# Generalized Bowen-Franks Groups and Profinite Conjugacy of Hyperbolic Toral
Automorphisms

Lennard F. Bakker  Department of Mathematics, Brigham Young University, Provo,
Utah, USA<EMAIL_ADDRESS>and  Pedro Martins Rodrigues  Department
of Mathematics, Instituto Superior Técnico, Univ. Tec. Lisboa, Lisboa,
Portugal<EMAIL_ADDRESS>
###### Abstract.

We show that a collection of generalized Bowen-Franks group, what we call the
principal Bowen-Franks $R$-modules, form a complete set of $R$-module
invariants for the equivalence relation of profinite conjugacy for similar
hyperbolic toral automorphisms. We also show that these principal generalized
Bowen-Franks $R$-modules are the principal invariants in a large class of
generalized Bowen-Franks $R$-module invariants for similar hyperbolic
automorphisms. We further show how the conjugacy invariant of cyclicity, when
applied to hyperbolic toral automorphisms, is realized for profinite
conjugacy.

###### Key words and phrases:

generalized Bowen-Franks groups, profinite conjugacy, hyperbolic toral
automorphism, cyclicity

###### 2010 Mathematics Subject Classification:

Primary: 37C15; Secondary: 37C79, 37D20, 20E18

##  1\. Introduction

Generalized Bowen-Franks groups are abelian group invariants for equivalence
relations in symbolic dynamics and toral endomorphisms. The original Bowen-
Franks group arose as an invariant of flow equivalence for suspension flows of
subshifts of finite type [1]. Following the notation of Lind and Marcus [2],
the Bowen-Franks group for $A\in M_{n}({\mathbb{Z}})$, an $n\times n$ integer
matrix representing the subshift of finite type, is

${\rm BF}(A)={\mathbb{Z}}^{n}/{\mathbb{Z}}^{n}(I-A),$

where $I-A$ acts on the right and the elements of ${\mathbb{Z}}^{n}$ are row
vectors. Franks showed that ${\rm BF}(A)$, together with the flow equivalence
invariant ${\rm det}(I-A)$ of Parry and Sullivan [3], are a complete set of
computable invariants of shift equivalence for irreducible subshifts of finite
type [4]. The Bowen-Franks group ${\rm BF}(A)$ has been an important tool in
the classification of reducible subshifts [5] as well as general subshifts [6,
7]. More invariants of shift equivalence for subshifts of finite type were
introduced by Kitchens that generalized the Bowen-Franks group [8]. These
generalized Bowen-Franks groups are of the form

${\rm BF}_{g}(A)={\mathbb{Z}}^{n}/{\mathbb{Z}}^{n}g(A),$

where $g\in{\mathbb{Z}}[t]$ is a polynomial with $g(0)$ being a unit in
${\mathbb{Z}}$, i.e., $g(0)=\pm 1$ [8, 2]. The original Bowen-Franks group
${\rm BF}(A)$ corresponds to ${\rm BF}_{g}(A)$ for $g(t)=1-t$. Another
generalization of the Bowen-Franks group ${\rm BF}(A)$ was introduced by one
of the authors and Sousa Ramos for toral endomorphisms of ${\mathbb{T}}^{n}$
which are represented by $A\in M_{n}({\mathbb{Z}})$ [9]. These generalized
Bowen-Frank groups ${\rm BF}_{g}(A)$ are invariants of topological conjugacy
for $A$ where $g\in{\mathbb{Z}}[t]$. Topological conjugacy of two toral
endomorphisms $A,B\in M_{n}({\mathbb{Z}})$ is equivalent to conjugacy by an
element of ${\rm GL}_{n}({\mathbb{Z}})$, i.e., there exists $C\in{\rm
GL}_{n}({\mathbb{Z}})$ such that $AC=CB$ (cf [10]). For a toral endomorphism
$A\in M_{n}({\mathbb{Z}})$, each generalized Bowen-Franks group ${\rm
BF}_{g}(A)$ has the additional structure of a right $R$-module for a ring $R$
isomorphic to the matrix ring

