#  Finite quotients, arithmetic invariants,
and hyperbolic volume

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

For any pair of orientable closed hyperbolic $3$–manifolds, this paper shows
that any isomorphism between the profinite completions of their fundamental
groups witnesses a bijective correspondence between the Zariski dense
$\mathrm{PSL}(2,\mathbb{Q}^{\mathtt{ac}})$–representations of their
fundamental groups, up to conjugacy; moreover, corresponding pairs of
representations have identical invariant trace fields and isomorphic invariant
quaternion algebras. (Here, $\mathbb{Q}^{\mathtt{ac}}$ denotes an algebraic
closure of $\mathbb{Q}$.) Next, assuming the $p$–adic Borel regulator
injectivity conjecture for number fields, this paper shows that uniform
lattices in $\mathrm{PSL}(2,\mathbb{C})$ with isomorphic profinite completions
have identical invariant trace fields, isomorphic invariant quaternion
algebras, identical covolume, and identical arithmeticity.

###### Key words and phrases:

profinite completion, hyperbolic geometry, 3-manifolds, fixed point theory

###### 2010 Mathematics Subject Classification:

Primary 57M50; Secondary 57M10, 30F40

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

##  1\. Introduction

Hyperbolic $3$–manifolds of finite volume have abundant finite covers, and
have strong rigidity. The latter fact opens the door for studying fundamental
groups of those manifolds by means of geometry and arithmetics. The hyperbolic
volume, the invariant trace field, and the invariant quaternion algebra are
among the most important invariants, not only in theory, but also in practical
computation. Arithmetic hyperbolic $3$–manifolds and non-arithmetic ones
exhibit contrast behaviors in many group theoretic aspects. The former fact,
in addition, makes it reasonable to ask if the fundamental group could
possibly be inferred from knowledge of its finite quotients. This challenging
question has attracted growing interest over the past decade. Its pertaining
topic is called profinite properties of $3$–manifold groups.

For any (abstract) group $G$, the _profinite completion_ of $G$ refers to the
group $\widehat{G}=\varprojlim_{N}G/N,$ where $N$ ranges over the inverse
system of all finite-index normal subgroups of $G$. When $G$ is finitely
generated and residually finite, the completion homomorphism $G\to\widehat{G}$
is injective, and the profinite topology on $\widehat{G}$ can be characterized
purely in abstract group theoretic terms [NikS07]; moreover, the group
isomorphism type of $\widehat{G}$ can be recovered purely from the set of
isomorphism classes of all the finite quotient groups of $G$ [Rei18, Theorem
2.2]. These facts make finitely generated residually finite groups arguably
the most comfortable class for speaking of profinite properties. There are
some smaller classes of groups, restricted to which one might expect richer
harvest. Finitely generated virtually nilpotent groups, semisimple Lie group
lattices, and finitely generated $3$–manifold groups are a few candidates on
top of our mind.

Some striking examples of arithmetic lattices in
$\mathrm{PSL}(2,{\mathbb{C}})$ and $\mathrm{O}^{+}(3,1)$ with absolute
profinite rigidity have been confirmed recently by M. R. Bridson, D. B.
McReynolds, A. W. Reid, and R. Spitler [BriMRS20]. Their examples include the
Bianchi group $\mathrm{PSL}(2,{\mathbb{Z}}[(-1+\sqrt{-3})/2])$, and the
fundamental group of the Fomenko–Matveev–Weeks manifold, which is the unique
closed orientable hyperbolic $3$–manifold of the smallest volume. Their work
shows that the isomorphism types of those groups are uniquely determined by
the isomorphism types of their profinite completions, among finitely
generated, residually finite groups. Restricted to the class of finitely
generated $3$–manifold groups, (which can be characterized as fundamental
groups of connected compact $3$–manifolds,) there are examples with profinite
rigidity [BriMRS20, BriRW17, BriR20], and also examples without profinite
rigidity [Fun13, Hem14, Ste72]. However, all known examples satisfy a weaker
condition, which we call _profinite almost rigidity_. This condition says
that, among finitely generated $3$–manifold groups, the isomorphism type of
the group is determined up to finitely many possibilities by the isomorphism
of the profinite completion. G. Wilkes shows that graph manifold groups are
profinitely almost rigid [Wlk17, Wlk18, Wlk19]. In a recent work [Liu20b], the
author shows that finite-volume hyperbolic $3$–manifold groups are profinitely
almost rigid. It seems plausible that finitely generated $3$–manifold groups
are all profinitely almost rigid. On the other hand, we recall the following
question raised by A. W. Reid [Rei15, Question 9], which remains widely open.
Its torsion-free case is equivalent to the (unknown) profinite rigidity among
orientable finite-volume hyperbolic $3$–manifolds.

######  Question 1.1.

If a pair of lattices in $\mathrm{PSL}(2,{\mathbb{C}})$ have isomorphic
profinite completions, must they be isomorphic to each other?

The present paper may be viewed as a continuation of [Liu20b] and an attempt
toward Question 1.1. We seek for a connection between dynamics of pseudo-
Anosov suspension flow and arithmetic invariants of
$\mathrm{SL}(2,{\mathbb{C}})$–representations. In the end, we are able to use
this connection, and reduce the profinite invariance problem of hyperbolic
volume and arithmecity to a long-standing conjecture of number theoretic
nature, that is, the $p$–adic Borel regulator injectivity conjecture for
number fields.

For any number field $k$ with $r_{1}$ real places and $r_{2}$ complex places,
the zeroth K-group $K_{0}(O_{k})$ of the ring of integers $O_{k}$ can be
identified with the ideal class group of $O_{k}$ direct sum with
${\mathbb{Z}}$, and the first K-group $K_{1}(O_{k})$ can be identified with
the multiplicative unit group $O_{k}^{\times}$. In particular, the real
dimension of $K_{n}(O_{k})\otimes_{\mathbb{Z}}{\mathbb{R}}$ equals $1$ for
$n=0$, by the class number finiteness, and equals $r_{1}+r_{2}-1$ for $n=1$,
by the Dirichlet unit theorem. A. Borel shows that the rank of higher K-groups
obeys periodicity, and indeed, when $n\geq 2$, the real dimension of
$K_{n}(k)\otimes_{\mathbb{Z}}{\mathbb{R}}\cong
K_{n}(O_{k})\otimes_{\mathbb{Z}}{\mathbb{R}}$ equals $0,r_{1}+r_{2},0,r_{2}$
for $n\equiv 0,1,2,3$ modulo $4$, respectively. For the nonvanishing
dimensions, the classical Borel regulator maps associated to $k$ can be
reinterpreted as ${\mathbb{R}}$–linear homomorphisms
$K_{4m+1}(k)\otimes_{\mathbb{Z}}{\mathbb{R}}\to{\mathbb{R}}^{r_{1}}\oplus{\mathbb{C}}^{r_{2}}$
and $K_{4m-1}(k)\otimes_{\mathbb{Z}}{\mathbb{R}}\to{\mathbb{C}}^{r_{2}}$,
which are actually injective onto
${\mathbb{R}}^{r_{1}}\oplus{\mathbb{R}}^{r_{2}}$ and ${\mathbb{R}}^{r_{2}}$,
respectively. For any rational prime $p$, the $p$–adic Borel regulator maps
associated to $k$ can be defined analogously, and can be thought of as
${\mathbb{Q}}_{p}$–linear homomorphisms
$K_{2m-1}(k)\otimes_{\mathbb{Z}}{\mathbb{Q}}_{p}\to\bigoplus_{v|p}k_{v}$. We
state the _$p$ –adic Borel regulator injectivity conjecture_ as follows. The
conjecture is expected to be true to the generality as stated. (See Section 2
for more detail.)

######  Conjecture 1.2.

For any number field $k$ and any rational prime $p$, the $p$–adic Borel
regulator map is injective on
$K_{2m-1}(k)\otimes_{\mathbb{Z}}{\mathbb{Q}}_{p}$ for every odd dimension
$2m-1\geq 3$.

######  Theorem 1.3.

If Conjecture 1.2 holds true, then, among uniform lattices in
$\mathrm{PSL}(2,{\mathbb{C}})$, the following lattice invariants are all
determined by the group isomorphism type of the profinite completion: the
invariant trace field in ${\mathbb{C}}$, the conjugacy class of the invariant
quaternion algebra in $\mathrm{Mat}(2,{\mathbb{C}})$, the covolume, and
arithmeticity.

See Theorem 11.2 in Section 11 for the precise statement. The proof of Theorem
1.3 actually only needs the injectivity assumption on
$K_{3}(k)\otimes_{\mathbb{Z}}{\mathbb{Q}}_{p}$ and for infinitely many $p$
(Hypothesis 2.8). It should be possible to extend Theorem 1.3 to the non-
uniform case, if one invokes works of H. Wilton and P. A. Zalesskii [WltZ17a,
WltZ17b], and keeps track of the parabolic subgroups more closely. However, we
do not pursue the generalization in this paper, as the conclusion of Theorem
1.3 is already hypothetical. Note that Theorem 1.3 can be derived immediately
from its torsion-free case, so it is essentially about orientable closed
hyperbolic $3$–manifolds with profinitely isomorphic fundamental groups.

######  Theorem 1.4.

If a pair of orientable closed hyperbolic $3$–manifolds have profinitely
isomorphic fundamental groups, then the Zariski dense
$\mathrm{PSL}(2,{\mathbb{Q}}^{\mathrm{ac}})$–representations of their
fundamental groups can be paired up bijectively up to conjugacy. Moreover,
corresponding pairs of representations have identical invariant trace fields
in ${\mathbb{Q}}^{\mathrm{ac}}$, and have conjugate invariant quaternion
algebras in $\mathrm{Mat}(2,{\mathbb{Q}}^{\mathrm{ac}})$.

The notation ${\mathbb{Q}}^{\mathrm{ac}}$ means an algebraic closure of
${\mathbb{Q}}$. Note that Theorem 1.4 does not rely on Conjecture 1.2. Theorem
1.4 is a main step toward Theorem 1.3. See Theorem 11.1 in Section 11 for the
precise statement; an effective description of the correspondence can be found
in the beginning paragraph of the proof of Theorem 11.1.

Theorem 1.4 is inspired by [BriMRS20, Theorem 4.8]. The Zariski denseness
assumption is needed for legitimating the invariant quaternion algebra. What
we are able to get rid of is the Galois rigidity assumption from [BriMRS20,
Theorem 4.8]. The basic reason is that we employ a bijective correspondence
between periodic trajectories for profinitely isomorphic pseudo-Anosov mapping
tori, as established in [Liu20b].

As one should be warned, Theorem 1.4 does not tell whether the discrete
faithful representations (or their Galois conjugates) are paired up by the
asserted correspondence. Nor does [BriMRS20, Theorem 4.8]. In fact, the
desired injectivity in [BriMRS20] is eventually obtained using the Hopfian
property of topologically finitely generated profinite groups, and moreover,
the argument thereof appeals to concrete features of particular lattices, such
as lists of their small index subgroups. In the proof of Theorem 1.3, we are
only able to infer from Conjecture 1.2 that discrete faithfulness should be
preserved under the correspondence, and this is a consequence of the volume
rigidity for $\mathrm{SL}(2,{\mathbb{C}})$–representations of uniform
$\mathrm{PSL}(2,{\mathbb{C}})$–lattices (see Example 2.1).

### Ingredients

In the rest of the introduction, we explain our strategy for proving Theorems
1.3 and 1.4. We also point out some new ideas of the present paper.

Let $\pi_{A}$ and $\pi_{B}$ be a pair of finitely generated, residually finite
groups. Suppose that $\Psi\colon\widehat{\pi}_{A}\to\widehat{\pi}_{B}$ is an
isomorphism between their profinite completions. In order to set up a
correspondence of representations
$\pi_{B}\to\mathrm{PSL}(2,{\mathbb{Q}}^{\mathrm{ac}})$ and
$\pi_{A}\to\mathrm{PSL}(2,{\mathbb{Q}}^{\mathrm{ac}})$ up to conjugacy, one
would naturally look for certain kind of pullback operation with respect to
$\Psi$. Assuming Zariski denseness of the representations will allow us to
speak of the invariant quaternion algebras (of the images), and will also make
it convenient for passing to finite-index subgroups.

For simplicity, let us consider a Zariski dense
$\mathrm{PSL}(2,{\mathbb{Q}}^{\mathrm{ac}})$–representation of $\pi_{B}$ that
comes from a representation $\rho_{B}\colon\pi_{B}\to\mathrm{SL}(2,K)$, where
$K$ is subfield finite over ${\mathbb{Q}}$. For all but finitely many non-
Archimedean places $v$ of $K$, the induced representation
$\rho_{B,v}\colon\pi_{B}\to\mathrm{SL}(2,K_{v})$ is bounded. For any such $v$,
we obtain a representation
$\rho_{A,v}^{\Psi}\colon\pi_{A}\to\mathrm{SL}(2,K_{v})$, as the
$\Psi$–pullback of the continuous extension
$\rho_{B,v}\colon\widehat{\pi}_{B}\to\mathrm{SL}(2,K_{v})$ to
$\widehat{\pi}_{A}$ and restricted to $\pi_{A}$. The same construction has
already appeared in the proof of [BriMRS20, Theorem 4.8]. However, we do not
automatically know whether the characters of $\rho_{A,v}^{\Psi}$ are valued in
${\mathbb{Q}}^{\mathrm{ac}}$, or whether the characters are the same for
various $v$. Certainly one would hope they are, and then, there will be some
representation
$\rho_{A}\colon\pi_{A}\to\mathrm{SL}(2,{\mathbb{Q}}^{\mathrm{ac}})$ with the
same character. If so, $\rho_{A}$ should be the right candidate for
corresponding to $\rho_{B}$, but how could we possibly read off the invariant
quaternion algebra of $\rho_{A}$ from the construction, and especially, its
ramification behavior over the Archimedean places of the invariant trace
field?

We are able to address the above issues in the case of closed hyperbolic
$3$–manifold groups $\pi_{A/B}=\pi_{1}(M_{A/B})$. (The notation $A/B$ means
$A$ or $B$ respectively.) We may assume $M_{A/B}$ fibering over a circle as a
surface bundle with pseudo-Anosov monodromy, because of the virtual fibering
theorem [Ago13], (see also [AscFW15] for a survey on virtual properties of
$3$–manifold groups). It is proved in [Liu20b] that $\Psi$ witnesses a
bijective correspondence of the surface fiber subgroups; and moreover, fixing
a corresponding pair of surface fibers $S_{A}$ and $S_{B}$, $\Psi$ witnesses a
bijective correspondence between the periodic trajectories of pseudo-Anosov
suspension flows on $M_{A}$ and on $M_{B}$. More precisely, there exists an
invertible profinite integer $\mu\in\widehat{{\mathbb{Z}}}^{\times}$, which
depends only on $(\pi_{A},\pi_{B},\Psi)$. For any corresponding pair of
periodic trajectories $\mathbf{c}_{A}$ and $\mathbf{c}_{B}$ (as conjugacy
classes in $\pi_{A}$ and $\pi_{B}$), the $\mu$–power of $\mathbf{c}_{A}$
projects $\mathbf{c}_{B}$ under $\Psi$ (as conjugacy classes in
$\widehat{\pi}_{A}$ and $\widehat{\pi}_{B}$). It is this trajectory
correspondence that allows us to compare the characters of $\rho_{A}$ and
$\rho_{B}$ on _some_ pairs of conjugacy classes in $\pi_{A}$ and $\pi_{B}$. We
use this partial information to infer about the characters of $\rho_{A,v}$,
and about the invariant quaternion algebra of $\rho_{A}$.

In order to compute the characters of $\rho_{A,v}^{\Psi}$ on the periodic
trajectories of $M_{A}$ from data with $\rho_{B}$ and $M_{B}$, we improve a
main result regarding $\widehat{{\mathbb{Z}}}^{\times}$–regularity in
[Liu20b]. We show that $\mu$ has multiplicative order $2$ (Lemma 7.1). Writing
$\mu=(\mu_{p})_{p}$ with respect to
$\widehat{{\mathbb{Z}}}^{\times}\cong\prod_{p}{\mathbb{Z}}_{p}^{\times}$, this
means $\mu_{p}\in\\{\pm 1\\}$ for each rational prime $p$. The topological
ingredient for the improvement comes from the resolution of McMullen’s
conjecture on virtual homological spectral radii of pseudo-Anosov
automorphisms [Liu20a]. That conjecture asserts that every pseudo-Anosov
automorphism $f\colon S\to S$ admits a lift $f^{\prime}\colon S^{\prime}\to
S^{\prime}$ to some finite cover $S^{\prime}\to S$, such that
$f^{\prime}_{*}\colon H_{1}(S^{\prime};{\mathbb{C}})\to
H_{1}(S^{\prime};{\mathbb{C}})$ has a complex eigenvalue outside the unit
circle. As a consequence, we can identify certain twisted Alexander
polynomials of $M_{A}$ and of $M_{B}$ (associated to some $\Psi$–corresponding
finite quotients $\pi_{A}\to\Gamma$ and $\pi_{B}\to\Gamma$), such that the
polynomials are monic and reciprocal over ${\mathbb{Z}}$, but are not product
of cyclotomic factors. With these polynomials, we show that $\mu_{p}$ is
rational for each $p$, using Mahler’s $p$–adic analogue of the
Gelfond–Schneider theorem [Mah35]. Then we deduce $\mu_{p}\in\\{\pm 1\\}$ by
simple observation. (See Sections 6 and 7.)

In order to determine the invariant quaternion algebra of $\rho_{A}$, we
introduce two new criteria, in terms of the character $\chi_{A}$ of $\rho_{A}$
evaluated at the periodic trajectories of $M_{A}$. The first criterion says
that the trace field of $\rho_{A}$ is generated over ${\mathbb{Q}}$ by the
values of $\chi_{A}$ on the periodic trajectories (Lemma 4.1). The second
criterion says that the quaternion algebra of $\rho_{A}$ at a real place of
the trace field is ramified or unramified, according as the real values of
$\chi_{A}$ on the periodic trajectories are uniformly bounded or unbounded
(Lemma 4.3). (See Section 4.) We can identify the invariant trace fields of
$\rho_{A}$ and $\rho_{B}$ with the first criterion (applied restricting to
finite-index subgroups), and identify the Archimedean places of ramifications
with the second criterion, and identify the non-Archimedean places of
ramification easily from the construction. Since quaternion algebras over
number fields are isomorphically classified by their places of ramification,
we can identify the invariant trace field of $\rho_{A}$ isomorphically with
that of $\rho_{B}$. (See Section 9.)

The above discussion covers all the essential points for proving Theorem 1.4.
As to Theorem 1.3, essentially we need to show that the induced homomorphisms
$\rho_{A*}\colon H_{3}(\pi_{A};{\mathbb{Z}})\to
H_{3}(\mathrm{SL}(K);{\mathbb{Z}})\otimes_{\mathbb{Z}}{\mathbb{Q}}$ and
$\rho_{B*}\colon H_{3}(\pi_{B};{\mathbb{Z}})\to
H_{3}(\mathrm{SL}(K);{\mathbb{Z}})\otimes_{\mathbb{Z}}{\mathbb{Q}}$ map the
fundamental classes $[M_{A}]$ and $[M_{B}]$ to the same group homology class
up to sign. (There are natural isomorphisms
$K_{3}(K)\otimes_{\mathbb{Z}}{\mathbb{Q}}\cong
H_{3}(\mathrm{SL}(K);{\mathbb{Z}})\otimes_{\mathbb{Z}}{\mathbb{Q}}\cong
H_{3}(\mathrm{SL}(N,K);{\mathbb{Z}})\otimes_{\mathbb{Z}}{\mathbb{Q}}$ for
sufficiently large $N$.) Then Borel’s theorem will tell us that $M_{A}$ and
$M_{B}$ have identical hyperbolic volume, and the rest assertions of Theorem
1.3 will become easy consequences of Theorem 1.4. With the profinite
isomorphism $\Psi$, it is only obvious to identify $[M_{A}]$ with $\pm[M_{B}]$
in the continuous group homology (also known as the profinite group homology)
$H_{3}^{\mathtt{cont}}(\widehat{\pi}_{A};{\mathbb{Z}}_{p})\cong
H_{3}^{\mathtt{cont}}(\widehat{\pi}_{B};{\mathbb{Z}}_{p})$. This forces us, in
the present paper, to work with suitable versions of structured topological
group homology or cohomology, (for example, the locally analytic group
cohomology of $\mathrm{GL}(N,O_{\mathfrak{P}})$ where $O_{\mathfrak{P}}$ is
the completion of the ring of integers in $K$ at some prime $\mathfrak{P}$).
We have to transfer information back to the abstract group homology at some
point. It turns out that Conjecture 1.2 (or Hypothesis 2.8) is exactly the
missing piece for completing the whole argument. (See Section 10.)

