\overline{q/g_{\mathtt{B}}}$,

$\|\xi\|_{\mathtt{HJ}}^{2}=\sum_{j=1}^{p}\int_{X}\mu\theta_{j}\,\overline{\mu\theta_{j}}.$

###### Proof.

In fact, the formula for $g_{\mathtt{B}}$ in Lemma 4.1 is a usual definition
of the Bergman metric on $X$, and its geometric interpretation is our above
description with the Abel–Jacobi map. The formula for $\|\\_\|_{\mathtt{HJ}}$
in Lemma 4.1 follows immediately from the formula for $g_{\mathtt{B}}$ and the
defining expression in (4.2). Note that $\overline{\mu\theta_{j}}$ are
differentials of type $(1,0)$, but not holomorphic in general. ∎

####  4.1.3. The Royden–Siegel metric

The natural map $J\colon\mathrm{Teich}(S)\to\mathfrak{H}(S)$ and the Siegel
metric on $\mathfrak{H}(S)$ induces a canonical Riemannian pseudometric on
$\mathrm{Teich}(S)$, degenerating exactly on the hyperelliptic locus. This
construction results in what we call the Royden–Siegel metric. In the survey
[Roy74], H. L. Royden discusses several such pullback metrics on
$\mathrm{Teich}(S)$ using canonical differential metrics on $\mathfrak{H}(S)$.

For any $\xi\in T_{X}\mathrm{Teich}(S)$, the _Royden–Siegel metric_ is the
Riemannian pseudometric determined by

(4.3) $\|\xi\|_{\mathtt{RS}}^{2}=\|J_{*}(\xi)\|_{\mathtt{S}}^{2},$

according to our notations in (2.1) and (2.3).

######  Lemma 4.2.

Let $\theta_{1},\cdots,\theta_{p}$ be a Hermitian orthonormal basis of
$\Omega^{1}(X)$, where $p=\dim_{\mathbb{C}}\Omega^{1}(X)$ is equal to the
genus of $S$. Then, for any $\xi\in T_{X}\mathrm{Teich}(S)$ represented by any
$\mu\in B(X)$, the following formula holds,

$\|\xi\|_{\mathtt{RS}}^{2}=4\times\sum_{j=1}^{p}\sum_{k=1}^{p}\,\left|\int_{X}\mu\theta_{j}\theta_{k}\right|^{2}.$

###### Proof.

The formula is equivalent to [Roy74, Theorem 2], except a factor $4$ due to
normalization. To be precise, Royden uses a rescaled metric on the generalized
upper half plane that is half of Siegel’s in length, (compare [Roy74, Eq. (15)
on p. 397] and [Sie64, Eq. (2) on p. 3]). With our notations (following
Siegel’s normalization), [Roy74, Theorem 2] can be rewritten as

$\|\xi\|_{\mathtt{RS}}^{2}=4\times\int_{X}\int_{X}\mu(x^{\prime})K(x^{\prime},x^{\prime\prime})\overline{\mu(x^{\prime\prime})},$

where $x^{\prime}$ parametrizes the first $X$ and $x^{\prime\prime}$
parametrizes the second $X$. The kernel $K(x^{\prime},x^{\prime\prime})$ is a
holomorphic quadratic differential in the first variable, and an anti-
holomorphic quadratic differential in the second variable, and is Hermitian
symmetric under switching of the variables. It is explicitly constructed as

$K(x^{\prime},x^{\prime\prime})=\left(\sum_{l}\theta_{l}(x^{\prime})\overline{\theta_{l}(x^{\prime\prime})}\right)^{2}=\sum_{j}\sum_{k}\theta_{j}(x^{\prime})\theta_{k}(x^{\prime})\overline{\theta_{j}(x^{\prime\prime})\theta_{k}(x^{\prime\prime})},$

indices all ranging over $\\{1,\cdots,p\\}$. Plug the last expression into the
above double integral, and move the summations out of the integrations. Then
the variables separate, and the asserted formula follows. ∎

###  4.2. Virtual versions

Any regular finite covering map $S^{\prime}\to S$ induces a canonical
embedding

$\mathrm{Teich}(S)\to\mathrm{Teich}(S^{\prime}),$

such that any Riemann surface with marking $S\to X$ goes to $S^{\prime}\to
X^{\prime}$ where $X^{\prime}\to X$ is holomorphic and equivariant with
$S^{\prime}\to S$.