${\mathbb{Z}}[A]=\\{p(A):p\in{\mathbb{Z}}[t]\\}$

(see Subsection 4.2). The structure of the finitely generated abelian group
${\rm BF}_{g}(A)$, with $A$ representing a subshift of finite type or a toral
endomorphism, can be determined by computing the Smith normal form of $g(A)$.
One of the authors and Sousa Ramos showed, for a given irreducible monic
polynomial $f\in{\mathbb{Z}}[t]$ satisfying $f(0)=\pm 1$ and having square-
free discriminant, that all $A\in{\rm GL}_{n}({\mathbb{Z}})$ having
characteristic polynomial $f(t)$ have $R$-module isomorphic ${\rm BF}_{g}(A)$
for each $g\in{\mathbb{Z}}[t]$ [9]. Chen showed in some cases that ${\rm
BF}_{g}(A)$, as an abelian group, is determined by the zeta function of $A$
for those $g\in{\mathbb{Z}}[t]$ with $g(0)=\pm 1$, without assuming
irreducibility of the characteristic polynomial of $A$ [11]. When $A\in{\rm
GL}_{n}({\mathbb{Z}})$ is a hyperbolic toral automorphism (no eigenvalue of
$A$ has modulus $1$) and $g_{k}(t)=t^{k}-1$, $k\in{\mathbb{N}}$, each
generalized Bowen-Franks group,

${\rm
BF}_{k}(A)={\mathbb{Z}}^{n}/{\mathbb{Z}}^{n}g_{k}(A)={\mathbb{Z}}^{n}/{\mathbb{Z}}^{n}(A^{k}-I),$

is a finite abelian group that is isomorphic by Pontryagin duality to the set
${\rm Per}_{k}(A)$ of $k$-periodic points of the left action of $A$ on
${\mathbb{T}}^{n}$, where the elements of ${\mathbb{T}}^{n}$ are column
vectors. Note that the original Bowen-Franks group ${\rm BF}(A)$,
corresponding to $g(t)=1-t$, is isomorphic to ${\rm BF}_{1}(A)$, corresponding
to $g_{1}(t)=t-1$. The main purpose of this paper is to show, when $A\in{\rm
GL}_{n}({\mathbb{Z}})$ is hyperbolic, that the collection of principal
generalized Bowen-Franks $R$-modules ${\rm BF}_{k}(A)$, $k\in{\mathbb{N}}$, is
a complete set of invariants for an equivalence relation called profinite
conjugacy.

Profinite conjugacy is a topological $R$-module isomorphism between two
profinite $R$-modules. For a hyperbolic $A\in{\rm GL}_{n}({\mathbb{Z}})$, the
principal Bowen-Franks $R$-modules ${\rm BF}_{k}(A)$, $k\in{\mathbb{N}}$, are
parameterized by the collection ${\mathcal{Q}}^{\rm p}$ of polynomials
$g_{k}$, $k\in{\mathbb{N}}$, and have an inverse limit $G_{{\mathcal{Q}}^{\rm
p},A}$ that is a profinite $R$-module, whose Pontryagin dual is the direct
limit of ${\rm Per}_{k}(A)$, $k\in{\mathbb{N}}$, which is the rational torus.
For another hyperbolic $B\in{\rm GL}_{n}({\mathbb{Z}})$ that is similar to $A$
over ${\mathbb{Q}}$ i.e., there exists $P\in{\rm GL}_{n}({\mathbb{Q}})$ such
that $AP=PB$, so it has the same characteristic polynomial as that of $A$, the
principal Bowen-Franks groups ${\rm BF}_{k}(B)$, $k\in{\mathbb{N}}$, have an
inverse limit $R$-module $G_{{\mathcal{Q}}^{\rm p},B}$. Similar hyperbolic
$A,B\in{\rm GL}_{n}({\mathbb{Z}})$ have a common ring $R$ to which both
${\mathbb{Z}}[A]$ and ${\mathbb{Z}}[B]$ are isomorphic (see Subsection 4.2),
and are said to be strongly ${\rm BF}$-equivalent over ${\mathcal{Q}}^{\rm p}$
if ${\rm BF}_{k}(A)$ and ${\rm BF}_{k}(B)$ are $R$-module isomorphic, written
${\rm BF}_{k}(A)\cong_{R}{\rm BF}_{k}(B)$, for all $k\in{\mathbb{N}}$. We
showed that if similar hyperbolic $A,B\in{\rm GL}_{n}({\mathbb{Z}})$ are
strongly ${\rm BF}$-equivalent over ${\mathcal{Q}}^{\rm p}$, then
$G_{{\mathcal{Q}}^{\rm p},A}$ and $G_{{\mathcal{Q}}^{\rm p},B}$ are
topologically $R$-module isomorphic, i.e., there exists a map
$\Psi_{{\mathcal{Q}}^{\rm p}}:G_{{\mathcal{Q}}^{\rm p},A}\to
G_{{\mathcal{Q}}^{\rm p},B}$ that is an $R$-module isomorphism and a
homeomorphism [12]. The existence of such a map $\Psi_{{\mathcal{Q}}^{\rm
p}}:G_{{\mathcal{Q}}^{\rm p},A}\to G_{{\mathcal{Q}}^{\rm p},B}$ we call a
profinite conjugacy over ${\mathcal{Q}}^{\rm p}$ between similiar hyperbolic
$A,B\in{\rm GL}_{n}({\mathbb{Z}})$. The main result of this paper includes the
converse: for similar hyperbolic $A,B\in{\rm GL}_{n}({\mathbb{Z}})$, if $A$
and $B$ are profinitely conjugate over ${\mathcal{Q}}^{\rm p}$, then $A$ and
$B$ are strongly ${\rm BF}$-equivalent over ${\mathcal{Q}}^{\rm p}$. This
establishes the equivalence of strong ${\rm BF}$-equivalence over
${\mathcal{Q}}^{\rm p}$ and profinite conjugacy over ${\mathcal{Q}}^{\rm p}$.