### Organization

The preliminary of this paper is divided into two sections. The first part
(Section 2) contains a quick review of Conjecture 1.2. We also include
necessary background regarding volume of
$\mathrm{SL}(2,{\mathbb{C}})$–representations, but the materials in Section 2
are not needed until Section 10. The second part (Section 3) is directly
related to the next few sections. We review general facts about quaternion
algebras and pseudo-Anosov automorphism dynamics.

In Section 4, we prove the two criteria regarding periodic trajectories in a
pseudo-Anosov mapping torus and arithmetic invariants associated to an
irreducible $\mathrm{SL}(2,{\mathbb{C}})$–representation. In Section 5, we
recall the profinite correspondence of periodic trajectories, and extract a
precise statement from results in [Liu20b]. In Section 6, we provide some
explicit description about the profininte power operation on profinite Laurent
polynomial rings and profinite matrix groups. In Section 7, we prove our
improvement $\mu^{2}=1$ based on algebraic lemmas in Section 6.

In Section 8, we develop some ad hoc theory regarding group representations in
quaternion algebras, in order to facilitate subsequent disposal. In Section 9,
we prove an essential case of Theorem 1.4, that is, the case with pseudo-
Anosov mapping tori. In Section 10, we prove the same essential case of
Theorem 1.3. In Section 11, we deduce the general cases of Theorem 1.4
(formally stated as Theorem 11.1) and Theorem 1.3 (formally stated as Theorem
11.2) from the above essential cases. and we actually restate the theorems
formally, respectively.

### Acknowledgement

The author thanks Ian Agol, Frank Calegari, and Shengkui Ye for valuable
communications.

##  2\. Preliminary I

In this preliminary section, we review volume of representations in
$\mathrm{SL}(2,{\mathbb{C}})$ and its relation with the classical Borel
regulator. We review the $p$–adic Borel regulator injectivity conjecture
(Conjecture 1.2) in a special case as Hypothesis 2.8. For convenience of our
application, we reformulate the materials on the level of suitable versions of
group homology and cohomology, and include the algebraic K-theoretic versions
in Remarks 2.7 and 2.9. We could have placed this section right before Section
10 without affecting the logical order, but we decide to place it here as
background of Conjecture 1.2.

###  2.1. Volume of representations in $\mathrm{SL}(2,{\mathbb{C}})$

Let $M$ be an oriented connected closed (smooth) $3$–manifold. Fixing a
universal covering space $\widetilde{M}_{\mathrm{univ}}$ of $M$, we denote by
$\pi=\pi_{1}(M)$ the universal deck transformation group, which we also
frequently refer to as the fundamental group of $M$. For any group
homomorphism $\rho\colon\pi\to\mathrm{SL}(2,{\mathbb{C}})$, there is a well-
defined value
$\mathrm{Vol}_{\mathrm{SL}(2,{\mathbb{C}})}\left(M,\rho\right)\in{\mathbb{R}}$,
called the _volume_ of $M$ with respect to the
$\mathrm{SL}(2,{\mathbb{C}})$–representation $\rho$. We recall a description
of $\mathrm{Vol}_{\mathrm{SL}(2,{\mathbb{C}})}(M,\rho)$ as follows.

Denote by
${\mathbb{H}}^{3}=\\{(x+\mathbf{i}y,h)\in{\mathbb{C}}\times{\mathbb{R}}\colon
h>0\\}$ the upper half space model of $3$–dimensional hyperbolic geometry. The
model is furnished with the Riemannian metric
$({\mathrm{d}}x^{2}+{\mathrm{d}}y^{2}+{\mathrm{d}}h^{2})^{1/2}/h$, which has
constant sectional curvature $-1$. The orientation-preserving isometry group
$\mathrm{Isom}_{+}({\mathbb{H}}^{3})$ is identified with
$\mathrm{PSL}(2,{\mathbb{C}})$, agreeing with the natural identification of
the sphere at infinity $\partial_{\infty}{\mathbb{H}}^{3}$ with the complex
projective line ${\mathbb{C}}\mathbf{P}^{1}={\mathbb{C}}\cup\\{\infty\\}$. If
$D_{\rho}\colon\widetilde{M}_{\mathrm{univ}}\to{\mathbb{H}}^{3}$ is a
$(\pi,\mathrm{PSL}(2,{\mathbb{C}}))$–equivariant (smooth) map with respect to
the projectivized representation
$\mathbf{P}(\rho)\colon\pi\to\mathrm{PSL}(2,{\mathbb{C}})$, the volume of
$(M,\rho)$ can be expressed with an integral

(2.1)
$\mathrm{Vol}_{\mathrm{SL}(2,{\mathbb{C}})}\left(M,\rho\right)=\int_{\mathcal{F}}D^{*}_{\rho}\,{\mathrm{d}}\mathrm{Vol}_{{\mathbb{H}}^{3}}.$

Here, $\mathcal{F}\subset\widetilde{M}_{\mathrm{univ}}$ is a fundamental
domain of $\pi$, and where
${\mathrm{d}}\mathrm{Vol}_{{\mathbb{H}}^{3}}=h^{-3}{\mathrm{d}}x\wedge{\mathrm{d}}y\wedge{\mathrm{d}}h$
denotes the volume form on ${\mathbb{H}}^{3}$.

Any map $D_{\rho}$ as above is called a _developing map_ for $\rho$. Since
${\mathbb{H}}^{3}$ is contractible, developing maps always exist, and the
resulting volume value does not depend on particular choices of $\mathcal{F}$
or $D_{\rho}$. Moreover, the volume of $(M,\rho)$ is invariant under
continuous deformations of $\rho$. In particular, it is
$\mathrm{SL}(2,{\mathbb{C}})$–conjugation invariant in $\rho$. It takes only
finitely many values as $\rho$ ranges over all the representations
$\pi\to\mathrm{SL}(2,{\mathbb{C}})$. We refer the reader to [DerLSW19] for an
exposition of volume of representations in a general setting, and for further
references to many important results in the literature.

######  Example 2.1.

Let $M$ be an oriented closed hyperbolic $3$–manifold. For any representation
$\rho\colon\pi_{1}(M)\to\mathrm{SL}(2,{\mathbb{C}})$, the following inequality
holds

$\left|\mathrm{Vol}_{\mathrm{SL}(2,{\mathbb{C}})}(M,\rho)\right|\leq\mathrm{Vol}_{{\mathbb{H}}^{3}}(M).$

Moreover, the equality is achieved if and only if $\rho$ is a discrete
faithful representation. See [KimK14, Theorem 1.1] or [BesCG07, Theorem 1.4].

######  Remark 2.2.

It is also possible to define the volume of any
$\mathrm{PSL}(2,{\mathbb{C}})$–representation of $\pi_{1}(M)$. For any
representation $\rho\colon\pi_{1}(M)\to\mathrm{SL}(2,{\mathbb{C}})$, the
volume $\mathrm{Vol}_{\mathrm{PSL}(2,{\mathbb{C}})}(M,\mathbf{P}(\rho))$ of
the projectivized representation
$\mathbf{P}(\rho)\colon\pi_{1}(M)\to\mathrm{PSL}(2,{\mathbb{C}})$ is equal to
$\mathrm{Vol}_{\mathrm{SL}(2,{\mathbb{C}})}(M,\rho)$, by definition (see
[DerLSW19, Section 2]). When $M$ is hyperbolic, $\rho$ is discrete and
faithful if and only if $\mathbf{P}(\rho)$ is discrete and faithful. In this
case, $\mathbf{P}(\rho)$ is unique up to a real Lie group automorphism of
$\mathrm{PSL}(2,{\mathbb{C}})$. It follows that $\rho$ is unique up to an
inner automorphism, and a complex conjugation, and a scalar representation
$\pi_{1}(M)\to\\{\pm 1\\}$. It is also known that any discrete faithful
representation of $\pi_{1}(M)$ in $\mathrm{PSL}(2,{\mathbb{C}})$ lifts to a
representation in $\mathrm{SL}(2,{\mathbb{C}})$ [Cul86]. In general, a
representation $\rho^{\flat}\colon\pi_{1}(M)\to\mathrm{PSL}(2,{\mathbb{C}})$
is not necessarily liftable to $\mathrm{SL}(2,{\mathbb{C}})$. The obstruction
can be characterized as some second Stiefel–Whitney class
$w_{2}(\rho^{\flat})\in H^{2}(\pi_{1}(M);{\mathbb{Z}}/2{\mathbb{Z}})$; in the
liftable case, the $\mathrm{SL}(2,{\mathbb{C}})$–representation lifts of
$\rho^{\flat}$ differ from each other by a scalar representation factor
$\varepsilon\colon\pi_{1}(M)\to\\{\pm 1\\}$, so they form an affine
$H^{1}(\pi_{1}(M);{\mathbb{Z}}/2{\mathbb{Z}})$. (See [HeuP04, Remark 4.1];
compare [DerLSW19, Proposition 5.1].)

###  2.2. Borel regulators: from classical to $p$–adic

Volume of $\mathrm{SL}(2,{\mathbb{C}})$–representations can be reinterpreted
in terms of the (second) complex Borel regulator. We briefly recall some
relevant facts, and explain a precise formula (2.3). We recall analogous Borel
regulators in the $p$–adic setting, following Huber and Kings [HubK11].

For every dimensions $n$, the $n$–th differentiable group cohomology
$H^{n}_{\mathtt{diff}}(\mathrm{GL}(N,{\mathbb{C}});{\mathbb{C}})$ of the
complex Lie group $\mathrm{GL}(N,{\mathbb{C}})$ stabilizes for all
sufficiently large $N$, with respect to the natural inclusions
$\mathrm{GL}(1,{\mathbb{C}})\to\mathrm{GL}(2,{\mathbb{C}})\to\cdots\to\mathrm{GL}(N,{\mathbb{C}})\to\cdots$,
(inserting extra diagonal entries $1$ at the lower right corner of square
matrices). In fact, via the van Est isomorphism [BorW00, Chapter X, Corollary
5.6], $H^{n}_{\mathtt{diff}}(\mathrm{GL}(N,{\mathbb{C}});{\mathbb{C}})$
identifies with the complex Lie algebra cohomology
$H^{n}(\mathfrak{gl}(N,{\mathbb{C}}),\mathrm{U}(N);{\mathbb{C}})\cong
H^{n}(\mathfrak{gl}(N,{\mathbb{C}});{\mathbb{C}})$ for all $N$. The cohomology
ring $H^{*}(\mathfrak{gl}(N,{\mathbb{C}});{\mathbb{C}})$ is an exterior
${\mathbb{C}}$–algebra on canonical primitive elements $p_{{\mathbb{C}},m}\in
H^{2m-1}(\mathfrak{gl}(N,{\mathbb{C}});{\mathbb{C}})$, where $m$ ranges in
$\\{1,2,\cdots,N\\}$, (see Remark 2.3 below about the normalization
convention).

######  Remark 2.3.

The canonical primitive elements $p_{{\mathbb{C}},m}$ may be chosen
differently by different authors. Although not quite important for our
discussion, we fix our normalization in this paper following Huber and Kings
[HubK11, Definition 0.4.5]. For any field $F$ of characteristic $0$ and any
index $m\geq 1$, we fix the canonical class $p_{F,m}\in
H^{2m-1}(\mathfrak{gl}(N,F);F)$ as represented by the alternating
$F$–multilinear form

$p_{F,m}(X_{1},X_{2}\cdots,X_{2m-1})=\frac{((m-1)!)^{2}}{(2m-1)!}\cdot\sum_{\sigma}\mathrm{sgn}(\sigma)\cdot\mathrm{tr}\left(X_{\sigma(1)}\cdot
X_{\sigma(2)}\cdot\cdots\cdot X_{\sigma(2m-1)}\right),$

for all $X_{1},X_{2},\cdots,X_{2m-1}$ in
$\mathfrak{gl}(N,F)=\mathrm{Mat}(N,F)$, where $\sigma$ ranges over all
permutations on the indices $\\{1,2,\cdots,2m-1\\}$. Note that these forms all
come from canonical forms over ${\mathbb{Q}}$ on by scalar extension.

In this paper, we declare the $m$–th _complex Borel regulator_ as the
canonical differentiable group cohomology class $p_{F,m}\in
H^{2m-1}_{\mathtt{diff}}(\mathrm{GL}(N,{\mathbb{C}});{\mathbb{C}})$, for any
$N$ in the stable range. One may equavalently declare the $m$–th complex Borel
regulator as a continuous group cohomology class, because the natural
homomorphism
$H^{*}_{\mathtt{diff}}(\mathrm{GL}(N,{\mathbb{C}});{\mathbb{C}})\to
H^{*}_{\mathtt{cont}}(\mathrm{GL}(N,{\mathbb{C}});{\mathbb{C}})$ is isomorphic
[BorW00, Chapter IX, Section 5].

######  Example 2.4.

  1. (1)

The first complex Borel regulator can be characterized as an abelian group
homomorphism $H_{1}(\mathrm{GL}(N,{\mathbb{C}});{\mathbb{Z}})\to{\mathbb{C}}$.
One may identify the (abstract) group homology
$H_{1}(\mathrm{GL}(N,{\mathbb{C}});{\mathbb{Z}})$ with the abelian group
$\mathrm{GL}(N,{\mathbb{C}})/[\mathrm{GL}(N,{\mathbb{C}}),\mathrm{GL}(N,{\mathbb{C}})]\cong\mathrm{GL}(N,{\mathbb{C}})/\mathrm{SL}(N,{\mathbb{C}})\cong{\mathbb{C}}^{\times}$.
By simple computation, the first complex Borel regulator is explicitly twice
the logarithmic modulus function on ${\mathbb{C}}^{\times}$, namely, $z\mapsto
2\cdot\log|z|$.

  2. (2)

The second complex Borel regulator lives in
$H^{3}_{\mathtt{diff}}(\mathrm{GL}(N,{\mathbb{C}});{\mathbb{C}})\cong
H^{3}_{\mathtt{diff}}(\mathrm{SL}(2,{\mathbb{C}});{\mathbb{C}})$. One may
identify $H^{3}_{\mathtt{diff}}(\mathrm{SL}(2,{\mathbb{C}});{\mathbb{C}})$
with $H^{3}(\mathfrak{sl}(2,{\mathbb{C}}),\mathrm{SU}(2);{\mathbb{C}})$, or
equivalently, with the cohomology of
$\mathrm{Isom}_{+}({\mathbb{H}}^{3})$–invariant complex valued differential
forms on ${\mathbb{H}}^{3}$ [BorW00, Chapter I, Sections 1.6 and 5.1]. Then,
the second complex Borel regulator is explicitly represented by twice the
volume form, namely,
$2h^{-3}{\mathrm{d}}x\wedge{\mathrm{d}}y\wedge{\mathrm{d}}h$. (The coeffiicent
$2$ can be easily checked at $(x,y,h)=(0,0,1)$ in ${\mathbb{H}}^{3}$,
according to Remark 2.3).

In this paper, we are the most interested in the second complex Borel
regulator. We rewrite the notation $p_{{\mathbb{C}},2}$ particularly as

(2.2) $b_{\mathbb{C}}\in
H^{3}_{\mathtt{diff}}(\mathrm{GL}(N,{\mathbb{C}});{\mathbb{C}}),$

with any $N$ in the stable range, (such as $N\geq 2$; see Example 2.4). We
simply refer to $b_{\mathbb{C}}$ as _the_ complex Borel regulator.

With the above description, we obtain a formula for volume of
$\mathrm{SL}(2,{\mathbb{C}})$–representations as follows. For any oriented
connected closed $3$–manifold $M$ and any representation
$\rho\colon\pi_{1}(M)\to\mathrm{SL}(2,{\mathbb{C}})$,

(2.3)
$2\cdot\mathrm{Vol}_{\mathrm{SL}(2,{\mathbb{C}})}(M,\rho)=\left\langle\rho^{*}_{{\mathtt{st}}}(b_{{\mathbb{C}}}),\,[M]\right\rangle.$

Here, $\langle\\_,\\_\rangle$ denotes the natural pairing
$H^{3}(M;{\mathbb{C}})\times H_{3}(M;{\mathbb{Z}})\to{\mathbb{C}}$, and
$\rho^{*}_{\mathtt{st}}$ denotes the homomorphism
$H^{3}_{\mathtt{diff}}(\mathrm{GL}(N,{\mathbb{C}});{\mathbb{C}})\to
H^{3}(\pi_{1}(M);{\mathbb{C}})\to H^{3}(M;{\mathbb{C}})$ induced by the
composite $\mathrm{GL}(N,{\mathbb{C}})$–representation
$\rho_{\mathtt{st}}\colon\pi_{1}(M)\to\mathrm{SL}(2,{\mathbb{C}})\to\mathrm{GL}(2,{\mathbb{C}})\to\mathrm{GL}(N,{\mathbb{C}})$.

Let $p$ be a prime in ${\mathbb{Q}}$. Let $L$ be a field extension over the
$p$–adic field ${\mathbb{Q}}_{p}$ of finite algebraic degree, with the
valuation ring denoted as $R$. Just as complex Borel regulators arise in
$H^{*}(\mathfrak{gl}(N,{\mathbb{C}});{\mathbb{C}})$, $L$–Borel regulators
arise in the cohomology ring $H^{*}(\mathfrak{gl}(N,L);L)$ of the Lie algebra
$\mathfrak{gl}(N,L)$ over $L$. The Lazard isomorphism plays the role instead
of the van Est isomorphism, identifying $H^{n}(\mathfrak{gl}(N,L);L)$ with the
locally analytic group cohomology $H^{n}_{\mathtt{la}}(\mathrm{GL}(N,R);L)$ of
the compact $L$–Lie group $\mathrm{GL}(N,R)$. The cohomology ring
$H^{*}(\mathfrak{gl}(N,L);L)$ of the Lie algebra $\mathfrak{gl}(N,L)$ over $L$
is an exterior $L$–algebra on primitive elements of odd dimensions up to
$2N-1$, just as in the complex case. (See [HubK11, Section 1].)

We declare the $m$–th _$L$ –Borel regulator_ as the canonical locally analytic
group cohomology class in $p_{L,m}\in
H^{2m-1}_{\mathtt{la}}(\mathrm{GL}(N,R);L)$, for any $N$ in the stable range.
By relaxing the regularity condition on the space of functions, one obtains
natural homomorphisms of $L$–modules

$H^{2m-1}_{\mathtt{la}}(\mathrm{GL}(N,R);L)\to
H^{2m-1}_{\mathtt{cont}}(\mathrm{GL}(N,R);L)\to H^{2m-1}(\mathrm{GL}(N,R);L).$

Therefore, it makes sense to declare $L$–Borel regulators less restrictively
as continuous or abstract group cohomology classes, by natural transformations
of cohomology theories.

In this paper, we only need in the second $L$–Borel regulator, so we simply
call it _the_ $L$–Borel regulator. Therefore, we denote the (second) $L$–Borel
regulator as

(2.4) $b_{L}\in H^{3}_{\mathtt{la}}(\mathrm{GL}(N,R);L),$

for any $N$ in the stable range. (For example, one may take any $N\geq 3$,
because of the canonical isomorphism
$H^{3}_{\mathtt{la}}(\mathrm{GL}(N,R);L)\cong
H^{3}_{\mathtt{la}}(\mathrm{GL}(3,R);L)$, by the Lazard isomorphism; see
[HubK11, Theorem 1.2.1]).

######  Remark 2.5.

As ${\mathbb{Q}}_{\infty}={\mathbb{R}}$ has exactly two finite algebraic field
extensions ${\mathbb{R}}$ and ${\mathbb{C}}$, we briefly describe real Borel
regulators for completeness of the classical versus $p$–adic analogy. For
every dimension $n$, the $n$–th differentiable group cohomology
$H^{n}_{\mathtt{diff}}(\mathrm{GL}(N,{\mathbb{R}});{\mathbb{R}})$ of the real
Lie group $\mathrm{GL}(N,{\mathbb{R}})$ stabilizes for all sufficiently large
$N$, and there are van Est isomorphisms
$H^{n}_{\mathtt{diff}}(\mathrm{GL}(N,{\mathbb{R}});{\mathbb{R}})\cong
H^{n}(\mathfrak{gl}(N,{\mathbb{R}}),\mathrm{O}(N);{\mathbb{R}})$ [BorW00,
Chapter X, Section 5]. The stable cohomology is an exterior
${\mathbb{R}}$–algebra on primitive elements $q_{{\mathbb{R}},m}$ of each
degree $4m-3$, (see [Bor74, Section 10.6]). Upon suitable normalization, one
may declare $q_{{\mathbb{R}},m}\in
H^{4m-3}_{\mathtt{diff}}(\mathrm{GL}(N,{\mathbb{R}});{\mathbb{R}})$ as the
$m$–th real Borel generator, with $N$ in the stable range.