The Royden–Siegel metric on $\mathrm{Teich}(S^{\prime})$ induces a
pseudometric on $\mathrm{Teich}(S)$ by restriction. To normalize, we introduce
the _virtual Royden–Siegel metric_ on $\mathrm{Teich}(S)$ with respect to the
regular finite cover $S^{\prime}$ as the Riemannian pseudometric, such that
for any $\xi\in T_{X}\mathrm{Teich}(S)$, we define

(4.4)
$\|\xi\|^{2}_{\mathtt{RS}^{\prime}}=\|\xi^{\prime}\|^{2}_{\mathtt{RS}}\times\frac{1}{[S^{\prime}:S]}=\|J^{\prime}_{*}(\xi^{\prime})\|^{2}_{\mathtt{S}}\times\frac{1}{[S^{\prime}:S]},$

where $\xi^{\prime}\in T_{X^{\prime}}\mathrm{Teich}(S^{\prime})$ denotes the
tangent map image of $\xi$, and where $J^{\prime}$ denotes the natural map
$\mathrm{Teich}(S^{\prime})\to\mathfrak{H}(S^{\prime})$ as in (2.1).

We introduce the _virtual Haberman–Jost metric_ on $\mathrm{Teich}(S)$ with
respect to any regular finite cover $S^{\prime}$ as follows. For any
$X\in\mathrm{Teich}(S)$, since the deck transformations on the corresponding
$X^{\prime}$ are all biholomorphic, they all preserve the Bergman metric
$g^{\prime}_{\mathtt{B}}$ on $X^{\prime}$. Therefore,
$g^{\prime}_{\mathtt{B}}$ is the pullback of a unique conformal metric
$g_{\mathtt{B}^{\prime}}$ on $X$. We notice the relation

$\mathrm{Area}_{\mathtt{B}^{\prime}}(X)=\frac{\mathrm{Area}_{\mathtt{B}}(X^{\prime})}{[S^{\prime}:S]}=\mathrm{genus}(S)-1+\frac{1}{[S^{\prime}:S]}.$

For any $\xi\in T_{X}\mathrm{Teich}(S)$, we define

$\|\xi\|_{\mathtt{HJ}^{\prime}}^{2}=\int_{X}|\mu|^{2}\,{\mathrm{d}}\mathrm{Area}_{\mathtt{B}^{\prime}}=\int_{X}\frac{q\bar{q}}{g_{\mathtt{B}^{\prime}}},$

where $\mu=\overline{q/g_{\mathtt{B}^{\prime}}}$ denotes the Beltrami
differential representative of $\xi$ that is harmonic with respect to
$g_{\mathtt{B}^{\prime}}$. The last expression implies a relation

$\|\xi\|_{\mathtt{HJ}^{\prime}}^{2}=\|\xi^{\prime}\|_{\mathtt{HJ}}^{2}\times\frac{1}{[S^{\prime}:S]},$

where $\xi^{\prime}\in T_{X^{\prime}}\mathrm{Teich}(S^{\prime})$ denotes the
tangent map image of $\xi$.

There is no need to introduce any “virtual Weil–Petersson metric”, as it would
be the same thing as the Weil–Petersson metric. This is because the conformal
hyperbolic metric $g_{\mathtt{h}}^{\prime}$ on $X^{\prime}$ agrees with the
pullback of $g_{\mathtt{h}}$ on $X$. We only mention the obvious relation

$\|\xi\|_{\mathtt{WP}}^{2}=\|\xi^{\prime}\|_{\mathtt{WP}}^{2}\times\frac{1}{[S^{\prime}:S]},$

where $\xi^{\prime}\in T_{X^{\prime}}\mathrm{Teich}(S^{\prime})$ denotes the
tangent map image of $\xi\in T_{X}\mathrm{Teich}(S)$.

##  5\. Comparison and convergence of virtual metrics

In this section, we prove two lemmas regarding relation of virtual metrics on
the Teichmüller space. The first is an inequality due to Habermann and Jost
[HabJ96] in the case of ordinary metrics, and we only carry it onto the
virtual version (Lemma 5.1). The second is an application of a well-known
theorem due to D. A. Kazhdan [Kaz75], regarding convergence of the Bergmann
metrics on finite coverings of Riemann surfaces (Lemma 5.2).

######  Lemma 5.1.

Let $S$ be a closed orientable surface of genus at least $2$. With respect to
any finite regular cover $S^{\prime}$ over $S$, and for all
$X\in\mathrm{Teich}(S)$ and $\xi\in T_{X}\mathrm{Teich}(S)$, the following
inequality holds,

$\|\xi\|_{\mathtt{RS}^{\prime}}\leq 2\times\|\xi\|_{\mathtt{HJ}^{\prime}}.$

###### Proof.

In the ordinary case, the inequality $\|\xi\|_{\mathtt{RS}}\leq
2\times\|\xi\|_{\mathtt{HJ}}$ is the same as [HabJ96, Lemma 6.3], except a
constant factor due to normalization; (check the formula (6.5) therein, which
has to agree with Royden’s normalization of the Siegel metric [Roy74, Eq. (15)
on p. 397], rather than Siegel’s [Sie64, Eq. (2) on p. 3]). For the reader’s
convenience, below we derive the inequality quickly from Lemmas 4.1 and 4.2,
following the same idea as in [HabJ96].