One of the authors and Sousa Ramos proved a result in [9] that implies for
similar hyperbolic automorphisms $A,B\in{\rm GL}_{n}({\mathbb{Z}})$ with
irreducible characteristic polynomial that if $A$ and $B$ are strongly BF-
equivalent over ${\mathcal{Q}}^{\rm p}$, then $A$ and $B$ are strongly BF-
equivalent over the collection ${\mathcal{Q}}^{\rm a}$ of polynomials
$g\in{\mathbb{Z}}[t]$ satisfying ${\rm det}(g(A))\neq 0$, i.e., ${\rm
BF}_{g}(A)\cong_{R}{\rm BF}_{g}(B)$ for all $g\in{\mathcal{Q}}^{a}$. We show,
as a corollary of the main result, that strong BF-equivalence of $A$ and $B$
over ${\mathcal{Q}}^{\rm p}$ implies strong BF-equivalence of $A$ and $B$ over
${\mathcal{Q}}^{\rm a}$ without assuming irreducibility of the characteristic
polynomial of the similar hyperbolic $A$ and $B$. Hence, in the irreducible
and reducible cases for similar hyperbolic toral automorphisms, the principal
Bowen-Franks $R$-modules are the principal invariants in the class of
generalized Bowen-Franks $R$-module invariants parameterized by
${\mathcal{Q}}^{\rm a}$.

For irreducible similar hyperbolic toral automorphisms, the principal
generalized Bowen-Franks $R$-modules are a complete set of invariants for the
equivalence relations of weak equivalence of the associated ideals and block
conjugacy of the matrices as well as for profinite conjugacy. Here irreducible
means the characteristic polynomial $f(t)$ for the similar $A,B\in{\rm
GL}_{n}({\mathbb{Z}})$ is irreducible in ${\mathbb{Z}}[t]$. Fixing a root
$\beta$ of $f(t)$, the selection of a right eigenvector
$u=(u_{1},\dots,u_{n})^{\rm T}$ of $A$, i.e., $Au=\beta u$, with entries
$u_{i}$ in the algebraic number field ${\mathbb{Q}}(\beta)$ determines a
(fractional) ideal