###  2.3. Borel regulator maps associated to number fields

Suppose that $k/{\mathbb{Q}}$ is finite algebraic field extension. As
${\mathbb{Q}}$ has a unique Archimedean completion
${\mathbb{Q}}_{\infty}={\mathbb{R}}$, there is a canonical ring isomorphism
$k_{\infty}=k\otimes_{\mathbb{Q}}{\mathbb{Q}}_{\infty}\cong\prod_{v|\infty}k_{v}$.
The ring $k_{\infty}$ is isomorphic to
$\prod^{r_{1}}{\mathbb{R}}\times\prod^{r_{2}}{\mathbb{C}}$, where
$r_{1}+2r_{2}=[k:{\mathbb{Q}}]$. We introduce an element

(2.5)
$b_{\infty}\in\bigoplus_{v|\infty}H^{3}_{\mathtt{diff}}(\mathrm{GL}(N,k_{v});k_{v})$

as the sum of the Borel regulators $b_{v}=b_{\mathbb{C}}$ in the direct
summands with $k_{v}\cong{\mathbb{C}}$, for $N$ in the stable range. Note that
$H^{3}_{\mathtt{diff}}(\mathrm{GL}(N,k_{v});k_{v})$ vanishes when
$k_{v}\cong{\mathbb{R}}$ (Remark 2.5). The completion embeddings $k\to k_{v}$
induce subgroup embeddings $\mathrm{SL}(2,k)\to\mathrm{SL}(2,k_{v})$. Summing
up the $k_{v}$–valued natural pairings with the pull-backs of
$b_{v}=b_{\mathbb{C}}$ associated to the complex places $v$, we obtain a real
linear homomorphism, also denoted as

(2.6) $b_{\infty}\colon
H_{3}\left(\mathrm{SL}(N,k);{\mathbb{Z}}\right)\otimes_{\mathbb{Z}}{\mathbb{R}}\to
k_{\infty},$

with $N$ sufficiently large. In this paper, we refer to $b_{\infty}$ in (2.6)
as the _classical Borel regulator map_ associated to the number field $k$. In
(2.6), one may actually take any $N\geq 7$, and the following theorem is due
to A. Borel [Bor74], (see Remark 2.7).

######  Theorem 2.6.

Let $k$ be a finite algebraic field extension over ${\mathbb{Q}}$. Then, the
classical Borel regulator map $b_{\infty}$, as defined in (2.6) is injective.
Moreover, $b_{\infty}$ has image of real dimension $r_{2}$, where $r_{2}$
denotes the number of complex places of $k$.

######  Remark 2.7.

For any (unital, associative) algebra $R$ and for $n\geq 1$, recall that the
$n$-th K-group of $R$ is defined as $K_{n}(R)=\pi_{n}(\mathrm{BGL}(R)^{+})$,
via Quillen’s plus construction [Ros94, Chapter 5, Definition 5.2.6]. If
$k/{\mathbb{Q}}$ is a finite algebraic field extension with $r_{1}$ real
places and $r_{2}$ complex places, Borel shows that
$K_{n}(O)\otimes_{\mathbb{Z}}{\mathbb{R}}$ has real dimension
$0,r_{1}+r_{2},0,r_{2}$ for $n\equiv 0,1,2,3$ modulo $4$ and $n\geq 2$, where
$O$ denotes the ring of integers in $K$ [Bor74, Proposition 12.2]. More
precisely, Borel considers the composite homomorphism

$\textstyle{K_{n}(O)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{v|\infty}K_{n}(O_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{*}}$$\textstyle{\bigoplus_{v|\infty}H_{n}(\mathrm{BGL}(k_{v})^{+};{\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\bigoplus_{v|\infty}H_{n}(\mathrm{BGL}(k_{v});{\mathbb{Z}}),}$

where $h_{*}=\bigoplus_{v|\infty}h_{v*}$ denotes the Hurewicz homomorphism. By
pairing with the complex or real Borel regulators of the correct dimension,
one obtains canonical abelian group homomorphisms of $K_{n}(O)$ to
$\\{0\\},{\mathbb{R}}^{r_{1}}\oplus{\mathbb{C}}^{r_{2}},\\{0\\},{\mathbb{C}}^{r_{2}}$,
for $n\equiv 0,1,2,3$ modulo $4$, respectively. Borel shows that these
homomorphisms induce isomorphisms of
$K_{n}(O)\otimes_{\mathbb{Z}}{\mathbb{R}}$ onto
$\\{0\\},{\mathbb{R}}^{r_{1}}\oplus{\mathbb{R}}^{r_{2}},\\{0\\},{\mathbb{R}}^{r_{2}}$,
respectively. The ring inclusion $O\to k$ induces an isomorphism
$K_{n}(O)\otimes_{\mathbb{Z}}{\mathbb{R}}\cong
K_{n}(k)\otimes_{\mathbb{Z}}{\mathbb{R}}$, by the localization theorem [Ros94,
Chapter 5, Corollary 5.3.28; see also Theorem 5.3.2]). When $n=3$ and $N\geq
7$, there are natural isomorphisms
$K_{3}(k)\otimes_{\mathbb{Z}}{\mathbb{R}}\cong
H_{3}(\mathrm{EL}(k);{\mathbb{Z}})\otimes_{\mathbb{Z}}{\mathbb{R}}\cong
H_{3}(\mathrm{EL}(N,k);{\mathbb{Z}})\otimes_{\mathbb{Z}}{\mathbb{R}}\cong
H_{3}(\mathrm{SL}(N,k);{\mathbb{Z}})\otimes_{\mathbb{Z}}{\mathbb{R}}$; see
[Sah89, Proposition 2.5 (a)], [VdK80, Theorem 4.11] and [Mil71, §16, Corollary
16.3 and Remark] for the steps, respectively.

For any prime $p$ in ${\mathbb{Q}}$, there is a canonical ring isomorphism
$k_{p}=k\otimes_{\mathbb{Q}}{\mathbb{Q}}_{p}\cong\prod_{v|p}k_{v}$. Denote by
$O$ the ring of integers in $k$. We denote
$O_{p}=O\otimes_{\mathbb{Z}}{\mathbb{Z}}_{p}\cong\prod_{v|p}O_{v}$, and
$O_{\\{p\\}}=k\cap O_{p}$ as viewed in $k_{p}$. Analogous to (2.5) and (2.6),
we introduce

(2.7)
$b_{p}\in\bigoplus_{v|p}H^{3}_{\mathtt{la}}(\mathrm{GL}(N,O_{v});k_{v}),$

and obtain a ${\mathbb{Q}}_{p}$–linear homomorphism

(2.8) $b_{p}\colon
H_{3}\left(\mathrm{SL}(N,O_{\\{p\\}});{\mathbb{Z}}\right)\otimes_{\mathbb{Z}}{\mathbb{Q}}_{p}\to
k_{p}.$

In this paper, we refer to $b_{p}$ in (2.8) as the _$p$ –adic Borel regulator
map_ associated to the number field $k$. In (2.8), one may actually take any
$N\geq 7$, and the following hypothesis is a special case of the $p$–adic
Borel regulator injectivity conjecture (Conjecture 1.2), parallel to Theorem
2.6, (see Remark 2.9).

######  Hypothesis 2.8.

For every finite algebraic field extension $k$ over ${\mathbb{Q}}$ with the
ring of integers $O$, there exists infinitely many primes $p$ in
${\mathbb{Q}}$, such that the following condition on $k$ and $p$ holds true:

  * •

The $p$–adic Borel regulator map $b_{p}$ associated to $k$, as defined in
(2.8), is injective.

######  Remark 2.9.

By the same procedure as in Remark 2.7, there is a canonical abelian group
homomorphism $K_{2m-1}(O)\to\bigoplus_{v|p}k_{v}$, obtained using the Hurewicz
homomorphism and the $m$–th $k_{v}$–Borel regulators. Replacing with the
natural isomorphisms $K_{n}(O)\otimes_{\mathbb{Z}}{\mathbb{Q}}_{p}\cong
K_{n}(k)\otimes_{\mathbb{Z}}{\mathbb{Q}}_{p}$ with $n\geq 2$ [Ros94, Chapter
5, Corollary 5.3.28 and Theorem 5.3.2], we obtain the
${\mathbb{Q}}_{p}$–linear homomorphism
$K_{2m-1}(k)\otimes_{\mathbb{Z}}{\mathbb{Q}}_{p}\to\bigoplus_{v|p}k_{v}$. This
is what we call the $m$–th _$p$ –adic Borel regulator map_ associated to $k$,
which makes precise the statement of Conjecture 1.2. This way our Conjecture
1.2 paraphrases [Cal15, Conjecture 2.9]. While the conjecture remains widely
open in general, some hypothetical criterion or partial evidence has been
pointed out by F. Calegari, see [Cal15, Lemma 2.11 and Proposition 2.14]. When
$n=3$ and $N\geq 7$, there are natural isomorphisms
$K_{3}(O)\otimes_{\mathbb{Z}}{\mathbb{Q}}_{p}\cong
K_{3}(O_{\\{p\\}})\otimes_{\mathbb{Z}}{\mathbb{Q}}_{p}\cong
H_{3}(\mathrm{EL}(O_{\\{p\\}});{\mathbb{Z}})\otimes_{\mathbb{Z}}{\mathbb{Q}}_{p}\cong
H_{3}(\mathrm{EL}(N,O_{\\{p\\}});{\mathbb{Z}})\otimes_{\mathbb{Z}}{\mathbb{Q}}_{p}\cong
H_{3}(\mathrm{SL}(N,O_{\\{p\\}});{\mathbb{Z}})\otimes_{\mathbb{Z}}{\mathbb{Q}}_{p}$,
which justify our Hypothesis 2.8 as an implication of Conjecture 1.2; see
[Ros94, Chapter 5, Corollary 5.3.28 and Theorem 5.3.2], [Sah89, Proposition
2.5 (a)], [VdK80, Theorem 4.11], and [Mil71, §16, Corollary 16.3 and Remark]
for the steps, respectively.

##  3\. Preliminary II

In this preliminary section, we review quaternion algebras, mostly following
the textbook of Maclachlan and Reid [MacR03]. Next, we review some dynamical
aspects of pseudo-Anosov automorphisms and their suspension flows, following
the perspective of [Liu20a, Section 5].

###  3.1. Quaternion algebras

Let $F$ be a field of characteristic $0$. A _quaternion algebra_ $\mathscr{A}$
over $F$ is an (associative unital) algebra which is isomorphic to a
$4$–dimensional $F$–vector space $F\mathbf{1}\oplus F\mathbf{i}\oplus
F\mathbf{j}\oplus F\mathbf{k}$ enriched with the multiplication rules
$\mathbf{i}^{2}=a\mathbf{1}$, $\mathbf{j}^{2}=b\mathbf{1}$, and
$\mathbf{i}\mathbf{j}=-\mathbf{j}\mathbf{i}=\mathbf{k}$, where $a,b\in
F^{\times}$ are nonzero elements of $F$. Adopting the Hilbert symbol, this is
written as

(3.1) $\mathscr{A}\cong\left(\frac{a,b}{F}\right).$

Hence $\mathbf{k}^{2}=-ab\mathbf{1}$,
$\mathbf{j}\mathbf{k}=-\mathbf{k}\mathbf{j}=-b\mathbf{i}$, and
$\mathbf{k}\mathbf{i}=-\mathbf{i}\mathbf{k}=-a\mathbf{j}$. Note that different
Hilbert symbols may present isomorphic quaternion algebras, for example,
$(a,b/F)\cong(b,a/F)\cong(a,-ab/F)\cong(b,-ab/F)\cong(ax^{2},by^{2}/F)$, where
$a,b,x,y\in F^{\times}$.

######  Example 3.1.

  1. (1)

Any quaternion algebra $\mathscr{A}$ over a field $F$ of characteristic $0$
(or more generally, $\neq 2$) is either a division algebra or isomorphic to
the algebra $\mathrm{Mat}(2,F)$ of $2\times 2$–matrices with entries in $F$
[MacR03, Chapter 2, Theorem 2.1.7]. In particular,
$\mathscr{A}\cong\mathrm{Mat}(2,F)\cong(1,1/F)$ if $F$ is algebraically
closed.

  2. (2)

There are exactly two quaternion algebras over ${\mathbb{R}}$ up to
isomorphism: the matrix algebra
$\mathrm{Mat}(2,{\mathbb{R}})\cong(1,1/{\mathbb{R}})$ and the division algebra
of Hamilton’s quaternions $\mathcal{H}\cong(-1,-1/{\mathbb{R}})$. The similar
dichotomy occurs for quaternion algebras over $\mathfrak{p}$–adic number
fields $k_{\mathfrak{p}}$, where $k$ is an algebraic field extension over
${\mathbb{Q}}$ of finite degree, and where $\mathfrak{p}$ is a prime, (that
is, a nonzero prime ideal of the subring of integers $O_{k}$). Besides the
matrix algebra $\mathrm{Mat}(2,k_{\mathfrak{p}})\cong(1,1/k_{\mathfrak{p}})$,
there is a unique division algebra up to isomorphism. It has a Hilbert symbol
$(\pi,u/k_{\mathfrak{p}})$, such that $\pi$ is a uniformizer of the
$\mathfrak{p}$–adic valuation ring $O_{k,\mathfrak{p}}$, and $k(\sqrt{u})$ is
the unique unramified quadratic extension of $k$, up to isomorphism. In all
the above cases, the division algebra is said to be _ramified_ , over
${\mathbb{R}}$ or $k_{\mathfrak{p}}$, while the matrix algebra is _unramified_
(or _split_). (See [MacR03, Chapter 2, Sections 2.5 and 2.6].)

  3. (3)

Let $k$ is a finite algebraic field extension over ${\mathbb{Q}}$, and
$\mathscr{B}$ be a quaternion algebra over $k$. Then the Archimedean places
correspond naturally with the real embeddings $\sigma\colon k\to{\mathbb{R}}$
and the complex-conjugate pairs of the (imaginary) complex embeddings
$\\{\sigma,\bar{\sigma}\colon k\to{\mathbb{C}}\\}$; the non-Archimedean places
correspond naturally with the primes $\mathfrak{p}$ of $k$. The complex
quaternion algebras
$\mathscr{B}\otimes_{\sigma}{\mathbb{C}}\cong\mathscr{B}\otimes_{\bar{\sigma}}{\mathbb{C}}$
at the complex places are always unramified, but the real quaternion algebras
$\mathscr{B}\otimes_{\sigma}{\mathbb{R}}$ at the real places may ramify or
split. Ramification or splitting may also occur with the $\mathfrak{p}$–adic
quaternion algebra $\mathscr{B}\otimes_{k}k_{\mathfrak{p}}$ at a
$\mathfrak{p}$–adic place. The classification of quaternion algebras over $k$
demonstrates that the isomorphism type of $\mathscr{B}$ is uniquely determined
by its locus of ramification, that is, the set of places where the completion
quaternion algebra ramifies. Moreover, the locus of ramification is always a
finite set, consisting of an even number of real or non-Archmedean places, and
any such configuration can be realized with a quaternion algebra over $k$.
(See [MacR03, Chapter 2, Section 2.7, and Chapter 7, Theorem 7.3.6].)

More intrinsically, a quaternion algebra $\mathscr{A}$ can be characterized as
a simple, central, $4$–dimensional algebra over $F$. The $F$–vector subspaces
$F\mathbf{1}$ and $F\mathbf{i}\oplus F\mathbf{j}\oplus F\mathbf{k}$ can be
characterized respectively as the center of $\mathscr{A}$ and the pure
quaternion complement of the center, whose nonzero elements are noncentral and
have central squares. The _quaternion conjugation_ on $\mathscr{A}$ is thereby
intrinsically defined, as

$\overline{t\mathbf{1}+x\mathbf{i}+y\mathbf{j}+z\mathbf{k}}=t\mathbf{1}-x\mathbf{i}-y\mathbf{j}-z\mathbf{k},$

where $t,x,y,z\in F$. It allows one to introduce the (reduced) _trace_ and the
(reduced) _norm_ of any quaternion $q\in\mathscr{A}$:

(3.2) $\mathrm{Tr}_{\mathscr{A}/F}(q)=q+\bar{q},\mbox{ and
}\mathrm{Nr}_{\mathscr{A}/F}(q)=q\bar{q},$

respectively, both valued in $F$. The functions
$\mathrm{Tr}_{\mathscr{A}/F}\colon\mathscr{A}\to F$ and
$\mathrm{Nr}_{\mathscr{A}/F}\colon\mathscr{A}^{\times}\to F^{\times}$ are
homomorphisms of the $F$–vector space $\mathscr{A}$ and the multiplicative
group of invertibles $\mathscr{A}^{\times}$, respectively. The trace and the
norm fit into a neat quadratic equation:

(3.3) $q^{2}-\mathrm{Tr}_{\mathscr{A}/F}(q)\cdot
q+\mathrm{Nr}_{\mathscr{A}/F}(q)=0.$

(See [MacR03, Chapter 2, Section 2.1].)

######  Proposition 3.2.

Let $\mathscr{A}$ be a quaternion algebra over a field $F$ of characteristic
$0$. For any $A,B,C,D\in\mathscr{A}$ with
$\mathrm{Nr}(A)=\mathrm{Nr}(B)=\mathrm{Nr}(C)=\mathrm{Nr}(D)=1$, the following
identities all hold true:

  1. (1)

$\mathrm{Tr}(1)=2$;
$\mathrm{Tr}(A^{-1})=\mathrm{Tr}(A)$;
$\mathrm{Tr}(A^{2})=\mathrm{Tr}(A)^{2}-2$.

  2. (2)

$\mathrm{Tr}(BA)=\mathrm{Tr}(AB)$;
$\mathrm{Tr}(BAB^{-1})=\mathrm{Tr}(A)$;
$\mathrm{Tr}(AB^{-1})=\mathrm{Tr}(A)\cdot\mathrm{Tr}(B)-\mathrm{Tr}(AB)$;
$\mathrm{Tr}(ABA^{-1}B^{-1})=\mathrm{Tr}(A)^{2}+\mathrm{Tr}(B)^{2}+\mathrm{Tr}(AB)^{2}-\mathrm{Tr}(A)\cdot\mathrm{Tr}(B)\cdot\mathrm{Tr}(AB)-2$.

  3. (3)

$2\cdot\mathrm{Tr}(ABCD)=\\\
\mathrm{Tr}(A)\cdot\mathrm{Tr}(BCD)+\mathrm{Tr}(B)\cdot\mathrm{Tr}(ACD)+\mathrm{Tr}(C)\cdot\mathrm{Tr}(ABD)+\mathrm{Tr}(D)\cdot\mathrm{Tr}(ABC)+\mathrm{Tr}(AB)\cdot\mathrm{Tr}(CD)-\mathrm{Tr}(AC)\cdot\mathrm{Tr}(BD)+\mathrm{Tr}(AD)\cdot\mathrm{Tr}(BC)-\mathrm{Tr}(A)\cdot\mathrm{Tr}(B)\cdot\mathrm{Tr}(CD)-\mathrm{Tr}(C)\cdot\mathrm{Tr}(D)\cdot\mathrm{Tr}(AB)-\mathrm{Tr}(A)\cdot\mathrm{Tr}(D)\cdot\mathrm{Tr}(BC)-\mathrm{Tr}(B)\cdot\mathrm{Tr}(C)\cdot\mathrm{Tr}(AD)+\mathrm{Tr}(A)\cdot\mathrm{Tr}(B)\cdot\mathrm{Tr}(C)\cdot\mathrm{Tr}(D).$

Here we drop the subscripts in $\mathrm{Tr}_{\mathscr{A}/F}$ and
$\mathrm{Nr}_{\mathscr{A}/F}$ for simplicity.