Note that it suffices to prove the ordinary case $S^{\prime}=S$, otherwise
arguing with $S^{\prime}$ and dividing both sides by $[S^{\prime}:S]$. Fix an
orthonormal basis of holomorphic differentials $\theta_{1},\cdots,\theta_{p}$
in $\Omega^{1}(X)$. Let $\mu=q/g_{\mathtt{B}}$ be a Bergmann harmonic Beltrami
differential representative of $\xi$. We denote by
$\eta_{j}(\mu)\in\Omega^{1}(X)$ the orthogonal summand of the
$(1,0)$–differential $\overline{\mu\theta_{j}}\in A^{1,0}(X)$, namely,

$\eta_{j}(\mu)=\sum_{l}\theta_{l}\int_{X}\mu\theta_{j}\theta_{l},$

for any $j=1,\cdots,p$. Since $\theta_{1},\cdots,\theta_{j}$ are orthonormal,
we obtain by Lemma 4.2,

$\sum_{j}\int_{X}\eta_{j}(\mu)\overline{\eta_{j}(\mu)}=\sum_{j}\sum_{k}\left(\int_{X}\mu\theta_{j}\theta_{k}\right)\overline{\left(\int_{X}\mu\theta_{j}\theta_{k}\right)}=\|\xi\|_{\mathtt{RS}}^{2}\times
1/4,$

and by Lemma 4.1,

$\sum_{j}\int_{X}\eta_{j}(\mu)\overline{\eta_{j}}\leq\sum_{j}\int_{X}\mu\theta_{j}\overline{\mu\theta_{j}}=\|\xi\|_{\mathtt{HJ}}^{2}.$

Then the asserted inequality follows. ∎

######  Lemma 5.2.

Let $S$ be a closed orientable surface of genus at least $2$. Suppose
$(S^{\prime}_{n}\to S)_{n\in{\mathbb{N}}}$ is a sequence of finite regular
covers converging to the universal cover. Then, for all
$X\in\mathrm{Teich}(S)$ and $\xi\in T_{X}\mathrm{Teich}(S)$, the following
convergence holds,

$\lim_{n\to\infty}\|\xi\|_{\mathtt{HJ}^{\prime}_{n}}=\frac{1}{\sqrt{4\pi}}\times\|\xi\|_{\mathtt{WP}},$

where the notation $n$ in the subscript indicates the respective cover
$S^{\prime}_{n}$. Morover, the convergence is uniform on any compact set of
tangent vectors $(X,\xi)$ on $\mathrm{Teich}(S)$.

###### Proof.

For any compact Riemann surface $X$, Kazhdan [Kaz75] shows that the sequence
of virtual Bergmann metrics $g_{\mathtt{B}^{\prime}_{n}}$ on $X$ converges
uniformly everywhere to the hyperbolic metric $g_{\mathtt{h}}$ rescaled by
$1/4\pi$ (that is, by $1/\sqrt{4\pi}$ in length); see [McM13, Appendix] for a
clear statement and a short proof. See also Remark 5.3 regarding the
assumption on the covering sequence.

For any cotangent vector $q$ in $T^{*}_{X}\mathrm{Teich}(S)=Q(X)$, the dual
norm squares of $q$ converge as

$\lim_{n\to\infty}\|q\|_{\mathtt{HJ}^{\prime}_{n}}^{2}=\lim_{n\to\infty}\int_{X}\frac{q\bar{q}}{g_{\mathtt{B}^{\prime}_{n}}}=4\pi\times\int_{X}\frac{q\bar{q}}{g_{\mathtt{h}}}=4\pi\times\|q\|_{\mathtt{WP}}^{2}.$

This implies the asserted convergence of norms

$\lim_{n\to\infty}\|\xi\|_{\mathtt{HJ}^{\prime}_{n}}=\frac{1}{\sqrt{4\pi}}\times\|\xi\|_{\mathtt{WP}},$

for any tangent vector $\xi\in T_{X}\mathrm{Teich}(S)$.