$I_{A}=u_{1}{\mathbb{Z}}+\cdots+u_{n}{\mathbb{Z}}$

in the ring ${\mathbb{Z}}[\beta]$. By the Lattimer-MacDuffey-Taussky one-to-
one correspondence [13, 14], different choices of right eigenvectors determine
arithmetically equivalent ideals in ${\mathbb{Z}}[\beta]$. Similarly the
selection of a right eigenvector $v=(v_{1},\dots,v_{n})^{\rm T}$ for $B$
determines a (fractional) ideal $I_{B}$ in ${\mathbb{Z}}[\beta]$. Two
(fractional) ideals $I$ and $J$ of ${\mathbb{Z}}[\beta]$ are weakly equivalent
if there exists (fractional) ideals $X$ and $Y$ of ${\mathbb{Z}}[\beta]$ such
that $IX=J$ and $JY=I$ [15]. We showed that for similar hyperbolic $A,B\in{\rm
GL}_{n}({\mathbb{Z}})$ with an irreducible characteristic polynomial, if
$I_{A}$ and $I_{B}$ are weakly equivalent, then $A$ and $B$ are strongly BF-
equivalent over ${\mathcal{Q}}^{\rm a}$ [12]. Since $A$ is hyperbolic, each
$g_{k}(A)=A^{k}-I$ is invertible, and so $I_{A}$ and $I_{B}$ being weakly
equivalent implies that $A$ and $B$ are strongly ${\rm BF}$-equivalent over
${\mathcal{Q}}^{\rm p}$. We showed that for similar hyperbolic $A,B\in{\rm
GL}_{n}({\mathbb{Z}})$ with an irreducible characteristic polynomial, if $A$
and $B$ are profinitely conjugate over ${\mathcal{Q}}^{\rm p}$, then $I_{A}$
and $I_{B}$ are weakly equivalent [16]. Similar $A,B\in{\rm
GL}_{n}({\mathbb{Z}})$ (not assumed hyperbolic) with an irreducible
characteristic polynomial are called block conjugate if there exists
$A^{\prime},B^{\prime}\in{\rm GL}_{n}({\mathbb{Z}})$ and $M,N\in{\rm
GL}_{2n}({\mathbb{Z}})$ such that the block diagonal matrices $A\oplus
A^{\prime}$, $B\oplus B$, $A\oplus A$, and $B\oplus B^{\prime}$ satisfy

$(A\oplus A^{\prime})M=M(B\oplus B){\rm\ and\ }(A\oplus A)N=N(B\oplus
B^{\prime}).$

We showed that similar $A,B\in{\rm GL}_{n}({\mathbb{Z}})$ with irreducible
characteristic polynomial are block conjugate if and only if the associated
ideals $I_{A}$ and $I_{B}$ are weakly equivalent [17]. Applying this to
similar hyperbolic $A,B\in{\rm GL}_{n}({\mathbb{Z}})$ with irreducible
characteristic polynomial gives $I_{A}$ and $I_{B}$ are weakly equivalent if
and only if $A$ and $B$ are block conjugate. Putting these results together
gives that the following are equivalent for irreducible similar hyperbolic
$A,B\in{\rm GL}_{n}({\mathbb{Z}})$:

  1. (a)

$I_{A}$ and $I_{B}$ are weakly equivalent,

  2. (b)

$A$ and $B$ are block conjugate

  3. (c)

$A$ and $B$ are profinitely conjugate over ${\mathcal{Q}}^{\rm p}$,

  4. (d)

$A$ and $B$ are strongly ${\rm BF}$-equivalent over ${\mathcal{Q}}^{\rm p}$.

This establishes the principal generalized Bowen-Franks $R$-modules as a set
of complete invariants for the equivalence relations listed in items (a), (b),
and (c) when $A$ and $B$ are irreducible similar hyperbolic toral
automorphisms.

The structure of this paper is as follows. Section 2 states some basic
definitions and theory for profinite groups, as found in [18], that will be
used in the proof of the main result. In particular profinite topologies,
profinite completions, and cofinality are detailed. Section 3 applies the
results from Section 2 to profinite completions of ${\mathbb{Z}}^{n}$ with
particular attention to the full profinite topology of ${\mathbb{Z}}^{n}$ and
its accompanying completion. In Section 4 we prove for ${\mathcal{Q}}$ being
either ${\mathcal{Q}}^{p}$ or ${\mathcal{Q}}^{a}$ the main result that
profinite conjugacy over ${\mathcal{Q}}$ implies strong ${\rm BF}$-equivalence
over ${\mathcal{Q}}$. Prior to giving the proof in this section, the
construction of the profinite $R$-modules $G_{{\mathcal{Q}},A}$ for
${\mathcal{Q}}={\mathcal{Q}}^{\rm p}$ and ${\mathcal{Q}}^{\rm a}$ is given. We
already have a proof of the main result over ${\mathcal{Q}}^{p}$ for
irreducible similar hyperbolic toral automorphisms, as this follows from the
implications (c) $\Rightarrow$ (a) $\Rightarrow$ (d) stated in the previous
paragraph. The proof presented in Section 4 does not assume irreducibility and
does not use weak equivalence. In Section 5 we present some properties of
profinite conjugacies for similar hyperbolic toral automorphisms. We correct a
characterization from [12] of when profinitely conjugate similar hyperbolic
toral automorphisms are conjugate. We also show how profinite conjugacy
realizes the conjugacy invariant of cyclicity when applied to hyperbolic toral
automorphisms.