######  Remark 3.3.

These trace relations are all proved in [MacR03, Chapter 3, Section 3.4] for
matrices $A,B,C,D\in\mathrm{SL}(2,{\mathbb{C}})$. They also hold in the
$\mathscr{A}/F$ setting. In fact, one may take a suitable subfield
$F^{\prime}$ of $F$ finitely generated over ${\mathbb{Q}}$, and a quaternion
subalgebra $\mathscr{A}^{\prime}$ over $F^{\prime}$ containing the present
quaternions $A,B,C,D\in\mathscr{A}$. By embedding $F^{\prime}$ into
${\mathbb{C}}$, there is an algebra embedding of $\mathscr{A}^{\prime}$ into
$\mathscr{\mathscr{A}}^{\prime}\otimes_{F^{\prime}}{\mathbb{C}}\cong\mathrm{Mat}(2,{\mathbb{C}})$.
The determinant function on complex $2\times 2$–matrices agrees with
$\mathrm{Nr}_{\mathscr{A}^{\prime}/F^{\prime}}=\mathrm{Nr}_{\mathscr{A}/F}$.
This adapts the $\mathrm{SL}(2,{\mathbb{C}})$ case to our setting.

###  3.2. Pseudo-Anosov automorphisms

Let $S$ be an orientable connected closed surface of genus $\geq 2$. A
_pseudo-Anosov_ automorphism $f\colon S\to S$ refers to an orientation-
preserving homeomorphism with the following property: There exist measured
foliations $(\mathscr{F}^{\mathtt{s}},\mu^{\mathtt{s}})$ and
$(\mathscr{F}^{\mathtt{u}},\mu^{\mathtt{u}})$ on $S$, having identical
singular points and transverse anywhere else, such that
$f\cdot(\mathscr{F}^{\mathtt{u}},\mu^{\mathtt{u}})=(\mathscr{F}^{\mathtt{u}},\lambda\mu^{\mathtt{u}})$
and
$f\cdot(\mathscr{F}^{\mathtt{s}},\mu^{\mathtt{s}})=(\mathscr{F}^{\mathtt{s}},\lambda^{-1}\mu^{\mathtt{s}})$
hold for some constant $\lambda>1$.

We recall that a _measured foliation_ on $S$ is a foliation $\mathscr{F}$ with
prong singularities together with a transverse measure $\mu$ that is invariant
under the holonomy of $\mathscr{F}$. A _prong singularity_ is modeled on the
singularity of a holomorphic quadratic differential $z^{k-2}{\mathrm{d}}z^{2}$
on ${\mathbb{C}}$, where $k=3,4,5,\cdots$ is called the _prong number_. (See
[FatLP12, Exposé 1].) The constant $\lambda$ is called the _stretch factor_.
The measured foliations $(\mathscr{F}^{\mathtt{s}},\mu^{\mathtt{s}})$ and
$(\mathscr{F}^{\mathtt{u}},\mu^{\mathtt{u}})$ are called the _stable_ and the
_unstable_ measured foliations, respectively. They all uniquely determined by
$f$. Moreover, any other pseudo-Anosov automorphism in the isotopy class of
$f$ is a conjugate of $f$ by a self-homeomorphism of $S$ that is isotopic to
the identity, (see [FatLP12, Exposé 12]).

Let $f\colon S\to S$ be a pseudo-Anosov automorphism. We obtain an orientable
connected closed $3$–manifold

(3.4) $M_{f}=\frac{S\times{\mathbb{R}}}{(x,r+1)\sim(f(x),r)}.$

This is called the _mapping torus_ of $(S,f)$. The inclusion of
$S\times\\{0\\}$ identifies $S$ as a distinguished surface fiber in $M_{f}$.
There is a the continuous family of homeomorphisms

(3.5) $\theta_{t}\colon M_{f}\to M_{f}$

parametrized by $t\in{\mathbb{R}}$ and determined by $(x,r)\mapsto(x,r+t)$,
This is called the _suspension flow_ on $M_{f}$. There is a distinuished
cohomology class of $M_{f}$, denoted as

(3.6) $\phi_{f}\in H^{1}(M_{f};{\mathbb{Z}}).$

It is represented by the distinguished projection $M_{f}\to{\mathbb{S}}^{1}$,
where ${\mathbb{S}}^{1}$ is identified with ${\mathbb{R}}/{\mathbb{Z}}$ and
the projection is determined by $(x,r)\mapsto r$.

For any (nonzero) natural number $m\in{\mathbb{N}}$, an _$m$ –periodic point_
of $f$ refers to a fixed point $p\in S$ of $f^{m}$. We obtain a line
${\mathbb{R}}\to S\times{\mathbb{R}}\colon r\mapsto(p,r)$, which descends to a
loop ${\mathbb{R}}/m{\mathbb{Z}}\to M_{f}$. This gives rise to an $m$–periodic
trajectory of the suspension flow. The construction sets up a natural
bijective correspondence between the $f$–iteration orbits of $m$–periodic
points on the surface and the $m$–periodic trajectories of the suspension flow
on the mapping torus. As $f$ is pseudo-Anosov, no periodic trajectories in
$M_{f}$ are freely homotopic to any other [JiaG93, Corollary 2.3] (see also
[Liu20a, Remark 2.1]). Therefore, the periodic orbits of $f$ can be identified
with mutually distinct conjugate classes in the fundamental group of $M_{f}$.
For all $m\in{\mathbb{N}}$, the numbers of $m$–periodic orbit of $f$ are
always finite, denoted as $N_{m}(f)$. They are called the _Nielsen numbers_
(of periodic orbit classes), and can be characterized purely in terms of the
homotopy class of $f$.

In this paper, we often fix a universal covering space
$\widetilde{S}_{\mathrm{univ}}\to S$, and denote by $\pi_{1}(S)$ the deck
transformation group acting on $\widetilde{S}_{\mathrm{univ}}$. Then the
composite map

$\textstyle{\widetilde{S}_{\mathrm{univ}}\times{\mathbb{R}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{cov}\times\mathrm{id}}$$\textstyle{S\times{\mathbb{R}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{quot}}$$\textstyle{M_{f}}$

is a universal covering projection onto $M_{f}$. We denote by $\pi_{1}(M_{f})$
the deck transformation group acting on
$\widetilde{S}_{\mathrm{univ}}\times{\mathbb{R}}$, and by
$\mathrm{Orb}(\pi_{1}(M_{f}))$ the set of conjugacy classes of
$\pi_{1}(M_{f})$. For any $m\in{\mathbb{N}}$, denote by
$\mathrm{Per}_{m}(f)\subset S$ the set of $m$–periodic points of $f$, and by
$\mathrm{Orb}_{m}(f)=\mathrm{Per}_{m}(f)/\langle f\rangle$ the set of
$m$–periodic orbits. For any $\mathbf{O}\in\mathrm{Orb}_{m}(f)$, the
$m$–periodic trajectory in $M_{f}$ represents a free-homotopy loop. We treat
the free-homotopy loop equivalently as a conjugacy class of $\pi_{1}(M_{f})$,
and denote it as $\ell_{m}(f,\mathbf{O})$. As mentioned, we obtain an
injective map

(3.7)
$\begin{array}[]{cc}\mathrm{Orb}_{m}(f)\to\mathrm{Orb}(\pi_{1}(M_{f}))\colon&\mathbf{O}\mapsto\ell_{m}(f,\mathbf{O})\end{array}$

for every $m\in{\mathbb{N}}$. Note that the homology class
$[\ell_{m}(f,\mathbf{O})]\in H_{1}(M_{f};{\mathbb{Z}})$ satisfies
$\phi_{f}(\ell_{m}(f,\mathbf{O}))=m$, treating $\phi_{f}$ naturally as a
${\mathbb{Z}}$–linear homomorphism $H_{1}(M_{f};{\mathbb{Z}})\to{\mathbb{Z}}$.

The measure theoretical dynamics of a pseudo-Anosov automorphism is equivalent
to a mixing Markov process. In this characterization, $f$ is comprehended as a
self-homeomorphism of $S$ that preserves the product measure
$\mu^{\mathtt{s}}\times\mu^{\mathtt{u}}$. With a Markov partition, the system
can be effectively by a symbolic approach, as a subshift of finite type. The
symbolic model is usually represented as a directed graph or its adjacency
matrix, called the _transition graph_ or the _transition matrix_. In this
paper, we adopt the transition graph model and use its dynamical cycles to
study periodic trajectories of the suspension flow.

A _directed graph_ is a cell $1$–complex with oriented $1$–cells. We call the
$0$–cells _vertices_ , and the oriented $1$–cells _directed edges_. A
_dynamical cycle_ is an immersed loop obtained by cyclically concatenating
finitely many directly consecutive directed edges. Unless otherwise declared,
we do not assume a base point or a preferred parametrization when using this
term. A directed graph is said to be _irreducible_ (also called _strongly
connected_) if every pair of (not necessarily distinct) vertices occurs in
some dynamical cycle. In particular, irreducible directed graphs are all
topologically connected.

######  Proposition 3.4.

Let $S$ be an orientable connected closed surface of genus $\geq 2$. Let
$f\colon S\to S$ be a pseudo-Anosov automorphism.

Then, there exist an irreducible directed graph $T$, and a continuous map
$T\to M_{f}$ which is surjective on the fundamental group level. Moreover, up
to free homotopy of loops, every dynamical cycle in $T$ projects a periodic
trajectory in $M_{f}$, and every periodic trajectory in $M_{f}$ admits a
finite cyclic cover which lifts to a dynamical cycle in $T$, among finitely
many choices.

In fact, $T$ can be construction as the transition graph $T_{f,\mathcal{R}}$
associated to a Markov partition $\mathcal{R}$ of $S$ with respect to $f$, and
there is a continuous map $T_{f,\mathcal{R}}\to M_{f}$, which is canonical up
to homotopy.

######  Remark 3.5.

See [Liu20a, Section 5]. In fact, Proposition 3.4 is a simplified summary of
Lemmas 5.2, 5.3, 5.6, and 5.7 thereof. See also [FatLP12, Exposés 9 and 10]
for exposition of the pseudo-Anosov theory from a Markov partition approach.

We briefly recall the construction of $T_{f,\mathcal{R}}$ in [Liu20a, Section
5] for the reader’s reference. Roughly speaking, a _Markov partition_
$\mathcal{R}=\\{R_{1},\cdots,R_{k}\\}$ with respect to a pseudo-Anosov
automorphism $f\colon S\to S$ is a collection of compact
$(\mathscr{F}^{\mathtt{s}},\mathscr{F}^{\mathtt{u}})$–rectangles (or
_birectangles_), which form a cover of $S$ without any overlap of positive
area. Moreover, $f$ streches any $R_{i}$ in the
$\mathscr{F}^{\mathtt{u}}$–transverse direction by the factor $\lambda$, and
shrinks it in the $\mathscr{F}^{\mathtt{s}}$–transverse direction by the same
factor, such that $f(R_{i})$ becomes a long and thin birectangle in $S$ whose
$\mathscr{F}^{\mathtt{u}}$–sides are contained, as subsegments, in some
$\mathscr{F}^{\mathtt{u}}$–sides of some birectangles from $\mathcal{R}$. It
is also required that $f(R_{i})$ intersects $R_{j}$ in at most one sub-
birectangle, ignoring any null area components. Hence, there is a completely
similar description of $f^{-1}(R_{i})$, switching the roles of
$\mathscr{F}^{\mathtt{u}}$ and $\mathscr{F}^{\mathtt{s}}$.

Given a Markov partition $\mathcal{R}=\\{R_{1},\cdots,R_{k}\\}$, the
_transition graph_ $T_{f,\mathcal{R}}$ is constructed as follows. The vertices
$v_{i}$ of $T_{f,\mathcal{R}}$ correspond bijectively with all the
birectangles $R_{i}\in\mathcal{R}$, and the directed edges $e_{ij}$ of
$T_{f,\mathcal{R}}$ correspond bijectively with all the pairs
$(R_{i},R_{j})\in\mathcal{R}\times\mathcal{R}$ with intersection $R_{i}\cap
f^{-1}(R_{j})$ of positive area. Obtain a subset
$\mathring{X}_{f,\mathcal{R}}$ of $M_{f}$ as the union of all the subsets
$\mathrm{int}(R_{i})\times\\{0\\}$ and $(\mathrm{int}(R_{i})\cap
f^{-1}(\mathrm{int}(R_{j})))\times(0,1)$, (represented with parameters in
$S\times{\mathbb{R}}$). Then there is an obvious collapsing map
$\mathring{X}_{f,\mathcal{R}}\to T_{f,\mathcal{R}}$, which is a homotopy
equivalence. Up to homotopy, this gives rise to a distinguished map
$T_{f,\mathcal{R}}\to M_{f}$, as the homotopy inverse followed by the
inclusion. In fact, the path compactification of
$\mathring{X}_{f,\mathcal{R}}$ can be identified with the flow-box complex
$X_{f,\mathcal{R}}$ as introduced in [Liu20a, Definition 5.1].

##  4\. Periodic trajectories and arithmetic invariants

In this section, we establish two criteria regarding irreducible
$\mathrm{SL}(2,{\mathbb{C}})$–representations of fundamental groups of pseudo-
Anosov mapping tori (Lemmas 4.1 and 4.3). These criteria relate the trace
field and the ramification type of the quaternion algebra (in the real case)
with character values on the periodic trajectories.

For any group $\pi$, an _irreducible representation_
$\rho\colon\pi\to\mathrm{SL}(2,{\mathbb{C}})$ is a group homomorphism such
that $\rho(\pi)$ preserves no nontrivial invariant subspace in
${\mathbb{C}}^{2}$ [CulS83, Section 1.2]. In this case, the _trace field_ of
$\rho$ refers to the subfield of ${\mathbb{C}}$ generated by
$\\{\mathrm{tr}(\rho(g))\colon g\in\pi\\}$ over ${\mathbb{Q}}$, and the
_quaternion algebra_ of $\rho$ refers to the subalgebra of
$\mathrm{Mat}(2,{\mathbb{C}})$ generated by $\\{\rho(g)\colon g\in\pi\\}$ over
the trace field, which is indeed a quaternion algebra. Note that if the trace
field of $\rho$ contained in ${\mathbb{R}}$, then the completion of the
quaternion algebra of $\rho$ is a quaternion algebra over ${\mathbb{R}}$. (See
also Lemma 8.2 and Definition 8.3 in Section 8 for justificaiton of the
terminology in a slightly more general setting.)

######  Lemma 4.1.

Let $S$ be an orientable connected surface of genus $\geq 2$. Let $f\colon
S\to S$ be a pseudo-Anosov automorphism. Denote by $M_{f}$ the mapping torus,
furnished with the suspension flow. Suppose that
$\rho\colon\pi_{1}(M_{f})\to\mathrm{SL}(2,{\mathbb{C}})$ is an irreducible
representation.

Then, any value of $\chi_{\rho}$ on $\pi_{1}(M_{f})$ lies in the
${\mathbb{Z}}$–subalgebra of ${\mathbb{C}}$ generated by the values of
$\chi_{\rho}$ on the set of periodic trajectories. In particular, the trace
field of $\rho$ is the field of fractions of that subalgebra.

###### Proof.

Take a Markov partition $\mathcal{R}$ of $S$ with respect to $f$. We obtain
the transition graph $T=T_{f,\mathcal{R}}$, whose dynamical cycles encode the
periodic trajectories of $M_{f}$ in the sense of Proposition 3.4.

We denote by $V=V(T)$ the set of vertices, and by $E=E(T)$ the set of directed
edges. For any directed edge $e\in E$, denote by $\bar{e}$ the orientation-
reversal of $e$. This operation also switches the roles of the initial and the
terminal vertices. Denote by $\bar{E}$ the set of the reversed edges. A
_combinatorial path_ in $T$ refers to the concatenation of a finite sequence
of consecutive directed edges or reversed edges. We denote a combinatorial
path as $\alpha=e_{1}e_{2}\cdots e_{n}$, where $e_{i}\in E\cup\bar{E}$, (so
the initial and the terminal vertices of $\alpha$ are the initial vertex of
$e_{1}$ and the terminal vertex of $e_{n}$, respectively). There is a reversed
combinatorial path $\bar{\alpha}=\bar{e}_{n}\cdots\bar{e}_{2}\bar{e}_{1}$. We
call $\alpha$ a _dynamical path_ if $e_{1},\cdots,e_{n}\in E$. When the
terminal vertex of a combinatorial path $\alpha$ coincides with the initial
vertex of another $\beta$, there is a concatenated combinatorial path
$\alpha\beta$. When the initial vertex of a combinatorial path $\gamma$
coincides with the terminal vertex, $\gamma$ is closed, then it makes sense to
evaluate $\chi_{\rho}$ on $\gamma$, treating $\gamma$ as a free-homotopy loop
in $T$ and using the homotopically distinguished map $T\to M_{f}$. The value
of $\chi_{\rho}$ is invariant under cyclical permutations of the edge sequence
of $\gamma$ and the reversion of $\gamma$, by the conjugacy invariance and the
inversion invariance of the trace on $\mathrm{SL}(2,{\mathbb{C}})$. Moreover,
any loop in $T$ can be freely homotoped to a closed combinatorial path in $T$,
(up to parametrization).

We claim that any combinatorial closed path $\gamma$ in $T$ is freely
homotopic to another one of the concatenated form

$\xi_{0}\bar{\eta}_{1}\xi_{1}\cdots\bar{\eta}_{n}\xi_{n},$

or possibly without $\xi_{0}$ or $\xi_{n}$, such that $\xi_{i},\eta_{i}$ are
all dynamical paths, and such that $\eta_{i}$ are all closed. This basically
follows from the irreducibility of the directed graph $T$ (Proposition 3.4).
In fact, $\gamma$ can be written as a unique concatenated form
$\alpha_{0}\bar{\beta}_{1}\alpha_{1}\cdots\bar{\beta}_{n}\alpha_{n}$, or
possibly without $\alpha_{0}$ or $\alpha_{n}$, such that
$\alpha_{i},\beta_{i}$ are all dynamical paths. For each
$\bar{\beta}=\bar{\beta}_{j}$, there is a dynamical cycle passing through the
initial vertex $v$ and the terminal vertex $w$ of $\bar{\beta}$, since $T$ is
irreducible. Then there is a dynamical path $\delta$ from $v$ to $w$. Replace
the term $\bar{\beta}$ in $\gamma$ with
$\delta\bar{\delta}\bar{\beta}=\delta\overline{(\beta\delta)}$. The modified
part becomes a dynamical path $\delta$ concatenated with a reversed closed
dynamical path $\overline{(\beta\delta)}$. Modifying all $\beta_{j}$ in
$\gamma$ this way, we end up getting a concatenated form as claimed.

Since $T\to M_{f}$ is $\pi_{1}$–surjective (Proposition 3.4), any conjugacy
class in $\pi_{1}(M_{f})$ admits a lift to $\pi_{1}(T)$, and can be
represented as a closed combinatorial path $\gamma$. We may assume $\gamma$
takes the claimed form
$\xi_{0}\bar{\eta}_{1}\xi_{1}\cdots\bar{\eta}_{n}\xi_{n}$, or possibly without
$x_{0}$ or $\xi_{n}$, as above. We show that the value of $\chi_{\rho}$ at
$\gamma$ can be expressed as a ${\mathbb{Z}}$–polynomial function of values of
$\chi_{\rho}$ at dynamical cycles. This will immediately imply Lemma 4.1,
because the dynamical cycles all project periodic trajectories in $M_{f}$ up
to free homotopy (Proposition 3.4).

We argue by induction on the number $n$ of reversed closed dynamical path
terms. Possibly after a cyclic permutation, we may rewrite $\gamma$ as
$\gamma^{\prime}\bar{\eta}_{n}$, where
$\gamma^{\prime}=(\xi_{n}\xi_{0})\bar{\eta}_{1}\xi_{1}\cdots\bar{\eta}_{n-1}\xi_{n-1}$.
By trace relations (Proposition 3.2), we obtain

$\chi_{\rho}(\gamma)=\chi_{\rho}(\gamma^{\prime}\bar{\eta}_{n})=\chi_{\rho}(\gamma^{\prime})\cdot\chi_{\rho}(\bar{\eta}_{n})-\chi_{\rho}(\gamma^{\prime}\eta_{n})=\chi_{\rho}(\gamma^{\prime})\cdot\chi_{\rho}(\eta_{n})-\chi_{\rho}(\gamma^{\prime}\eta_{n}).$

Note that $\gamma^{\prime}$ and $\gamma^{\prime}\eta_{n}$ are both in the
claimed form with $n-1$ reversed closed dynamical cycle terms. By induction,
$\chi_{\rho}(\gamma^{\prime})$ and $\chi_{\rho}(\gamma^{\prime}\eta_{n})$ are
both ${\mathbb{Z}}$–polynomial functions of values of $\chi_{\rho}$ at
dynamical cycles, so $\chi_{\rho}(\gamma)$ also has the same property. This
completes the induction. ∎

We also obtain the following byproduct of independent interest. It is probably
known to experts.