Moreover, the convergence is uniform on any compact subsets of tangent
vectors, because the norm $\|\\_\|_{\mathtt{HJ}^{\prime}_{n}}$ and
$\|\\_\|_{\mathtt{WP}}$ are continuous functions on the Teichmüller tangent
bundle. ∎

######  Remark 5.3.

In the literature, Kazhdan’s theorem is classically stated for cofinal
_towers_ of regular finite covers of a compact Riemann surface $X$. However,
we can replace ‘tower’ with ‘sequence’. In fact, as is evident from the proof
in [McM13, Appendix], the argument only relies on the properties that
$\pi_{1}(X)$ acts on all the covers $X^{\prime}_{n}$, and that
$X^{\prime}_{n}$ are all compact, and that the hyperbolic injectivity radii of
$X^{\prime}_{n}$ tend to infinity.

##  6\. Proof of the main theorem

This section is devoted to the proof of Theorem 1.1.

Let $f\colon S\to S$ be a pseudo-Anosov automorphism on a connected closed
orientable surface of genus at least $2$. Suppose that
$(S^{\prime}_{n},f^{\prime}_{n})_{n\in{\mathbb{N}}}$ is a cofinal sequence of
regular finite covers $(S^{\prime}_{n})_{n\in{\mathbb{N}}}$ of $S$ together
with pseudo-Anosov automorphisms $f^{\prime}_{n}\colon S^{\prime}_{n}\to
S^{\prime}_{n}$ lifting $f$.

Let $(S^{\prime},f^{\prime})=(S^{\prime}_{n},f^{\prime}_{n})$ be any term of
the above sequence. Denote by $J^{\prime}$ the canonical composite map
$\mathrm{Teich}(S)\to\mathrm{Teich}(S^{\prime})\to\mathfrak{H}(S^{\prime})$,
where the first map is the embedding induced by $S^{\prime}\to S$. By
definition, the action of $f^{\prime}\in\mathrm{Mod}(S^{\prime})$ on
$\mathrm{Teich}(S^{\prime})$ preserves $\mathrm{Teich}(S)$, and extends the
action of $f\in\mathrm{Mod}(S)$ on $\mathrm{Teich}(S)$.

For any regular path $\gamma\colon[0,1]\to\mathrm{Teich}(S)$ with the property
$\gamma(1)=f.\gamma(0)$, it follows that the induced action $f^{\prime}_{*}$
on $\mathfrak{H}(S)$ also moves $J^{\prime}(\gamma(0))$ to
$J^{\prime}(\gamma(1))$. We obtain by Theorem 3.1,

$\mathrm{Length}_{\mathtt{S}}(J^{\prime}\circ\gamma)\geq{\ell}_{\mathtt{S}}(f^{\prime}_{*})=2\times\sqrt{\mathbf{w}(P)},$

where $P$ denotes the characteristic polynomial of $f^{\prime}_{*}$ on
$H_{1}(S^{\prime};{\mathbb{C}})$. On the other hand, we obtain by Lemma 5.1
and (4.3),

$\mathrm{Length}_{\mathtt{S}}(J^{\prime}\circ\gamma)=\mathrm{Length}_{\mathtt{RS}^{\prime}}(\gamma)\times\sqrt{[S^{\prime}:S]}\leq
2\times\mathrm{Length}_{\mathtt{HJ}^{\prime}}(\gamma)\times\sqrt{[S^{\prime}:S]}.$

Retaining the subscript $n$, the above inequalities yield

$\sqrt{\frac{\mathbf{w}(P_{n})}{[S^{\prime}_{n}:S]}}\leq\mathrm{Length}_{\mathtt{HJ}^{\prime}_{n}}(\gamma).$

Passing to limit, we obtain by Lemma 5.2,

$\varlimsup_{n\to\infty}\sqrt{\frac{\mathbf{w}(P_{n})}{[S^{\prime}_{n}:S]}}\leq\frac{\mathrm{Length}_{\mathtt{WP}}(\gamma)}{\sqrt{4\pi}}.$

Apply the above inequality to a regular path $\gamma$ along the Weil–Petersson
axis of $f$. The Weil–Petersson translation length is realized as
${\ell}_{\mathtt{WP}}(f)=d_{\mathtt{WP}}(\gamma(0),\gamma(1))=\mathrm{Length}_{\mathtt{WP}}(\gamma)$.
In particular, we obtain the desired inequality,

$\varlimsup_{n\to\infty}\sqrt{\frac{\mathbf{w}(P_{n})}{[S^{\prime}_{n}:S]}}\leq\frac{{\ell}_{\mathtt{WP}}(f)}{\sqrt{4\pi}}.$

This completes the proof of Theorem 1.1.

## References

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