##  2\. Profinite Theory

###  2.1. Profinite Topologies

A profinite topology on an abelian group $G$ is a translation invariant
topology determined by selecting a neighbourhood basis of the identity element
$0$ of $G$ in the following way. Let ${\mathcal{N}}$ be a nonempty collection
of finite index subgroups of $G$ that is filtered from below, i.e., for any
$N_{1},N_{2}\in{\mathcal{N}}$ there exists $N_{3}\in{\mathcal{N}}$ such that
$N_{3}\subset N_{1}\cap N_{2}$. The filtered from below property makes
${\mathcal{N}}$ into a directed poset for the binary relation $\preceq$ on
${\mathcal{N}}$ defined by $N_{1}\preceq N_{2}$ if and only if $N_{2}\subset
N_{1}$. The “directed” part is that for any $N_{1},N_{2}\in{\mathcal{N}}$
there exists $N_{3}\in{\mathcal{N}}$ such that $N_{1}\preceq N_{3}$ and
$N_{2}\preceq N_{3}$. Viewing ${\mathcal{N}}$ as a fundamental system of
neighbourhoods (which is the same as a neighbourhood basis) of the identity
element $0$ of $G$ determines a profinite topology on $G$ (see [19]).

For a nonempty collection ${\mathcal{N}}$ of filtered from below finite index
subgroups of $G$, the type of profinite topology on $G$ determined by
${\mathcal{N}}$ is described by the class ${\mathcal{C}}$ of finite abelian
groups to which all the quotients $G/N$, $N\in{\mathcal{N}}$, belong. A
specified class ${\mathcal{C}}$ of finite abelian groups always contains all
isomorphic copies of the groups in it. The simplest structure on a class
${\mathcal{C}}$ is that of a formation. A class ${\mathcal{C}}$ of finite
abelian groups is called a formation if it satisfies the following two
properties: (1) if $K\in{\mathcal{C}}$ and $H\leq K$, then
$K/H\in{\mathcal{C}}$, and (2) if $H_{j}\in{\mathcal{C}}$ for $j=1,\dots,r$
and $K$ is a subgroup of the direct product $H_{1}\times\cdots\times H_{r}$
such that for the canonical projections (epimorphisms)

$q_{j}:H_{1}\times\cdots\times H_{j}\times\cdots\times H_{r}\to H_{j},$

the restrictions $q_{j}|_{K}:K\to H_{j}$ are surjections for all
$j=1,\dots,r$, then $K\in{\mathcal{C}}$. The first property is the closure of
${\mathcal{C}}$ under taking quotients. The second property is the closure of
${\mathcal{C}}$ under taking subdirect products. The second property implies
the closure of ${\mathcal{C}}$ under taking direct products. The class
${\mathcal{C}}$ of all finite abelian groups is readily verified as a
formation.

For a formation ${\mathcal{C}}$ of finite abelian groups, a
pro-${\mathcal{C}}$ topology on $G$ is a profinite topology determined by a
nonempty collection ${\mathcal{N}}$ of filtered from below finite indexed
subgroups of $G$ that satisfies $G/N\in{\mathcal{C}}$ for all
$N\in{\mathcal{N}}$. The full pro-${\mathcal{C}}$ topology on $G$ is the
pro-${\mathcal{C}}$ topology determined by the nonempty filtered from below
collection

${\mathcal{N}}_{\mathcal{C}}(G)=\\{N\leq_{f}G:G/N\in{\mathcal{C}}\\},$

where $\leq_{f}$ means finite index subgroup.

###  2.2. Profinite Completions

A profinite completion of $G$ is induced by a profinite topology on $G$. Let
${\mathcal{N}}$ be a nonempty collection of finite index subgroups of $G$ that
is filtered from below. For each $N\in{\mathcal{N}}$ equip the finite abelian
quotient group $G/N$ with the discrete topology. Each $G/N$ is a compact,
Hausdorff, totally disconnected topological group. For $N,M\in{\mathcal{N}}$
satisfying $M\preceq N$ and $M\neq N$, the natural epimorphism