######  Corollary 4.2.

For any pseudo-Anosov automorphism, the fundamental group of the mapping torus
is normally generated by the periodic trajectories of the suspension flow. In
other words, the union of the conjugacy classes represented by the periodic
trajectories forms a generating set.

###### Proof.

As shown in the proof Lemma 4.1, any closed combinatorial path $\gamma$ in an
irreducible directed graph $T$ is freely homotopic to a dynamical cycle
$\xi_{0}\cdots\xi_{n}$ inserted with finitely many reversed dynanical paths
$\bar{\eta}_{1},\cdots,\bar{\eta}_{n}$. It follows that attaching $2$–cells
along all the dynamical cycles in $T$ will result in a simply connected CW
$2$–complex, or equivalently, the union of the conjugacy classes represented
by the dynamical cycles forms a generating set of $\pi_{1}(T)$. Then the
normal generation of $\pi_{1}(M_{f})$ by periodic trajectories follows from
the $\pi_{1}$–surjectivity of the map $T\to M_{f}$ as provided in Proposition
3.4. ∎

######  Lemma 4.3.

Let $S$ be an orientable connected surface of genus $\geq 2$. Let $f\colon
S\to S$ be a pseudo-Anosov automorphism. Suppose that
$\rho\colon\pi_{1}(M_{f})\to\mathrm{SL}(2,{\mathbb{C}})$ is an irreducible
representation whose trace field is real.

Then, the following dichotomy holds true.

  1. (1)

If $\chi_{\rho}$ is uniformly bounded on the set of periodic trajectories,
then the real complete quaternion algebra of $\rho$ is ramified. In this case,
$\rho$ is conjugate to a representation in $\mathrm{SU}(2)$.

  2. (2)

If $\chi_{\rho}$ is unbounded on the set of periodic trajectories, then the
real complete quaternion algebra of $\rho$ is unramified. In this case, $\rho$
is conjugate to a representation in $\mathrm{SL}(2,{\mathbb{R}})$.

###### Proof.

The real complete quaternion algebra is isomorphic to either Hamilton’s
quaternion algebra $\mathcal{H}$ or the real $2\times 2$–matrix algebra
$\mathrm{Mat}(2,{\mathbb{R}})$, (see Example 3.1). The elements of norm $1$
form a real Lie group, isomorphic to $\mathrm{SU}(2)$ or
$\mathrm{SL}(2,{\mathbb{R}})$, respectively. The natural inclusion of the
image of $\rho$ into the real complete quaternion algebra can be regarded as a
representation of $\pi$ in $\mathrm{SU}(2)$ or $\mathrm{SL}(2,{\mathbb{R}})$,
which has the same character as $\chi_{\rho}$. Therefore, the conjugacy
characterizations in the dichotomy follow from the well-known fact that
irreducible $\mathrm{SL}(2,{\mathbb{C}})$–representations are uniquely
determined by their characters up to conjugacy [CulS83, Proposition 1.5.2].

Recall any matrix $A\in\mathrm{SL}(2,{\mathbb{R}})$ other than $\pm I$ falls
into one of three types, according to the sign of its discriminant
$\mathrm{tr}(A)^{2}-4$: elliptic if negative, parabolic if zero, or hyperbolic
if positive. In the hyperbolic case, $|\mathrm{tr}(A^{n})|$ grows
exponentially as $n$ tends to $\infty$. On the other hand,
$|\mathrm{tr}(A)|\leq 2$ holds for any matrix $A\in\mathrm{SU}(2)$. With these
facts, the proof of Lemma 4.3 essentially reduces to the following claim: _For
any representation $\rho\colon\pi_{1}(M_{f})\to\mathrm{SL}(2,{\mathbb{R}})$
that is irreducible in $\mathrm{SL}(2,{\mathbb{C}})$, there exists some
periodic trajectory which is, up to conjugacy, represented by a matrix of
hyperbolic type. _

To prove the claim, it suffices to assume that the image of $\rho$ is torsion-
free. In fact, if $G=\mathrm{Im}(\rho)$ contains nontrivial torsion, there is
a torsion-free finite-index normal subgroup $G^{\prime}$ of $G$, by Selberg’s
lemma [Rat06, Chapter 7, §7.6, Corollary 4]. The preimage of $G^{\prime}$ in
$\pi=\pi_{1}(M_{f})$ is a finite-index normal subgroup
$\pi^{\prime}=\pi_{1}(M^{\prime})$, which corresponds to a connected regular
finite cover $M^{\prime}\to M_{f}$. We can identify $M^{\prime}$ as the
mapping torus $M_{f^{\prime}}$ of a pseudo-Anosov automorphism
$f^{\prime}\colon S^{\prime}\to S^{\prime}$, where $S^{\prime}$ is a preimage
component in $M^{\prime}$ of the distingushed fiber $S$ in $M_{f}$, and where
$f^{\prime}$ is the return map of the lifted suspension flow on $M^{\prime}$.
The restriction of $\rho$ to $\pi^{\prime}$ becomes a representation
$\rho^{\prime}\colon\pi_{1}(M_{f^{\prime}})\to\mathrm{SL}(2,{\mathbb{R}})$. As
$\rho$ is irreducible in $\mathrm{SL}(2,{\mathbb{C}})$, $\rho^{\prime}$ is
also irreducible in $\mathrm{SL}(2,{\mathbb{C}})$. (This follows from the fact
that a subgroup in $\mathrm{SL}(2,{\mathbb{R}})$ has a fixed point in
${\mathbb{C}}\cup\\{\infty\\}$ if and only if it has a finite orbit in
${\mathbb{C}}\cup\\{\infty\\}$.) Therefore, we can deduce the general case
from the torsion-free case by arguing with $f^{\prime}\colon S^{\prime}\to
S^{\prime}$ and $\rho^{\prime}$.

Below we suppose that $\rho$ is torsion-free. Take a Markov partition
$\mathcal{R}$ of $S$ with respect to $f$. We obtain the transition graph
$T=T_{f,\mathcal{R}}$, whose dynamical cycles encode the periodic trajectories
of $M_{f}$ in the sense of Proposition 3.4. We adopt the notations and
terminology as in the proof of Lemma 4.1. Recall that $V=V(T)$ denotes the set
of vertices, and $E=E(T)$ the set of directed edges. To speak of fundamental
groups without ambiguity, we fix basepoints of $T$ and $M_{f}$ as follows.
Take a vertex $v\in V$ as the basepoint of $T$. Fix a map $T\to M_{f}$ among
its distinguished homotopy class (Proposition 3.4), and take the image $*$ of
$v$ in $M_{f}$ as the basepoint of $M_{f}$. Denote by
$\rho_{T}\colon\pi_{1}(T,v)\to\pi_{1}(M_{f},*)\to\mathrm{SL}(2,{\mathbb{R}})$
the pull-back representation of $\rho$.

We start by observing that there exists a pair of closed combinatorial paths
$\mu$ and $\nu$, both based at $v$, such that $\rho_{T}$ is
$\mathrm{SL}(2,{\mathbb{C}})$–irreducible restricted to the subgroup
$\langle\mu,\nu\rangle$ of $\pi_{1}(T,v)$. This is because $\rho_{T}$ is
$\mathrm{SL}(2,{\mathbb{C}})$–irreducible on $\pi_{1}(T,v)$. By inserting
terms of the form $\delta\bar{\delta}$, these combinatorial paths can be
modified into concatenation forms

$\begin{array}[]{cc}\mu=\xi_{0}\bar{\eta}_{1}\xi_{1}\cdots\bar{\eta}_{m}\xi_{m},&\nu=\psi_{0}\bar{\varphi}_{1}\psi_{1}\cdots\bar{\varphi}_{n}\psi_{n},\end{array}$

possibly without $\xi_{0}$, $\xi_{m}$, $\psi_{0}$, or $\psi_{n}$, such that
$\xi_{i},\psi_{j}$ are all dynamical paths, and $\eta_{i},\varphi_{j}$ are all
closed dyanmical paths. This is the same trick as explained in the proof of
Lemma 4.1, which relies on the irreducibility of the directed graph $T$
(Proposition 3.4). In particular, the modification does not change the
homotopy class of the closed paths relative to the base point $v$.

We show that there exists a pair of closed dynamical paths $\alpha$ and
$\beta$, both based at $v$, such that $\rho_{T}$ is
$\mathrm{SL}(2,{\mathbb{C}})$–irreducible restricted to the subgroup
$\langle\alpha,\beta\rangle$ of $\pi_{1}(T,v)$. This is done based on the
existence of closed combinatorial paths $\mu$ and $\nu$ as above, (already
modified into specified concatenation forms). We argue by induction on the
complexity $\max(m,n)$. In the induction step, we pass to other simpler
$(\mu,\nu)$ keeping the $\mathrm{SL}(2,{\mathbb{C}})$–irreducibility of
$\rho_{T}$ on $\langle\mu,\nu\rangle$.

When $\max(m,n)=0$, we can simply take $\alpha=\mu$ and $\beta=\nu$. Suppose
that $\alpha$ and $\beta$ can be constructed when $\max(m,n)=l$, for some
nonnegative integer $l$. When $\max(m,n)=l+1$, we may assume $m=l+1$, without
loss of generality. Write $\mu=\mu^{\prime}\bar{\eta}_{m}\xi_{m}$, where
$\mu^{\prime}=\xi_{0}\bar{\eta}_{1}\xi_{1}\cdots\bar{\eta}_{m-1}\xi_{m-1}$. We
observe that $\rho_{T}$ is $\mathrm{SL}(2,{\mathbb{C}})$–irreducible on at
least one the three subgroups $\langle\mu^{\prime}\eta_{m}\xi_{m},\nu\rangle$,
$\langle\mu^{\prime}\xi_{m},\nu\rangle$, or
$\langle\mu^{\prime}\eta_{m}\xi_{m},\mu^{\prime}\xi_{m}\rangle$. In fact,
$\rho_{T}(\nu)\in\mathrm{SL}(2,{\mathbb{R}})$ must be noncentral, since
$\rho_{T}$ is irreducible on $\langle\mu,\nu\rangle$. Hence, $\rho_{T}(\nu)$
has only one or two fixed points on ${\mathbb{C}}\cup\\{\infty\\}$. Suppose
$\rho_{T}$ is $\mathrm{SL}(2,{\mathbb{C}})$–reducible on both
$\langle\mu^{\prime}\eta_{m}\xi_{m},\nu\rangle$ and
$\langle\mu^{\prime}\xi_{m},\nu\rangle$, then both
$\rho_{T}(\mu^{\prime}\eta_{m}\xi_{m})$ and
$\rho_{T}(\mu^{\prime}\eta_{m}\xi_{m})$ fix some fixed point of
$\rho_{T}(\nu)$. If they fix a common fixed point
$p\in{\mathbb{C}}\cup\\{\infty\\}$ of $\rho_{T}(\nu)$, then $p$ is also fixed
by

$\rho_{T}(\mu)=\rho_{T}(\mu^{\prime}\bar{\eta}_{m}\xi_{m})=\rho_{T}\left((\mu^{\prime}\xi_{m})\overline{(\mu^{\prime}\eta_{m}\xi_{m})}(\mu^{\prime}\xi_{m})\right)=\rho_{T}(\mu^{\prime}\xi_{m})\cdot\rho_{T}(\mu^{\prime}\eta_{m}\xi_{m})^{-1}\cdot\rho_{T}(\mu^{\prime}\xi_{m}).$

However, this would contradict the assumed
$\mathrm{SL}(2,{\mathbb{C}})$–irreducibility of $\rho_{T}$ on
$\langle\mu,\nu\rangle$. Therefore, if $\rho_{T}$ is
$\mathrm{SL}(2,{\mathbb{C}})$–reducible on both
$\langle\mu^{\prime}\eta_{m}\xi_{m},\nu\rangle$ and
$\langle\mu^{\prime}\xi_{m},\nu\rangle$, we infer that
$\rho_{T}(\mu^{\prime}\eta_{m}\xi_{m})$ and
$\rho_{T}(\mu^{\prime}\eta_{m}\xi_{m})$ are hyperbolic or parabolic with fixed
point sets neither containing the other, so $\rho_{T}$ is
$\mathrm{SL}(2,{\mathbb{C}})$–irreducible on
$\langle\mu^{\prime}\eta_{m}\xi_{m},\mu^{\prime}\xi_{m}\rangle$. With the
above observation, we replace the pair $(\mu,\nu)$ with one of three pairs
$(\mu^{\prime}\eta_{m}\xi_{m},\nu)$, $(\mu^{\prime}\xi_{m},\nu)$, or
$(\mu^{\prime}\eta_{m}\xi_{m},\mu^{\prime}\xi_{m})$, denoted as
$(\tilde{\mu},\tilde{\nu})$, such that $\rho_{T}$ is
$\mathrm{SL}(2,{\mathbb{C}})$–irreducible on
$\langle\tilde{\mu},\tilde{\nu}\rangle$. Passing to
$(\tilde{\mu},\tilde{\nu})$ causes the numbers of reversed dynamical cycles
$(m,n)$ changing into $(m-1,n)$, $(m-1,n)$, or $(m-1,m-1)$, respectively. If
$n$ is smaller than $m=l+1$, we have decreased $\max(m,n)$ by $1$, so we can
construct the asserted $\alpha$ and $\beta$, by induction. If $n$ equals
$m=l+1$, we need to redo the above construction with respect to
$(\tilde{\mu},\tilde{\nu})$, working on $\tilde{\nu}$. This will decrease
$\max(m,n)$ by $1$, so we can still construct $\alpha$ and $\beta$.

To finish the proof, let $\alpha$ and $\beta$ be a pair of closed dynamical
paths in $T$ based at $v$, such that
$\rho_{T}\colon\pi_{1}(T,v)\to\mathrm{SL}(2,{\mathbb{R}})$ is
$\mathrm{SL}(2,{\mathbb{C}})$–irreducible on $\langle\alpha,\beta\rangle$. We
show that $\rho_{T}(\alpha^{r}\beta^{s})$ is hyperbolic for some nonnegative
integers $r$ and $s$. This will complete the proof since the dynamical cycle
$\alpha^{r}\beta^{s}$ in $T$ projects a periodic trajectory in $M_{f}$ as
claimed (Proposition 3.4).

When $\sigma_{\alpha}=\rho_{T}(\alpha)$ or $\sigma_{\beta}=\rho_{T}(\beta)$ is
hyperbolic, we can take $(r,s)$ as $(1,0)$ or $(0,1)$, respectively.

When $\sigma_{\alpha}$ is parabolic, it fixes some point
$p\in{\mathbb{R}}\cup\\{\infty\\}$. As $\rho_{T}$ is
$\mathrm{SL}(2,{\mathbb{C}})$–irreducible on $\langle\alpha,\beta\rangle$, the
point $p$ is distinct from $\sigma_{\beta}^{-1}(p)$. Take a small compact arc
$J\in{\mathbb{R}}\cup\\{\infty\\}$ containing $p$ in the interior, such that
$J$ and $\sigma_{\beta}^{-1}(J)$ are disjoint. Then for some sufficiently
large $r$, we can make $\sigma_{\alpha}^{r}\sigma_{\beta}(J)$ contained in
$J$, and $\sigma_{\beta}^{-1}\sigma_{\alpha}^{-r}(J)$ contained in
$\sigma_{\beta}^{-1}(J)$. In particular, $\sigma_{\alpha}^{r}\sigma_{\beta}$
has two distinct fixed points on ${\mathbb{R}}\cup\\{\infty\\}$, so it is
hyperbolic. We can take $(r,s)$ as $(r,1)$ for some large $r$. Similarly, when
$\sigma_{\beta}$ is parabolic, we can take $(r,s)$ as $(1,s)$ for some large
$s$.

When $\sigma_{\alpha}$ and $\sigma_{\beta}$ are both elliptic, they fix
distinct points $p$ and $q$ in ${\mathbb{H}}^{2}$, since $\rho_{T}$ is
$\mathrm{SL}(2,{\mathbb{C}})$–irreducible on $\langle\alpha,\beta\rangle$.
They act on ${\mathbb{H}}^{2}$ as rotations about those points, in view of
hyperbolic geometry. Note that composing the $\pi$–rotation about $p$ with the
$\pi$–rotation about $q$ yields a hyperbolic translation on ${\mathbb{H}}^{2}$
along the axis through $p$ and $q$. As we have assumed $\rho_{T}$ torsion-
free, the rotation angles of $\sigma_{\alpha}$ and $\sigma_{\beta}$ are
irrational multiples of $\pi$. There are positive integers $r$ and $s$, such
that the rotations angles of $\sigma_{\alpha}^{r}$ and $\sigma_{\beta}^{s}$
are very close to $\pi$. This way, $\sigma_{\alpha}^{r}\sigma_{\beta}^{s}$ is
hyperbolic. We can take any such $(r,s)$ as claimed. ∎

##  5\. Profinite correspondence of periodic trajectories

In this section, we explain a bijective correspondence between the periodic
trajectories in pseudo-Anosov mapping tori with profinitely isomorphic
fundamental groups. This correspondence has been established [Liu20b],
although it is not stated explicitly therein. Therefore, the purpose of this
section is largely expository, and we extract a precise form of the
correspondence (Lemma 5.2) from results in [Liu20b].

Let $M_{A}$ and $M_{B}$ be a pair of closed hyperbolic $3$–manifolds. If
$\Psi\colon\widehat{\pi}_{A}\to\widehat{\pi}_{B}$ is an isomorphism between
the profinite completions of their fundamental groups $\pi_{A}=\pi_{1}(M_{A})$
and $\pi_{B}=\pi_{1}(M_{B})$, then for any surface fiber $S_{A}$ of $M_{A}$,
there is a unique surface fiber $S_{B}$ of $M_{B}$, such that the closure of
the normal subgroup $\Sigma_{A}=\pi_{1}(S_{A})$ in $\widehat{\pi}_{A}$
projects isomorphically onto the closure of $\Sigma_{B}=\pi_{1}(S_{B})$ in
$\widehat{\pi}_{B}$, under $\Psi$. Note that the closures of $\Sigma_{A}$ and
$\Sigma_{B}$ can be identified with their own profinite completions
$\widehat{\Sigma}_{A}$ and $\widehat{\Sigma}_{B}$, respectively.

Fix transverse orientations of the surface fibers in the $3$–manifolds. Then
$M_{A}$ can be identified homeomorphically with the mapping torus of a pseudo-
Anosov automorphism $f_{A}\colon S_{A}\to S_{A}$, which is isotopically
unique, and which has a distinguished cohomology class $\phi_{A}\in
H^{1}(M_{A};{\mathbb{Z}})$ dual to $S_{A}$. Similarly, we obtain $f_{B}\colon
S_{B}\to S_{B}$ and $\phi_{B}\in H^{1}(M_{B};{\mathbb{Z}})$ associated with
$M_{B}$. There is a unique invertible profinite integer
$\mu\in\widehat{{\mathbb{Z}}}^{\times}$, (according to the fiber orientation),
such that $\mu$ fits into the following commutative diagram of profinite
groups:

(5.1)
$\textstyle{\\{1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\Sigma}_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\Psi|}$$\textstyle{\widehat{\pi}_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\widehat{\phi}_{A}}}$$\scriptstyle{\cong}$$\scriptstyle{\Psi}$$\textstyle{t^{\widehat{{\mathbb{Z}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pwr}_{\mu}}$$\scriptstyle{\cong}$$\textstyle{\\{1\\}}$$\textstyle{\\{1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\Sigma}_{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\pi}_{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\widehat{\phi}_{B}}}$$\textstyle{t^{\widehat{{\mathbb{Z}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{1\\}}$

where $t^{\widehat{{\mathbb{Z}}}}$ denotes the profinite completion of the
infinite cyclic group $\langle t\rangle=t^{{\mathbb{Z}}}$, and where we define
$\mathrm{pwr}_{\mu}\colon t^{\nu}\mapsto t^{\mu\nu}$ and
$t^{\widehat{\phi}}\colon g\mapsto t^{\widehat{\phi}(g)}$.