$\varphi_{NM}:G/N\to G/M{\rm\ given\ by\ }x+N\mapsto x+M$

is continuous. Denote the coset $x+N$ of $G/N$ by $[x]_{N}$. For each
$N\in{\mathcal{N}}$ we set $\varphi_{NN}$ to be the identity map from $G/N$ to
$G/N$. For all $N,M,P\in{\mathcal{N}}$ satisfying $N\preceq M\preceq P$ there
holds $\varphi_{PN}=\varphi_{MN}\varphi_{PM}$. The continuous epimorphisms
$\varphi_{NM}$ for $M\preceq N$ are called transitions maps. The collection
$\\{G/N,\varphi_{NM},{\mathcal{N}}\\}$ is a surjective inverse system of
topological finite abelian groups. The profinite completion of $G$ with
respect to the profinite topology on $G$ determined by ${\mathcal{N}}$ is the
inverse limit of the surjective inverse system,

${\mathcal{K}}_{\mathcal{N}}(G)=\varprojlim_{N\in{\mathcal{N}}}G/N,$

whose elements are the coherent tuples $\\{[x_{N}]_{N}\\}_{N\in{\mathcal{N}}}$
in $\prod_{N\in{\mathcal{N}}}G/N$, i.e., those tuples satisfying

$[x_{M}]_{M}=\varphi_{NM}([x_{N}]_{N})=[x_{N}]_{M}$

whenever $M\preceq N$. The profinite completion
${\mathcal{K}}_{\mathcal{N}}(G)$ is a nonempty, compact, Hausdorff, totally
disconnected topological abelian group. The profinite completion
${\mathcal{K}}_{\mathcal{N}}(G)$ is called a profinite abelian group.

Several natural continuous homomorphisms are associated to the profinite
abelian group ${\mathcal{K}}_{\mathcal{N}}(G)$. There are the continuous
epimorphisms

$\varphi_{N}:{\mathcal{K}}_{\mathcal{N}}(G)\to G/N,\ \
\varphi_{N}([x_{M}]_{M\in{\mathcal{N}}})=[x_{N}],\ N\in{\mathcal{N}}.$

The continuous epimorphisms $\iota_{N}:G\to G/N$, $N\in{\mathcal{N}}$, induce
the continuous homomorphism
$\iota_{\mathcal{N}}:G\to{\mathcal{K}}_{\mathcal{N}}(G)$ given by

$\iota_{\mathcal{N}}(x)=\\{[x]_{N}\\}_{N\in{\mathcal{N}}},$

which take each $x\in G$ to the constant tuple
$\\{[x]_{N}\\}_{N\in{\mathcal{N}}}$ in ${\mathcal{K}}_{\mathcal{N}}(G)$. The
map $\iota_{\mathcal{N}}$ has the property that $\iota_{\mathcal{N}}(G)$ is
dense in ${\mathcal{K}}_{\mathcal{N}}(G)$. Furthermore $\iota_{\mathcal{N}}$
is a monomorphism if and only if

$\bigcap_{N\in{\mathcal{N}}}N=\\{0\\}.$

For a fixed formation ${\mathcal{C}}$ of finite abelian groups, a
pro-${\mathcal{C}}$ completion of $G$ is the profinite abelian group
${\mathcal{K}}_{\mathcal{N}}(G)$ when $G/N\in{\mathcal{C}}$ for all
$N\in{\mathcal{N}}$. The profinite completion of $G$ induced by the full
profinite topology determined by ${\mathcal{N}}_{\mathcal{C}}(G)$ will be
denoted by ${\mathcal{K}}_{\mathcal{C}}(G)$ instead of the more cumbersome
${\mathcal{K}}_{{\mathcal{N}}_{\mathcal{C}}(G)}(G)$. The continuous
homomorphism
$\iota_{{\mathcal{N}}_{\mathcal{C}}}(G):G\to{\mathcal{K}}_{\mathcal{C}}(G)$
will be denoted by the simpler $\iota_{\mathcal{C}}(G)$ as well. The profinite
abelian group ${\mathcal{K}}_{\mathcal{C}}(G)$ is called residually
${\mathcal{C}}$ if

$\bigcap_{N\in{\mathcal{N}}_{\mathcal{C}}(G)}N=\\{0\\}.$

This condition is equivalent to the continuous homomorphism
$\iota_{\mathcal{C}}$ being a monomorphism.

We need a few additional notations to state the following result that relates
the full pro-${\mathcal{C}}$ topology on $G$ with the topology on the
profinite completion ${\mathcal{K}}_{\mathcal{C}}(G)$. The notation
$\\{U:U\leq_{o}G;{\mathcal{N}}\\}$ means the collection of all open subgroups
$U$ of $G$ with respect to the profinite topology determined by a nonempty
filtered from below collection ${\mathcal{N}}$ of finite index subgroups of
$G$. For groups $H$ and $K$ with $H\leq K$ the notation $[K:H]$ means the
index of $H$ in $K$. The notation $H\leq_{o}G$ means $H$ is an open subgroup
of a topological group $G$. The following result (Proposition 3.2.2 in [18])
is presented in the abelian category, and will be used to prove the main
result.

######  Theorem 2.1.

Let ${\mathcal{C}}$ be a formation of finite abelian groups and suppose $G$ is
a residually ${\mathcal{C}}$ group. Identify $G$ with its embedded image
$\iota_{\mathcal{C}}(G)$ in its pro-${\mathcal{C}}$ completion
${\mathcal{K}}_{\mathcal{C}}(G)$. Let $\overline{X}$ denote the closure in
${\mathcal{K}}_{\mathcal{C}}(G)$ of a subset of $X$ in $G$.

  1. (a)

Let

$\Upsilon:\\{U:U\leq_{o}G;{\mathcal{N}}_{\mathcal{C}}(G)\\}\to\\{V:V\leq_{o}{\mathcal{K}}_{\mathcal{C}}(G)\\}$

be the mapping that assigns to each open subgroup $H$ in $G$ $($with respect
to the full profinite topology$)$ the closure $\Upsilon(H)=\overline{H}$ in
${\mathcal{K}}_{\mathcal{C}}(G)$. Then $\Upsilon$ is a one-to-one
correspondence between the set of all open subgroups in the full
pro-${\mathcal{C}}$ topology of $G$ and the set of all open subgroups of
${\mathcal{K}}_{\mathcal{C}}(G)$. The inverse of $\Upsilon$ is

$V\to V\cap\iota_{\mathcal{C}}(G),$

where in particular $\overline{V\cap\iota_{\mathcal{C}}(G)}=V$ when
$V\leq_{o}{\mathcal{K}}_{\mathcal{C}}(G)$.

  2. (b)

The topology of ${\mathcal{K}}_{\mathcal{C}}(G)$ induces on $G$ its full
pro-${\mathcal{C}}$ topology via the map $V\to V\cap\iota_{\mathcal{C}}(G)$
and the identification of $G$ with $\iota_{\mathcal{C}}(G)$.

  3. (c)

If $H,K\in\\{U:U\leq_{o}G;{\mathcal{N}}_{\mathcal{C}}(G)\\}$ and $H\leq K$,
then $K/H\cong\overline{K}/\overline{H}$ which implies
$[K:H]=[\overline{K}:\overline{H}]$.

###  2.3. Cofinality

Different choices of fundamental systems of neighbourhoods of the identity
element may lead to topologically isomorphic profinite completions, where by
topological isomorphism is meant an isomorphism that is also a homeomorphism.
This can happen because of cofinality. Suppose ${\mathcal{N}}$ and
${\mathcal{M}}$ are nonempty filtered from below collections of finite index
subgroups of $G$. The collection ${\mathcal{M}}$ is cofinal in ${\mathcal{N}}$
if ${\mathcal{M}}\subset{\mathcal{N}}$ and for every $N\in{\mathcal{N}}$ there
exists $M\in{\mathcal{M}}$ such that $N\preceq M$, i.e., $M\subset N$. The
following result (Lemma 1.1.9 in [18]) is presented in the notation of
Subsection 2.2.

######  Proposition 2.2.

For nonempty collections ${\mathcal{N}}$ and ${\mathcal{M}}$ of finite index
subgroups of $G$ filtered from below, if ${\mathcal{M}}$ is cofinal in
${\mathcal{N}}$, then there exists a topological isomorphism
$\Theta_{\mathcal{NM}}:{\mathcal{K}}_{\mathcal{N}}(G)\to{\mathcal{K}}_{\mathcal{M}}(G)$,
given explicitly by

$\Theta_{\mathcal{NM}}(\\{[x_{N}]_{N}\\}_{N\in{\mathcal{N}}})=\\{[x_{N}]_{N}\\}_{N\in{\mathcal{M}}},$

that satisfies $\Theta_{\mathcal{NM}}\iota_{\mathcal{N}}=\iota_{\mathcal{M}}$.