######  Convention 5.1.

By declaring $(M_{A},M_{B},\Psi)$ as a _profinite isomorphism setting of
pseudo-Anosov mapping tori_ , we assume that $M_{A}$ and $M_{B}$ are a pair of
closed fibered hyperbolic $3$–manifolds, with fixed universal covering spaces
and fixed transversely oriented connected surface fibers, and we assume that
$\Psi\colon\widehat{\pi}_{A}\to\widehat{\pi}_{B}$ is an isomorphism between
the profinite completions of the universal deck transformation groups, fitting
into the diagram (5.1). We adopt the notations as involved in the diagram
(5.1), such as $\widehat{\phi}_{A}$, $\widehat{\phi}_{B}$, and $\mu$.

If $\pi$ is a finitely generated residually finite group with profinite
completion $\widehat{\pi}$, any conjugacy class $\mathbf{c}$ in $\pi$
determines a unique conjugacy class $\hat{\mathbf{c}}$ in $\widehat{\pi}$,
which is the closure of $\mathbf{c}$. (Different $\mathbf{c}$ determines
different $\hat{\mathbf{c}}$ if and only if $\pi$ is conjugacy separable.) For
any $n\in{\mathbb{Z}}$, there is a well-defined conjugacy class
$\mathbf{c}^{n}$ in $\pi$, represented by the $n$–th power of any
representative of $\mathbf{c}$. Similarly, for any
$\nu\in\widehat{{\mathbb{Z}}}$, there is a well-defined conjugacy class
$\hat{\mathbf{c}}^{\nu}$ in $\widehat{\pi}$. This can be defined as the
inverse limit of $\mathbf{c}_{\Gamma}^{n}$, where $\mathbf{c}_{\Gamma}$
denotes the residual conjugacy class of $\mathbf{c}$ in any finite quotient
group $\Gamma$ of $\pi$, and where $n\in{\mathbb{Z}}$ is congruent to $\nu$
modulo $|\Gamma|$.

######  Lemma 5.2.

Under any profinite isomorphism setting of pseudo-Anosov mapping tori
$(M_{A},M_{B},\Psi)$ (Convention 5.1), $\Psi$ witnesses a bijective
correspondence between the periodic trajectories in $M_{A}$ and in $M_{B}$ of
their suspension flows, as follows.

For any periodic trajactory $\mathbf{c}_{A}$ of $M_{A}$, there exists a unique
periodic trajectory $\mathbf{c}_{B}$ of $M_{B}$, such that
$\hat{\mathbf{c}}_{A}^{\mu}$ projects $\hat{\mathbf{c}}_{B}$ under $\Psi$.
Here, periodic trajectories are treated as conjugacy classes in the
fundamental group. Hence, $\hat{\mathbf{c}}_{B}^{1/\mu}$ projects
$\hat{\mathbf{c}}_{A}$ under the inverse of $\Psi$.

###### Proof.

In any profinite group $\Pi$, we say that two conjugacy classes $\mathbf{C}$
and $\mathbf{C}^{\prime}$ are _$\widehat{{\mathbb{Z}}}^{\times}$ –equivalent_
if $\mathbf{C}^{\mu^{\prime}}=\mathbf{C}^{\prime}$ holds for some
$\mu^{\prime}\in\widehat{{\mathbb{Z}}}^{\times}$. Denote by
$\mathrm{Orb}(\Pi)$ the set of conjugacy classes in $\Pi$, and denote by
$\Omega(\Pi)$ the set of $\widehat{{\mathbb{Z}}}^{\times}$–equivalence classes
in $\mathrm{Orb}(\Pi)$. Note that if $\Gamma$ is a finite quotient group of
$\Pi$, then every $\widehat{{\mathbb{Z}}}^{\times}$–equivalence class in
$\mathrm{Orb}(\Pi)$ projects a unique
$\widehat{{\mathbb{Z}}}^{\times}$–equivalence class in $\mathrm{Orb}(\Gamma)$,
so the quotient homomorphism $\Pi\to\Gamma$ naturally induces a quotient map
of sets $\Omega(\Pi)\to\Omega(\Gamma)$.

One may state the condition in Lemma 5.2 alternatively as
$\Psi(\hat{\mathbf{c}}_{A})$ to be
$\widehat{{\mathbb{Z}}}^{\times}$–equivalent to $\hat{\mathbf{c}}_{B}$. In
fact, if $\Psi(\hat{\mathbf{c}}_{A})^{\mu^{\prime}}=\hat{\mathbf{c}}_{B}$
holds for some $\mu^{\prime}\in\widehat{{\mathbb{Z}}}^{\times}$, the following
equation holds in $\widehat{{\mathbb{Z}}}$ by the commutative diagram (5.1):

$\mu\cdot\phi_{A}(\mathbf{c}_{A})=\mu\cdot\widehat{\phi}_{A}\left(\hat{\mathbf{c}}_{A}\right)=\widehat{\phi}_{B}\left(\Psi(\hat{\mathbf{c}}_{A})\right)=\widehat{\phi}_{B}\left(\mu^{\prime}\cdot\hat{\mathbf{c}}_{B}\right)=\mu^{\prime}\cdot\widehat{\phi}_{B}\left(\hat{\mathbf{c}}_{B}\right)=\mu^{\prime}\cdot\phi_{B}(\mathbf{c}_{B}).$

Note that $\phi_{A}(\mathbf{c}_{A})$ and $\phi_{B}(\mathbf{c}_{B})$ are
positive integers. Then $\phi_{A}(\mathbf{c}_{A})=\phi_{B}(\mathbf{c}_{B})$
holds in ${\mathbb{Z}}$, and $\mu^{\prime}=\mu$ holds in
$\widehat{{\mathbb{Z}}}^{\times}$.

For any finite group $\Gamma$ and any
$\widehat{{\mathbb{Z}}}^{\times}$–equivalence class $\omega\in\Omega(\Gamma)$,
we denote by $\chi_{\omega}$ the characteristic function defined on
$\mathrm{Orb}(\Gamma)$, such that $\chi_{\omega}(\mathbf{c})=1$ for all
$\mathbf{c}\in\omega$, and $\chi_{\omega}(\mathbf{c})=0$ for all
$\mathbf{c}\not\in\omega$. For any pseudo-Anosov automorphism $f\colon S\to S$
and any homomorphism $\gamma\colon\pi_{1}(M_{f})\to\Gamma$, we recall the
following defining expression of the analogous twisted Lefschetz number, as in
[Liu20b, Section 8, (8.7)]:

(5.2)
$L_{m}(f;\gamma^{*}\chi_{\omega})=\sum_{\mathbf{O}\in\mathrm{Orb}_{m}(f)}\chi_{\omega}(\gamma(\ell_{m}(f,\mathbf{O})))\cdot\mathrm{ind}_{m}(f;\mathbf{O}),$

(see (3.7)). The factor $\mathrm{ind}_{m}(f;\mathbf{O})\in{\mathbb{Z}}$
denotes the $m$–periodic index of $f$ at $\mathbf{O}$, (see [Liu20b, Section
2, (2.4)]); for current purpose, it suffices to mention that
$\mathrm{ind}_{m}(f;\mathbf{O})$ is nonzero for every
$\mathbf{O}\in\mathrm{Orb}_{m}(f)$. Note that the profinite isomorphism $\Psi$
witnesses a bijective correspondence between the homomorphisms
$\gamma_{A}\colon\pi_{A}\to\Gamma$ and $\gamma_{B}\colon\pi_{B}\to\Gamma$,
such that $\widehat{\gamma}_{A}\colon\widehat{\pi}_{A}\to\Gamma$ is the
$\Psi$–pullback of $\widehat{\gamma}_{B}\colon\pi_{B}\to\Gamma$. For any
$\omega\in\Omega(\Gamma)$, and $m\in{\mathbb{N}}$, and any
$\Psi$–corresponding homomorphism pair $\gamma_{A}$ and $\gamma_{B}$ as above,
the following version of profinite invariance of twisted Lefschetz numbers
follows from [Liu20b, Theorems 2.7, 7.2, and Lemma 8.4]:

(5.3)
$L_{m}\left(f_{A};\gamma_{A}^{*}\chi_{\omega}\right)=L_{m}\left(f_{B};\gamma_{B}^{*}\chi_{\omega}\right).$

For any $m\in{\mathbb{N}}$, there exists some finite group $\Gamma_{m}$ and
some quotient homomorphism $\gamma_{m,A}\colon\pi_{A}\to\Gamma_{m}$, such that
the $m$–periodic trajectories of $M_{A}$ project into mutually distinct
$\widehat{{\mathbb{Z}}}^{\times}$–equivalence classes in
$\mathrm{Orb}(\Gamma_{m})$ under $\gamma_{m,A}$ [Liu20b, Lemma 8.6]. Then
(5.2) and (5.3) imply that those $\widehat{{\mathbb{Z}}}^{\times}$–equivalence
classes also receive $m$–periodic trajectories of $M_{B}$ under the
$\Psi$–corresponding $\gamma_{m,B}$. Moreover, $M_{B}$ has the same number of
$m$–periodic trajectories as $M_{A}$ does [Liu20b, Theorem 8.1]. Then the
$m$–periodic trajectories of $M_{B}$ must project exactly the same set of
$\widehat{{\mathbb{Z}}}^{\times}$–equivalence classes in
$\mathrm{Orb}(\Gamma_{m})$ as those of $M_{A}$ do, (applying (5.3) for every
$\omega\in\Omega(\Gamma_{m})$). This property persists as we pass to any finer
$\Psi$–corresponding quotients, (that is, $\pi_{A}\to\Gamma$ and
$\pi_{B}\to\Gamma$ which descend to $\gamma_{m,A}$ and $\gamma_{m,B}$ under a
further quotient $\Gamma\to\Gamma_{m}$). Then the property also persists as we
pass to the profinite completions $\pi_{A}\to\widehat{\pi}_{A}$ and
$\pi_{B}\to\widehat{\pi}_{B}$.

To summarize, we have established bijective correspondences between the
$m$–periodic trajectories $\mathbf{c}_{A}=\ell_{m}(f_{A};\mathbf{O}_{A})$ of
$M_{A}$ and $\mathbf{c}_{B}=\ell_{m}(f_{B};\mathbf{O}_{B})$ of $M_{B}$, for
any $m\in{\mathbb{N}}$. They satisfy the condition that
$\Psi(\hat{\mathbf{c}}_{A})$ is $\widehat{{\mathbb{Z}}}^{\times}$–equivalent
to $\hat{\mathbf{c}}_{B}$, as asserted. ∎

##  6\. On the profinite power operation

In this section, we prove some algebraic lemmas (Lemmas 6.1 and 6.3),
regarding the profinite power operation $t\mapsto t^{\nu}$ in the profinite
cyclic group $t^{\widehat{{\mathbb{Z}}}}$ and $A\mapsto A^{\nu}$ in the matrix
group $\mathrm{SL}(2,R)$ over some profinite ring $R$, where $\nu$ is a
profinite integer, namely, $\nu\in\widehat{{\mathbb{Z}}}$. These lemmas are
prepared for Sections 7 and 9.

######  Lemma 6.1.

Let $f(t)\in{\mathbb{Z}}[t]$ be a monic reciprocal polynomial in an
indeterminate $t$. Denote by
$\llbracket\widehat{{\mathbb{Z}}}t^{\widehat{{\mathbb{Z}}}}\rrbracket$ the
profinite ring completion of the group ring
$[{\mathbb{Z}}t^{\mathbb{Z}}]={\mathbb{Z}}[t,t^{-1}]$. Suppose that the
principal ideals $(f(t))$ and $(f(t^{\mu}))$ are equal in
$\llbracket\widehat{{\mathbb{Z}}}t^{\widehat{{\mathbb{Z}}}}\rrbracket$ for
some $\mu\in\widehat{{\mathbb{Z}}}^{\times}$,

Then, either $f(t)$ is a product of cyclotomic factors, or $\mu^{2}=1$ holds
in $\widehat{{\mathbb{Z}}}^{\times}$.

###### Proof.

For any rational prime $p$, let $L$ be an algebraic field extension over
${\mathbb{Q}}_{p}$ of finite degree. Denote by $|\cdot|_{p}$ the unique
$p$–adic norm on $L$ extending that of ${\mathbb{Q}}_{p}$. Denote by
$R=\\{x\in L\colon|x|_{p}\leq 1\\}$ the valuation ring of $L$, and
$R^{\times}=\\{x\in L\colon|x|_{p}=1\\}$ the multiplicative group of units in
$R$.

For any $\alpha\in R^{\times}$, we obtain a unique continuous ring
homomorphism
$\mathrm{ev}_{\alpha}\colon\llbracket\widehat{{\mathbb{Z}}}t^{\widehat{{\mathbb{Z}}}}\rrbracket\to
R$, such that $t\mapsto\alpha$. The principal ideal $(f(t))$ is contained in
the kernel of the $\mathrm{ev}_{\alpha}$ if and only if $f(\alpha)=0$ holds in
$L$. Since $f$ is monic and reciprocal, any zero of $f$ in $L$ is an algebraic
integer, hence lying in $R^{\times}$. The ideal equation $(f(t))=(f(t^{\mu}))$
in $\llbracket\widehat{{\mathbb{Z}}}t^{\widehat{{\mathbb{Z}}}}\rrbracket$
implies that $\alpha^{1/\mu}$ is a zero of $f$ in $L$ if and only if $\alpha$
is a zero of $f$ in $L$. Since there are only finitely many zeros, there
exists some $d\in{\mathbb{N}}$, such that the group automorphism
$R^{\times}\to R^{\times}\colon\alpha\mapsto\alpha^{\mu^{d}}$ fixes every zero
of $f$ in $L$.

Below we assume that $f$ is not a product of cyclotomic factors. Fix a root
$\xi$ of $f(t)\in{\mathbb{Z}}[t]$ that is not a root of unity. We work with
the field $L={\mathbb{Q}}_{p}(\xi)$ for an arbitrary $p$, fixing an embedding
of $L$ into ${\mathbb{C}}_{p}$ (the $p$–adic completion of an algebraic
closure of ${\mathbb{Q}}_{p}$). There exists some $m\in{\mathbb{N}}$, such
that $|\alpha^{m}-1|_{p}<p^{-1}$ holds for all $\alpha\in R^{\times}$. (For
example, $m$ can be taken as $p^{k}\cdot(p^{l}-1)$ where
$l=[L:{\mathbb{Q}}_{p}]$ and $k=\lfloor\log l/\log p\rfloor+1$; see Remark
6.2.) Since the multiplicative subgroup $1+pR$ of $R^{\times}$ is pro–$p$, the
power action of $\widehat{{\mathbb{Z}}}$ on $1+pR$ factors through the natural
projection onto ${\mathbb{Z}}_{p}$. Therefore, $\xi^{ma}=\xi^{ma_{p}}$ holds
for any $a\in\widehat{{\mathbb{Z}}}$, where $a_{p}\in{\mathbb{Z}}_{p}$ denotes
the $p$–summand of $a$. Therefore, we obtain an equation in
${\mathbb{C}}_{p}$:

$\xi^{m}=\xi^{m\mu_{p}^{d}}.$

Since $\xi$ is not a root of unity, we infer $\mu_{p}^{d}=1$ in
${\mathbb{Z}}_{p}$. (Otherwise, the equation $(1+x)^{\delta}=1$ with
$\delta=\mu_{d}^{p}-1$ would have arbitrarily small solutions of the form
$x=\xi^{mp^{n}}-1$, where $n\in{\mathbb{N}}$, contradicting the invertibility
of $(1+x)^{\delta}$ near $x=0$ in ${\mathbb{C}}_{p}$.) This means $\mu_{p}$ is
a root of unity. (Compare [Uek21, Proposition 1.3].)

We have observed that $\eta=\xi^{\mu}$ is again some zero of $f$ in $L$.
Raising to the $m$–th power, we obtain

$\xi^{m\mu_{p}}=\eta^{m}.$

For any algebraic integers
$\alpha,\beta\in{\mathbb{C}}_{p}\setminus\\{0,1\\}$, such that
$|\alpha-1|_{p}<p^{-1}$ and $|\beta-1|_{p}<p^{-1}$, the $p$–adic logarithmic
ratio $\log_{p}\alpha/\log_{p}\beta$ is either rational or transcendental, (or
equivalently, $\alpha^{\theta}=\beta$ implies $\theta$ rational or
transcendental in ${\mathbb{C}}_{p}$). This is known as Mahler’s $p$–adic
analogue of the Gelfond–Schneider theorem, [Mah35] (see also [Vel40]). Since
$\xi$ is not a root of unity, we infer that
$\mu_{p}\in{\mathbb{Z}}_{p}^{\times}$ must be a rational integer, and hence
$\pm 1$. (Besides, $\eta$ should be $\xi$ or $\xi^{-1}$ times some root of
unity in $L$.) In particular, $\mu_{p}^{2}=1$ holds in
${\mathbb{Z}}_{p}^{\times}$. Since $p$ is arbitrary, we obtain $\mu^{2}=1$ in
$\widehat{{\mathbb{Z}}}^{\times}=\prod_{p}{\mathbb{Z}}_{p}^{\times}$, as
asserted. ∎

######  Remark 6.2.

Let $p$ be a rational prime and $L$ be a field extension $L$ over
${\mathbb{Q}}_{p}$ of finite algebraic degree. Denote by $R$ the $p$–adic
valuation ring of $L$ and $\mathfrak{M}$ the maximal ideal of $R$. For any
$\lambda\in R^{\times}$, there is a unique factorization
$\lambda=(1+\beta)\cdot\zeta$, where $|\beta|_{p}<1$, and where $\zeta$ is a
root of unity of order coprime to $p$. This agrees with the canonical
decomposition of the multiplicative group
$R^{\times}\cong(1+\mathfrak{M})\times(R/\mathfrak{M})^{\times}$. Note that
the additive coset $1+\mathfrak{M}$ forms a multiplicative pro–$p$ subgroup.
Then for any $a\in\widehat{{\mathbb{Z}}}$, we obtain a formula

$\lambda^{a}=(1+\beta)^{a_{p}}\cdot\zeta^{a\bmod(q-1)}=\left[\sum_{m=0}^{\infty}\frac{a_{p}\cdot(a_{p}-1)\cdot\cdots\cdot(a_{p}-m+1)}{m!}\cdot\beta^{m}\right]\cdot\zeta^{a\bmod(q-1)},$

where $q$ denotes the cardinality of the residue field $R/\mathfrak{M}$. Since
$a_{p}\in{\mathbb{Z}}_{p}$, the above $p$–adic binomial series converges to a
continuous function of $\beta\in\mathfrak{M}$, (see [Kob84, Chapter IV,
Section 1]). The function agrees with the ${\mathbb{Z}}_{p}$–power operation
$1+\beta\mapsto(1+\beta)^{a_{p}}$ on $1+\mathfrak{M}$, because for every
$a_{p}\in{\mathbb{N}}$, the agreement is obvious by the usual binomial
expansion.

######  Lemma 6.3.

Let $p$ be a rational prime and $L$ be a field extension $L$ over
${\mathbb{Q}}_{p}$ of finite algebraic degree. Denote by $R$ the $p$–adic
valuation ring of $L$ and $\mathfrak{M}$ the maximal ideal of $R$.

If $A\in\mathrm{SL}(2,R)$ and $\mu\in\widehat{{\mathbb{Z}}}^{\times}$ satisfy
$A\equiv I\bmod\mathfrak{M}$, and $\mu^{2}=1$, then
$\mathrm{tr}(A^{\mu})=\mathrm{tr}(A)$.

###### Proof.

The principal $\mathfrak{M}$–congruence subgroup
$I+\mathrm{Mat}(2,\mathfrak{M})$ of $\mathrm{GL}(2,R)$ is pro–$p$. In fact,
for all $n\in{\mathbb{N}}$, the quotient groups of
$I+\mathrm{Mat}(2,\mathfrak{M}^{n})$ by $I+\mathrm{Mat}(2,\mathfrak{M}^{n+1})$
are all cosets of finite-dimensional vector spaces over the residue field of
characteristic $p$, affinely isomorphic to
$\mathrm{Mat}(2,\mathfrak{M}^{n})\otimes_{R}(R/\mathfrak{M})$. It follows that
the intersection of $\mathrm{SL}(2,R)$ with $I+\mathrm{Mat}(2,\mathfrak{M})$
is also pro–$p$.

If $A\equiv I\bmod\mathfrak{M}$, we obtain $A^{\mu}=A^{\mu_{p}}$ in
$\mathrm{SL}(2,R)$, where $\mu_{p}$ denotes the ${\mathbb{Z}}_{p}$–summand of
$\mu$ in $\widehat{{\mathbb{Z}}}$. If $\mu^{2}=1$, we obtain $\mu_{p}=\pm 1$
in ${\mathbb{Z}}_{p}$. Then

$\mathrm{tr}(A^{\mu})=\mathrm{tr}(A^{\mu_{p}})=\mathrm{tr}(A^{\pm
1})=\mathrm{tr}(A),$

the last step using $\mathrm{det}(A)=1$. ∎

##  7\. Mu square equals one

In this section, we improve a main result in [Liu20b], obtaining more
information about the $\widehat{{\mathbb{Z}}}^{\times}$–regularity factor
$\mu$.

######  Lemma 7.1.

Let $(M_{A},M_{B},\Psi)$ be a profinite isomorphism setting of pseudo-Anosov
mapping tori, (Convention 5.1). Then, $\mu^{2}=1$ holds in
$\widehat{{\mathbb{Z}}}^{\times}$.

###### Proof.

Let $M_{f}$ be the mapping torus of any pseudo-Anosov automorphism $f\colon
S\to S$ of a closed orientable surface. For any quotient homomorphism
$\pi_{1}(M_{f})\to\Gamma$ onto a finite group $\Gamma$, the kernel of the
homomorphism corresponds to a regular finite cover $M^{\prime}\to M_{f}$. The
suspension flow lifts to $M^{\prime}$, transverse to any preimage component
$S^{\prime}$ of $S$, so the return map of the lifted flow gives rise to a
pseudo-Anosov automorphism $f^{\prime}\colon S^{\prime}\to S^{\prime}$. This
way, $M^{\prime}$ can be identified as the mapping torus $M_{f^{\prime}}$. We
denote by $\phi_{f}\in H^{1}(M_{f};{\mathbb{Z}})$ and $\phi_{f^{\prime}}\in
H^{1}(M_{f^{\prime}};{\mathbb{Z}})$ the distinguished cohomology classes dual
to the distinguished fibers $S$ and $S^{\prime}$, respectively. Then
$[M^{\prime}:M]/[S^{\prime}:S]$ times $\phi_{f^{\prime}}$ agrees with the
pullback of $\phi_{f}$ with respect to the covering projection
$M_{f^{\prime}}\to M_{f}$.

In [Liu20a], it is shown that for any pseudo-Anosov mapping torus $M_{f}$ as
above, there exists a covering mapping torus $M^{\prime}=M_{f^{\prime}}$, such
that the Alexander polynomial
$\Delta_{M^{\prime}}^{\phi^{\prime}}\in{\mathbb{Z}}[t,t^{-1}]$ (up to a factor
in the group of units $\\{\pm 1\\}\times t^{{\mathbb{Z}}}$) with respect to
$\phi^{\prime}=\phi_{f^{\prime}}$ has a root in ${\mathbb{C}}$ outside the
unit circle. In the case of mapping tori, the Alexander polynomial
$\Delta_{M^{\prime}}^{\phi^{\prime}}(t)$ can be identified with the
characteristic polynomial of the cohomological action $f^{\prime}_{*}\colon
H_{1}(S^{\prime};{\mathbb{Z}})\to H_{1}(S^{\prime};{\mathbb{Z}})$, namely,

$\Delta_{M^{\prime}}^{\phi^{\prime}}(t)\doteq\mathrm{det}_{{\mathbb{Z}}[t]}\left(t\cdot\mathrm{id}-f^{\prime}_{*}\right),$

where $\doteq$ means an equality in ${\mathbb{Z}}[t,t^{-1}]$ up to a factor in
$\\{\pm 1\\}\times t^{{\mathbb{Z}}}$. In particular, we may represent
$\Delta_{M^{\prime}}^{\phi^{\prime}}(t)$ with a unique monic reciprocal
polynomial in ${\mathbb{Z}}[t]$ of degree $\chi(S^{\prime})+2$. The
Reidemeister torsion $\tau_{M^{\prime}}^{\phi^{\prime}}(t)$ lives in the field
of fractions ${\mathbb{Q}}(t)=\mathrm{Frac}({\mathbb{Z}}[t,t^{-1}])$ up to a
factor in $\\{\pm 1\\}\times t^{{\mathbb{Z}}}$. In the mapping torus case,
$\tau_{M^{\prime}}^{\phi^{\prime}}(t)$ can be identified with the alternating
product of the characteristic polynomials of $f^{\prime}_{*}\colon
H_{*}(S^{\prime};{\mathbb{Z}})\to H_{*}(S^{\prime};{\mathbb{Z}})$, namely,

$\tau_{M^{\prime}}^{\phi^{\prime}}(t)\doteq\Delta_{M^{\prime}}^{\phi^{\prime}}(t)\cdot(t-1)^{-2},$

(see [Liu20b, Section 2.3 and Lemma 7.7], for example).

Under a profinite isomorphism setting $(M_{A},M_{B},\Psi)$ of pseudo-Anosov
mapping tori (Convention 5.1), we obtain a similar profinite isomorphism
setting of the covering mapping tori
$(M^{\prime}_{A},M^{\prime}_{B},\Psi^{\prime})$, associated to any
$\Psi$–corresponding pair of finite quotients $\pi_{A}\to\Gamma$ and
$\pi_{B}\to\Gamma$. (Being $\Psi$–corresponding means that the completion
homomorphism $\widehat{\pi}_{A}\to\Gamma$ is the $\Psi$–pullback of
$\widehat{\pi}_{B}\to\Gamma$; so $M^{\prime}_{A}$ and $M^{\prime}_{B}$ are the
finite covers corresponding to the quotient kernels $\pi^{\prime}_{A}$ and
$\pi^{\prime}_{B}$, and
$\Psi^{\prime}\colon\widehat{\pi}^{\prime}_{A}\to\widehat{\pi}^{\prime}_{B}$
is the restriction of $\Psi$ to their profinite completions.) The covering
setting can be summarized with a similar diagram as follows:

(7.1)
$\textstyle{\\{1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\Sigma}^{\prime}_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\Psi^{\prime}|}$$\textstyle{\widehat{\pi}^{\prime}_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\widehat{\phi}^{\prime}_{A}}}$$\scriptstyle{\cong}$$\scriptstyle{\Psi^{\prime}}$$\textstyle{t^{\widehat{{\mathbb{Z}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pwr}_{\mu^{\prime}}}$$\scriptstyle{\cong}$$\textstyle{\\{1\\}}$$\textstyle{\\{1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\Sigma}^{\prime}_{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\pi}^{\prime}_{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\widehat{\phi}^{\prime}_{B}}}$$\textstyle{t^{\widehat{{\mathbb{Z}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{1\\}}$

We can identify $\mu^{\prime}\in\widehat{{\mathbb{Z}}}^{\times}$ in the
diagram (7.1), with $\mu\in\widehat{{\mathbb{Z}}}^{\times}$ in the diagram
(5.1), because they are related via the following parallel commutative
diagrams:

$\textstyle{\widehat{\pi}^{\prime}_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\widehat{\phi}^{\prime}_{A}}}$$\scriptstyle{\mathrm{incl}}$$\textstyle{t^{\widehat{{\mathbb{Z}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pwr}_{k}}$$\textstyle{\widehat{\pi}^{\prime}_{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\widehat{\phi}^{\prime}_{B}}}$$\scriptstyle{\mathrm{incl}}$$\textstyle{t^{\widehat{{\mathbb{Z}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pwr}_{k}}$$\textstyle{\widehat{\pi}_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\widehat{\phi}_{A}}}$$\textstyle{t^{\widehat{{\mathbb{Z}}}}}$$\textstyle{\widehat{\pi}_{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\widehat{\phi}_{B}}}$$\textstyle{t^{\widehat{{\mathbb{Z}}}}}$

where
$k=[M_{A}^{\prime}:M_{A}]/[S_{A}^{\prime}:S_{A}]=[M_{B}^{\prime}:M_{B}]/[S_{B}^{\prime}:S_{B}]$.

By [Liu20b, Theorem 7.2], applied with the trivial representation
$\Gamma\to\mathrm{GL}(1,{\mathbb{Z}})$, the Reidemeister torsions of
profinitely corresponding pairs of pseudo-Anosov mapping tori are equal. We
obtain

$\tau^{\phi^{\prime}_{A}}_{M^{\prime}_{A}}\doteq\tau^{\phi^{\prime}_{B}}_{M^{\prime}_{B}},$

and hence

$\Delta^{\phi^{\prime}_{A}}_{M^{\prime}_{A}}\doteq\Delta^{\phi^{\prime}_{B}}_{M^{\prime}_{B}}.$

(Note that our notation $\phi^{\prime}_{A}\in
H^{1}(M^{\prime}_{A};{\mathbb{Z}})$ corresponds to the notation
$\psi^{\prime}_{A}\in H^{1}(M^{\prime}_{A};{\mathbb{Z}})$ in [Liu20b].) We
represent both sides with a common monic reciprocal polynomial
$P(t)\in{\mathbb{Z}}[t]$. Moreover, there is an equality of the principal
ideals in
$\llbracket\widehat{{\mathbb{Z}}}t^{\widehat{{\mathbb{Z}}}}\rrbracket$:

$(P(t^{\mu}))=(P(t)),$

(see [Liu20b, Lemmas 7.7 and 7.8]).

We apply the above results to some covering setting
$(M^{\prime}_{A},M^{\prime}_{B},\Psi^{\prime})$, where $P(t)$ has a root in
${\mathbb{C}}$ outside the unit circle. Then, by Lemma 6.1, we see that
$\mu\in\widehat{{\mathbb{Z}}}^{\times}$ is multiplicative $2$–torsion, as
asserted. ∎

##  8\. Representations in quaternions

In this section, we introduce some working terminology regarding group
representations in an abstract quaternion algebra. Most of the materials below
are well-known in the case of $\mathrm{SL}(2,{\mathbb{C}})$–representations.
The main purpose of this section is to justify the formulation of results in
Sections 9 and 10.

######  Definition 8.1.

Let $\mathscr{A}$ be a quaternion algebra over a field $F$ of characteristic
$0$. For any group $\pi$, we call any group homomorphism
$\rho\colon\pi\to\mathrm{SL}(1,\mathscr{A})$ a (_special linear_ or
_unimodular_) _representation_ of $\pi$ in $\mathscr{A}$. Here,
$\mathrm{SL}(1,\mathscr{A})$ denotes the normal subgroup of
$\mathrm{GL}(1,\mathscr{A})=\mathscr{A}^{\times}$ which consists of the
quaternions of norm $1$. The _character_ of $\rho$ refers to the
$\pi$–conjugacy invariant function $\chi_{\rho}\colon\pi\to F$, defined as
$\chi_{\rho}(g)=\mathrm{Tr}_{\mathscr{A}/F}(\rho(g))$ for all $g\in\pi$.

######  Lemma 8.2.

With $\mathscr{A}/F$ as in Definition 8.1, let $\pi$ be a group and
$\rho\colon\pi\to\mathrm{SL}(1,\mathscr{A})$ be a representation. Then the
following conditions are all equivalent.

  1. (1)

For some (hence any) field extension $E$ of $F$ over which
$\mathscr{A}\otimes_{F}E$ splits as a matrix algebra $\mathrm{Mat}(2,E)$, the
induced representation $\rho_{E}\colon\pi\to\mathrm{SL}(2,E)$ is irreducible
over $E$, that is, $\rho_{E}$ has no proper $E$–linear invariant subspace in
${E^{2}}$ other than $\\{0\\}$.

  2. (2)

The image of $\rho$ generates $\mathscr{A}$ as an algebra over $F$.

  3. (3)

The inequality $\chi_{\rho}([g,h])\neq 2$ holds for some $g,h\in\pi$, where
$[g,h]=ghg^{-1}h^{-1}$ denotes the commutator.

###### Proof.

The equivalence (1) $\Leftrightarrow$ (3) is well-known. We refer to [CulS83,
Lemma 1.2.1 and Corollary 1.2.2] for a standard proof. The implication (2)
$\Rightarrow$ (1) is obvious (see Example 3.1). It remains to prove the
implication (1) $\Rightarrow$ (2), or essentially, the following (well-known)
claim in linear algebra: If $E$ is an algebraically closed field of
characteristic $0$, then any proper $E$–subalgebra $B$ of $\mathrm{Mat}(2,E)$
preserves some $1$–dimensional $E$–linear subspace in ${E^{2}}$.

When $\mathrm{dim}_{E}B<3$, the claim is obvious, since $B$ is central, and
hence commutative. When $\mathrm{dim}_{E}B=3$, we take a basis
$I,X,Y\in\mathrm{Mat}(2,E)$ of $B$ over $E$, where $I$ is the identity matrix.
Possibly after a generic modification of $X$ within $B$, we may require
$\mathrm{det}(X)\neq 0$ and $\mathrm{tr}(X)^{2}-4\cdot\mathrm{det}(X)\neq 0$.
So, possibly after conjugating $B$ in $\mathrm{Mat}(2,E)$, we may assume that
$X$ is diagonal with distinct diagonal entries. It follows that $Y$ has some
nonzero off-diagonal entries. If the upper and lower off-diagonal entries of
$Y$ were both nonzero, we observe that $XY$ and $Y$ would be linearly
independent. Moreover, the matrices $I,X,Y,XY$ would span $\mathrm{Mat}(2,E)$
over $E$, contradicting the assumption $\mathrm{dim}_{E}B=3$. This means $Y$
is either upper triangular or lower triangular. Therefore, $B$ is conjugate to
a subalgebra of triangular matrices in $\mathrm{Mat}(2,E)$. It preserves a
$1$–dimensional $E$–linear subspace in ${E^{2}}$, as claimed. ∎

######  Definition 8.3.

With $\mathscr{A}/F$ as in Definition 8.1, we say that a representation
$\rho\colon\pi\to\mathrm{SL}(1,\mathscr{A})$ is _irreducible_ if $\rho$
satisfies any of the equivalent conditions as in Lemma 8.2.

######  Lemma 8.4.

With $\mathscr{A}/F$ as in Definition 8.1, let $\pi$ be a finitely generated
group and $\rho\colon\pi\to\mathrm{SL}(1,\mathscr{A})$ be an irreducible
representation. If $\tilde{\rho}\colon\pi\to\mathrm{SL}(1,\mathscr{A})$ is a
representation with character $\chi_{\tilde{\rho}}=\chi_{\rho}$ on $\pi$, then
$\tilde{\rho}$ is conjugate to $\rho$ in
$\mathrm{GL}(1,\mathscr{A})=\mathscr{A}^{\times}$, namely,
$\tilde{\rho}=\tau\circ\rho$ for some inner automorphism $\tau$ of
$\mathscr{A}^{\times}$. The converse also holds true.

###### Proof.

We only prove the main assertion, as the converse is obvious. Since $\pi$ is
finitely generated, we may assume $F$ finitely generated over ${\mathbb{Q}}$.
We fix a field embedding of $F$ into ${\mathbb{C}}$, and an isomorphism
$\mathscr{A}\otimes_{F}{\mathbb{C}}\cong\mathrm{Mat}(2,{\mathbb{C}})$. Since
the induced irreducible $\mathrm{SL}(2,{\mathbb{C}})$–representations
$\rho_{\mathbb{C}}$ and $\tilde{\rho}_{\mathbb{C}}$ of the finitely generated
group $\pi$ have identical characters, they are conjugate in
$\mathrm{GL}(2,{\mathbb{C}})$, and indeed, conjugate in
$\mathrm{SL}(2,{\mathbb{C}})$ [CulS83, Proposition 1.5.2].

Take a finite generating set $g_{1},\cdots,g_{r}$ of $\pi$. We set up $r$
equations in $\mathscr{A}$ with an unknown $T$, namely,
$T\cdot\rho(g_{i})=\tilde{\rho}(g_{i})\cdot T$ for all $i\in\\{1,\cdots,r\\}$.
It suffices to show that there exists some solution $T\in\mathscr{A}$ with
$\mathrm{Nr}_{\mathscr{A}/F}(T)\neq 0$. To this end, we take a basis
$X_{0},X_{1},X_{2},X_{3}$ of $\mathscr{A}$ over $F$. and write $T=u_{0}\cdot
X_{0}+u_{1}\cdot X_{1}+u_{2}\cdot X_{2}+u_{3}\cdot X_{3}$. Then the above
equations becomes a linear system of $4r$ equations over $F$ in the unknowns
$u_{0},u_{1},u_{2},u_{3}$. The $F$–solution space $V$ of these equations is a
linear subspace in $\mathscr{A}$ over $F$, and the ${\mathbb{C}}$–solution
space is the scalar extension $V\otimes_{F}{\mathbb{C}}$ in
$\mathrm{Mat}(2,{\mathbb{C}})$. Moreover, the norm
$\mathrm{Nr}_{\mathscr{A}/F}$ on $\mathscr{A}$ restricts to a $F$–quadratic
form on $V$, which agrees with the determinant function on
$\mathrm{Mat}(2,{\mathbb{C}})$ under our identification. Since
$\rho_{\mathbb{C}}$ and $\tilde{\rho}_{\mathbb{C}}$ are conjugate in
$\mathrm{GL}(2,{\mathbb{C}})$, we infer that $V$ is nontrivial and
$\mathrm{Nr}_{\mathscr{A}/F}$ is nonvanishing on $V$. This means there exists
some solution $T$ in $\mathrm{GL}(1,\mathscr{A})=\mathscr{A}^{\times}$ as
desired. ∎

Recall that the _$2$ –Frattini subgroup_ of any group $\pi$ is by definition
the characteristic subgroup generated by the commutators and the squares of
elements in $\pi$. In fact, the squares of elements already form a generating
set. (This is not true in general for Frattini subgroups associated with other
primes $p$.) In this paper, we denote the $2$–Frattini subgroup of any group
$\pi$ as $\mathrm{Fratt}_{2}(\pi)$.

######  Lemma 8.5.

With $\mathscr{A}/F$ as in Definition 8.1, let $\pi$ be a group and
$\rho\colon\pi\to\mathrm{SL}(1,\mathscr{A})$ be a representation. Then the
following conditions are all equivalent.

  1. (1)

For some (hence any) algebraically closed field extension $E$ of $F$ over
which $\mathscr{A}\otimes_{F}E$ splits as a matrix algebra
$\mathrm{Mat}(2,E)$, the induced representation
$\rho_{E}\colon\pi\to\mathrm{SL}(2,E)$ is Zariski dense, that is, the image of
$\rho$ is dense in the linear algebraic group $\mathrm{SL}(2,E)$ over $E$ with
respect to the Zariski topology.

  2. (2)

For every finite-index normal subgroup $\pi^{\prime}$ of $\pi$, the
representation $\rho^{\prime}\colon\pi^{\prime}\to\mathrm{SL}(1,\mathscr{A})$
restricting $\rho$ to $\pi^{\prime}$ is irreducible.

  3. (3)

The image of $\rho$ is infinite, and the representation
$\rho^{\natural}\colon\mathrm{Fratt}_{2}(\pi)\to\mathrm{SL}(1,\mathscr{A})$
restricting $\rho$ to the $2$–Frattini subgroup is irreducible.

###### Proof.

For any algebraically closed field $E$, the Zariski closure of the image of
$\rho(\pi)$ is a Zariski closed subgroup in $\mathrm{SL}(2,E)$, (see [Spr09,
Chapter 2, Lemma 2.2.4]). Therefore,
$\rho^{\prime}\colon\pi^{\prime}\to\mathrm{SL}(1,\mathscr{A})$ is reducible
for some finite index subgroup $\pi^{\prime}$ of $\pi$ if and only if the
Zariski closure of $\rho_{E}(\pi)$ has dimension at most $2$ over $E$ (Lemma
8.2). This proves the equivalence (1) $\Leftrightarrow$ (2).

If $\rho_{E}$ is Zarisk dense, the Zariski closure of
$\rho_{E}(\mathrm{Fratt}_{2}(\pi))$ is an infinite, normal, Zariski closed
subgroup of $\mathrm{SL}(2,E)$, which has to be $\mathrm{SL}(2,E)$ itself. It
follows that $\rho$ has infinite image and $\rho^{\natural}$ is irreducible.
On the other hand, if $\rho_{E}$ is not Zariski dense, there exists a finite-
index normal subgroup $\pi^{\prime}$, such that the restricted representation
$\rho^{\prime}\colon\pi^{\prime}\to\mathrm{SL}(1,\mathscr{A})$ is reducible.
Therefore, $\rho_{E}(\pi^{\prime})$ fixes some points on the projective line
$E\mathbf{P}^{1}$ over $E$, on which $\mathrm{SL}(2,E)$ acts by projective
$E$–linear transformations (Lemma 8.2). When $\rho_{E}(\pi^{\prime})$ fixes
only one or two points on $E\mathbf{P}^{1}$, we observe that $\rho_{E}(\pi)$
stabilizes the fixed point set. In this case, $\rho^{\natural}$ is reducible,
since $\rho_{E}(\mathrm{Fratt}_{2}(\pi))$ fixes those fixed points of
$\rho_{E}(\pi^{\prime})$ on $E\mathbf{P}^{1}$. When $\rho_{E}(\pi^{\prime})$
fixes at least three points on $E\mathbf{P}^{1}$, it fixes $E\mathbf{P}^{1}$,
so $\rho_{E}(\pi^{\prime})$ is contained in the center $\\{\pm I\\}$ of
$\mathrm{SL}(2,E)$. In this case, $\rho$ has finite image, since
$\pi^{\prime}$ has finite index in $\pi$. This proves the equivalence (2)
$\Leftrightarrow$ (3). ∎

######  Remark 8.6.

Considering $\mathrm{SL}(1,\mathscr{A})$ as an algebraic group over $F$, one
may actually replace $\mathrm{SL}(2,E)$ with $\mathrm{SL}(1,\mathscr{A})$ in
the condition (1). When $\pi$ is finitely generated, one may remove the
requirement of being normal in the condition (2), because any finite-index
subgroup of a finitely generated group contains a finite-index normal
subgroup.

######  Definition 8.7.

With $\mathscr{A}/F$ as in Definition 8.1, we say that a representation
$\rho\colon\pi\to\mathrm{SL}(1,\mathscr{A})$ is _Zariski dense_ if $\rho$
satisfies any of the equivalent conditions as in Lemma 8.5.

######  Definition 8.8.

With $\mathscr{A}/F$ as in Definition 8.1, let $\pi$ be a group, and
$\rho\colon\pi\to\mathrm{SL}(1,\mathscr{A})$ be a representation.

  1. (1)

When $\rho$ is irreducible, the _trace field_ of $\rho$ is defined as the
subfield of $F$ generated by $\\{\chi_{\rho}(g)\in F\colon g\in\pi\\}$ over
${\mathbb{Q}}$, and the _quaternion algebra_ of $\rho$ is defined as the
quaternion algebra of $\mathscr{A}$ generated by
$\\{\rho(g)\in\mathscr{A}\colon g\in\pi\\}$ over the trace field.

  2. (2)

When $\rho$ is Zariski dense, the _invariant trace field_ of $\rho$ is defined
as the subfield of $F$ generated by $\\{\chi_{\rho}(g)\in F\colon
g\in\mathrm{Fratt}_{2}(\pi)\\}$ over ${\mathbb{Q}}$, and the _invariant
quaternion algebra_ of $\rho$ is defined as the subalgebra of $\mathscr{A}$
generated by $\\{\rho(g)\in\mathscr{A}\colon g\in\mathrm{Fratt}_{2}(\pi)\\}$
over the invariant trace field.

######  Remark 8.9.

Denote by $\mathrm{PSL}(1,\mathscr{A})$ the quotient of
$\mathrm{SL}(1,\mathscr{A})$ by the center $\\{\pm\mathbf{1}\\}$. For any
_projective special linear representation_
$\rho^{\flat}\colon\pi\to\mathrm{PSL}(1,\mathscr{A})$, one may define the
_adjoint trace field_ of $\rho^{\flat}$ as the subfield of $F$ generated over
${\mathbb{Q}}$ by the traces of the adjoint representations
$\mathrm{Ad}(\rho^{\flat})\colon\pi\to\mathrm{Aut}_{F}(\mathfrak{sl}(1,\mathscr{A}))$,
where $\mathfrak{sl}(1,\mathscr{A})$ can be identified as the $3$–dimensional
$F$–linear subspace of $\mathscr{A}$ consisting of all the trace zero
elements. When $\rho^{\flat}$ comes from a representation
$\rho\colon\pi\to\mathrm{SL}(1,\mathscr{A})$ as the projectivization
$\mathbf{P}(\rho)$, one obtains the relation
$\mathrm{tr}_{F}(\rho^{\flat}(g))=\chi_{\rho}(g)^{2}-1$ for all $g\in\pi$.
Therefore, the adjoint trace field of any liftable
$\mathrm{PSL}(1,\mathscr{A})$–representation coincides with the trace field of
any of its $\mathrm{SL}(1,\mathscr{A})$–lifts restricted to the $2$–Frattini
subgroup. For any algebraically closed field extension $E$ over $F$, we can
regard
$\mathrm{PSL}(1,\mathscr{A}\otimes_{F}E)\cong\mathrm{PSL}(2,E)\cong\mathrm{SO}(3,E)$
as an algebraic group over $E$. Observe that a subgroup of
$\mathrm{PSL}(1,\mathscr{A}\otimes_{F}E)$ is Zariski closed if and only if its
preimage in $\mathrm{SL}(1,\mathscr{A}\otimes_{F}E)$ is Zariski closed.
Therefore, one may define a _Zariski dense_ representation
$\rho^{\flat}\colon\pi\to\mathrm{PSL}(1,\mathscr{A})$ by the property that
$\rho^{\flat}(\pi)$ is Zariski dense in
$\mathrm{PSL}(1,\mathscr{A}\otimes_{F}E)$ for some algebraically closed field
extension $E$ over $F$. In this case, the adjoint trace field of
$\rho^{\flat}$ coincides with the invariant trace field of $\Pi$, where $\Pi$
denotes the full preimage of $\rho^{\flat}(\pi)$ in
$\mathrm{SL}(1,\mathscr{A})$. Therefore, one may define the _invariant
quaternion algebra_ of a Zariski dense
$\mathrm{PSL}(1,\mathscr{A})$–representation $\rho^{\flat}$ as the invariant
quaternion algebra of $\Pi$, over the invariant trace field of $\Pi$. Compare
[Vin93].

######  Lemma 8.10.

With $\mathscr{A}/F$ as in Definition 8.1, let $\pi$ is a finitely generated
group and $\rho\colon\pi\to\mathrm{SL}(1,\mathscr{A})$ be a Zariski dense
representation. Then, the following conditions on a representation
$\tilde{\rho}\colon\pi\to\mathrm{SL}(1,\mathscr{A})$ are all equivalent.

  1. (1)

For some group homomorphism $\varepsilon\colon\pi\to\\{\pm 1\\}$,
$\tilde{\rho}$ is conjugate to $\varepsilon\cdot\rho$ in
$\mathrm{GL}(1,\mathscr{A})$.

  2. (2)

For some group homomorphism $\varepsilon\colon\pi\to\\{\pm 1\\}$,
$\chi_{\tilde{\rho}}=\varepsilon\cdot\chi_{\rho}$ holds on $\pi$.

  3. (3)

The equality $\chi_{\tilde{\rho}}=\chi_{\rho}$ holds on
$\mathrm{Fratt}_{2}(\pi)$.

  4. (4)

For some (hence any) finite-index subgroup $\pi^{\prime}$ of $\pi$,
$\chi_{\tilde{\rho}}=\chi_{\rho}$ holds on $\mathrm{Fratt}_{2}(\pi^{\prime})$.

###### Proof.

The equivalence (1) $\Leftrightarrow$ (2) follows from Lemma 8.4. The
implication (2) $\Rightarrow$ (3) follows immediately from the fact that
$\mathrm{Fratt}_{2}(\pi)$ is generated by squares of elements in $\pi$. The
implication (3) $\Rightarrow$ (4) with ‘any’ is trivial. It remains to prove
(4) with ‘some’ $\Rightarrow$ (2). To this end, we may assume without loss of
generality that $\pi^{\prime}$ is normal in $\pi$, and argue with
$\mathscr{A}=\mathrm{Mat}(2,{\mathbb{C}})$ over $F={\mathbb{C}}$, since $\pi$
is finitely generated. By Lemma 8.5, and Remark 8.6,
$\rho\colon\pi\to\mathrm{SL}(2,{\mathbb{C}})$ is irreducible restricted to the
finite-index subgroup $\mathrm{Fratt}(\pi^{\prime})$. Since $\tilde{\rho}$ has
the same character as that of $\rho$ restricted to
$\mathrm{Fratt}_{2}(\pi^{\prime})$, we may adjust $\tilde{\rho}$ with a
conjugation in $\mathrm{SL}(2,{\mathbb{C}})$, and assume henceforth
$\tilde{\rho}=\rho$ on $\mathrm{Fratt}_{2}(\pi^{\prime})$, (see [CulS83,
Proposition 1.5.2]).

Since $\rho$ is irreducible on $\mathrm{Fratt}_{2}(\pi^{\prime})$, there exist
$g_{1},g_{2},g_{3}\in\mathrm{Fratt}_{2}(\pi^{\prime})$, such that the matrices
$I,\rho(g_{1}),\rho(g_{2}),\rho(g_{3})$ span $\mathrm{Mat}(2,{\mathbb{C}})$
over ${\mathbb{C}}$. We denote
$X_{i}=2\cdot\rho(g_{i})-\chi_{\rho}(g_{i})\cdot I$ in
$\mathrm{Mat}(2,{\mathbb{C}})$, for $i=1,2,3$. Then the elements
$X_{1},X_{2},X_{3}$ span the complex Lie subalgebra
$\mathfrak{sl}(2,{\mathbb{C}})$, (which consists of all the traceless
matrices). It is well-known that the adjoint action
$\mathrm{Ad}_{g}(X)=gXg^{-1}$ defines a group homomorphism
$\mathrm{Ad}\colon\mathrm{SL}(2,{\mathbb{C}})\to\mathrm{Aut}(\mathfrak{sl}(2,{\mathbb{C}}))$
with kernel $\\{\pm I\\}$. As we have assumed $\pi^{\prime}$ normal in $\pi$,
its characteristic subgroup $\mathrm{Fratt}_{2}(\pi^{\prime})$ is also normal
in $\pi$. We observe
$\mathrm{Ad}_{\rho(u)}(X_{i})=2\cdot\rho(ug_{i}u^{-1})-\chi_{\rho}(g_{i})\cdot
I=2\cdot\tilde{\rho}(ug_{i}u^{-1})-\chi_{\rho}(g_{i})\cdot
I=2\cdot\tilde{\rho}(u)\tilde{\rho}(g_{i})\tilde{\rho}(u^{-1})-\chi_{\rho}(g_{i})\cdot
I=2\cdot\tilde{\rho}(u)\rho(g_{i})\tilde{\rho}(u^{-1})-\chi_{\rho}(g_{i})\cdot
I=\mathrm{Ad}_{\tilde{\rho}(u)}(X_{i})$, for $i=1,2,3$, and for all $u\in\pi$.
It follows that $\rho(u)=\varepsilon(u)\cdot\tilde{\rho}(u)$ holds for some
$\varepsilon(u)\in\\{\pm 1\\}$. Clearly $\varepsilon\colon\pi\to\\{\pm 1\\}$
must be a group homomorphism, so we obtain
$\chi_{\tilde{\rho}}=\varepsilon\cdot\chi_{\rho}$ on $\pi$, as desired. ∎

######  Definition 8.11.

With $\mathscr{A}/F$ as in Definition 8.1, let $\pi$ is a finitely generated
group. We say that a representation
$\tilde{\rho}\colon\pi\to\mathrm{SL}(1,\mathscr{A})$ is
_$\mathrm{Hom}(\pi;\\{\pm 1\\})$ –equivalent_ to a Zariski dense
representation $\rho\colon\pi\to\mathrm{SL}(1,\mathscr{A})$ if $\tilde{\rho}$
satisfies any of the equivalent conditions in Lemma 8.10. Note that
$\tilde{\rho}$ is necessarily Zariski dense in this case.

We draw the following immediate consequence from the above discussion,
(compare [MacR03, Chapter 3, Theorem 3.3.4]).

######  Corollary 8.12.

With $\mathscr{A}/F$ as in Definition 8.1, let $\pi$ be a finitely generated
group, and $\rho\colon\pi\to\mathrm{SL}(1,\mathscr{A})$ be a Zariski dense
representation. Then the invariant trace field and the invariant quaternion
algebra of $\rho$ depend only on the $\mathrm{Hom}(\pi,\\{\pm
1\\})$–equivalence class of $\rho$, and they remain unchanged under passage to
finite-index subgroups of $\pi$ and restrictions of $\rho$.

######  Definition 8.13.

With $\mathscr{A}/F$ as in Definition 8.1. We say that a representation
$\rho\colon\pi\to\mathrm{SL}(1,\mathscr{A})$ is _bounded_ at a place $v$, if
the image of the induced homomorphism
$\rho_{v}\colon\pi\to\mathrm{SL}(1,\mathscr{A}_{v})$ is bounded in
$\mathrm{Mat}(1,\mathscr{A}_{v})=\mathscr{A}_{v}$. Here,
$\mathscr{A}_{v}=\mathscr{A}\otimes_{F}F_{v}$ denotes the complete quaternion
algebra over the complete field $F_{v}$.

######  Lemma 8.14.

With $\mathscr{A}/F$ as in Definition 8.1, let $\pi$ be a group and
$\rho\colon\pi\to\mathrm{SL}(1,\mathscr{A})$ be an irreducible representation.
Then, $\rho$ is bounded at a non-Archimedean place $v$ if and only if
$\chi_{\rho}\colon\pi\to F$ has image contained in the valuation ring of $F$
at $v$.

###### Proof.

Denote by $R=\\{x\in F\colon|x|_{v}\leq 1\\}$ the valuation ring of $F$ at
$v$, fixing a representative norm $|\cdot|_{v}$. The trace relation
$\mathrm{Tr}_{\mathscr{A}/F}(\rho(g^{2}))=\mathrm{Tr}_{\mathscr{A}/F}(\rho(g))^{2}-2$
(Proposition 3.2) and the super-triangular inequality for non-Archimedean
valuations imply $|\chi_{\rho}(g^{2})|_{v}=|\chi_{\rho}(g)|_{v}^{2}$ whenever
$|\chi_{\rho}(g)|_{v}>1$. By iteration, we see that $\chi_{\rho}$ diverges
along the $2$–exponent powers of $g$ whenever $|\chi_{\rho}(g)|_{v}>1$.
Therefore, if $\rho$ is bounded at $v$, we obtain $\chi_{\rho}(g)\in R$ for
all $g\in\pi$.

Conversely, if $\chi_{\rho}(g)\in R$ holds for all $g\in\pi$, we show that
$\rho$ is bounded at $v$. We make use of the canonical _trace pairing_
$\mathscr{A}\times\mathscr{A}\to F$, defined as
$(A,B)\mapsto\mathrm{Tr}_{\mathscr{A}/F}(AB)$. The pairing is symmetric,
$F$–bilinear, and nondegenerate. (This can be easily seen with any Hilbert
symbol $\mathscr{A}\cong(a,b/F)$: The pairing matrix is diagonalized over the
standard basis $\mathbf{1},\mathbf{i},\mathbf{j},\mathbf{k}$ with diagonal
entries $2,2a,2b,-2ab$.) Since $\rho$ is irreducible, there exist elements
$g_{0},g_{1},g_{2},g_{3}\in\pi$, such that
$X_{0}=\rho(g_{0}),X_{1}=\rho(g_{1}),X_{2}=\rho(g_{2}),X_{3}=\rho(g_{3})$ form
a basis of $\mathscr{A}$ over $F$. Denote by
$X_{0}^{*},X_{1}^{*},X_{2}^{*},X_{3}^{*}$ the dual basis with respect to the
trace pairing. Then $\rho(u)=\chi_{\rho}(ug_{0})\cdot
X_{0}^{*}+\chi_{\rho}(ug_{1})\cdot X_{1}^{*}+\chi_{\rho}(ug_{2})\cdot
X_{2}^{*}+\chi_{\rho}(ug_{3})\cdot X_{3}^{*}$ holds in $\mathscr{A}$ for all
$u\in\pi$. Since $\chi_{\rho}$ has image in $R$, this means $\rho_{v}(u)$ lies
in the bounded subset
$R_{v}X_{0}^{*}+R_{v}X_{1}^{*}+R_{v}X_{2}^{*}+R_{v}X_{3}^{*}\subset\mathscr{A}_{v}$,
for all $u\in\pi$, so $\rho$ is bounded at $v$. ∎

##  9\. Profinite correspondence of representations

In this section, we prove an essential case of Theorem 1.4, namely, the case
with pseudo-Anosov mapping tori (Lemma 9.1).

######  Lemma 9.1.

Let ${\mathbb{Q}}^{{\mathrm{ac}}}$ be an algebraic closure of ${\mathbb{Q}}$.
Let $(M_{A},M_{B},\Psi)$ be a profinite isomorphism setting of pseudo-Anosov
mapping tori (Convention 5.1).

Then, for any Zariski dense representation
$\rho_{B}\colon\pi_{B}\to\mathrm{SL}(2,{\mathbb{Q}}^{{\mathrm{ac}}})$, there
exists a Zariski dense representation
$\rho_{A}\colon\pi_{A}\to\mathrm{SL}(2,{\mathbb{Q}}^{{\mathrm{ac}}})$ with all
the following properties:

  * •

The invariant trace fields $k_{A}$ of $\rho_{A}$ and $k_{B}$ of $\rho_{B}$ are
identical in ${\mathbb{Q}}^{{\mathrm{ac}}}$.

  * •

The invariant quaternion algebras $\mathscr{B}_{A}$ of $\rho_{A}$ and
$\mathscr{B}_{B}$ of $\rho_{B}$ are ramified over identical sets of places in
$k=k_{A}=k_{B}$. Hence, $\mathscr{B}_{A}$ and $\mathscr{B}_{B}$ are
$\mathrm{GL}(2,{\mathbb{Q}}^{{\mathrm{ac}}})$–conjugate in
$\mathrm{Mat}(2,{\mathbb{Q}}^{{\mathrm{ac}}})$.

  * •

For any prime $\mathfrak{p}$ in $k$, the restricted representation
$\rho^{\natural}_{A}\colon\mathrm{Fratt}_{2}(\pi_{A})\to\mathrm{SL}(1,\mathscr{B}_{A})$
is bounded at $\mathfrak{p}$ if and only if
$\rho^{\natural}_{B}\colon\mathrm{Fratt}_{2}(\pi_{B})\to\mathrm{SL}(1,\mathscr{B}_{B})$
is bounded at $\mathfrak{p}$.

  * •

For any prime $\mathfrak{p}$ in $k$ where $\rho^{\natural}_{A}$ and
$\rho^{\natural}_{B}$ are bounded, the equality
$\chi^{\natural}_{A,\mathfrak{p}}=\chi^{\natural}_{B,\mathfrak{p}}\circ\Psi$
holds on
$\mathrm{Fratt}_{2}(\widehat{\pi}_{A})=\Psi^{-1}(\mathrm{Fratt}_{2}(\widehat{\pi}_{B}))$.
Here, we denote by
$\chi^{\natural}_{B,\mathfrak{p}}\colon\mathrm{Fratt}_{2}(\widehat{\pi}_{B})\to
k_{\mathfrak{p}}$ the character of the continuous representation
$\rho^{\natural}_{B,\mathfrak{p}}\colon\mathrm{Fratt}_{2}(\widehat{\pi}_{B})\to\mathrm{SL}(1,\mathscr{B}_{B,\mathfrak{p}})$
obtained by completion, and similarly $\chi^{\natural}_{A,\mathfrak{p}}$ of
$\rho^{\natural}_{A,\mathfrak{p}}$.

Subject to the above properties, the $\mathrm{Hom}(\pi_{A},\\{\pm
1\\})$–equivalence class of $\rho_{A}$ depends only on $\Psi$ and the
$\mathrm{Hom}(\pi_{B},\\{\pm 1\\})$–equivalence class of $\rho_{B}$.

###### Proof.

Starting from any Zariski dense representation
$\rho_{B}\colon\pi_{B}\to\mathrm{SL}(2,{\mathbb{Q}}^{{\mathrm{ac}}})$, we
obtain its invariant trace field $k=k_{B}$ in ${\mathbb{Q}}^{{\mathrm{ac}}}$,
and its invariant quaternion algebra $\mathscr{B}_{B}$ in
$\mathrm{Mat}(2,{\mathbb{Q}}^{{\mathrm{ac}}})$. There is an associated
representation