The topological isomorphism $\Theta_{{\mathcal{N}}{\mathcal{M}}}$ in
Proposition 2.2 is the inverse of the map constructed in Lemma 1.1.9 in [18];
the map $\Theta_{{\mathcal{N}}{\mathcal{M}}}$ “forgets” those $[x_{N}]_{N}$ in
the coherent tuple $\\{[x_{N}]_{N}\\}_{N\in{{\mathcal{N}}}}$ for
$N\in{\mathcal{N}}\setminus{\mathcal{M}}$. A direct consequence of Proposition
2.2 is that the injectivity of the homomorphisms $\iota_{N}$ and $\iota_{M}$
are related when ${\mathcal{M}}$ is cofinal in ${\mathcal{N}}$.

######  Corollary 2.3.

For nonempty collections ${\mathcal{N}}$ and ${\mathcal{M}}$ of finite index
subgroups of $G$ filtered from below, suppose ${\mathcal{M}}$ is cofinal in
${\mathcal{N}}$. Then $\iota_{\mathcal{M}}$ is a monomorphism if and only if
$\iota_{\mathcal{N}}$ is a monomorphism.

A pro-${\mathcal{C}}$ topology on $G$ determined by ${\mathcal{N}}$ is the
full pro-${\mathcal{C}}$ topology whenever ${\mathcal{N}}$ is cofinal in
${\mathcal{N}}_{\mathcal{C}}(G)$. This follows from Proposition 2.2 and part
(b) of Theorem 2.1. We present a proof of this that doesn’t pass through the
profinite completions.

######  Proposition 2.4.

Let ${\mathcal{C}}$ be a formation of finite abelian groups. If
${\mathcal{N}}$ is cofinal in ${\mathcal{N}}_{\mathcal{C}}(G)$, then the
pro-${\mathcal{C}}$ topology on $G$ determined by ${\mathcal{N}}$ is the full
pro-${\mathcal{C}}$ topology on $G$.

###### Proof.

By translation invariance it suffices to show for a subset $U$ containing $0$
that $U$ is open with respect to the pro-${\mathcal{C}}$ topology determined
by ${\mathcal{N}}$ if and only if $U$ is open with respect to the full
pro-${\mathcal{C}}$ topology.

Suppose $U$ is an open subset of $G$ containing $0$ with respect to the
pro-${\mathcal{C}}$ topology determined by ${\mathcal{N}}$. For each $x\in U$
there exists $N_{x}\in{\mathcal{N}}$ such that $x\in x+N_{x}\subset U$, where
$x+N_{x}$ is the translation of $N_{x}$ by $x$. Since ${\mathcal{N}}$ is
cofinal in ${\mathcal{N}}_{\mathcal{C}}(G)$, we have
${\mathcal{N}}\subset{\mathcal{N}}_{\mathcal{C}}(G)$, so that
$N_{x}\in{\mathcal{N}}_{\mathcal{C}}(G)$ for all $x\in U$. Thus for every
$x\in U$ there exists $N_{x}\in{\mathcal{N}}_{\mathcal{C}}(G)$ such that $x\in
x+N_{x}\subset U$, and thus $U$ is open in the full pro-${\mathcal{C}}$
topology.

Now suppose that $U$ is an open subset of $G$ containing $0$ with respect to
the full pro-${\mathcal{C}}$ topology. For each $x\in U$ there exist
$F_{x}\in{\mathcal{N}}_{\mathcal{C}}(G)$ such that $x\in x+F_{x}\subset U$.
Since ${\mathcal{N}}$ is cofinal in ${\mathcal{N}}_{\mathcal{C}}(G)$ there
exist $N_{x}\in{\mathcal{N}}$ such that $0\in N_{x}\subset F_{x}$. This
implies that $x\in x+N_{x}\subset x+F_{x}\subset U$. Thus for every $x\in U$
there exist $N_{x}\in{\mathcal{N}}$ such that $x\in x+N_{x}\subset U$, and
thus $U$ is open in the pro-${\mathcal{C}}$ topology determined by
${\mathcal{N}}$. ∎

######  Remark 1.

The hypotheses for Theorem 2.1 are that $G$ is residually ${\mathcal{C}}$,
i.e., the map $\iota_{\mathcal{C}}:G\to{\mathcal{K}}_{\mathcal{C}}(G)$ is a
monomorphism, and that the pro-${\mathcal{C}}$ topology on $G$ is the full
pro-${\mathcal{C}}$ topology, i.e., determined by
${\mathcal{N}}_{\mathcal{C}}(G)$. For a pro-${\mathcal{C}}$ completion
${\mathcal{K}}_{\mathcal{N}}(G)$, if ${\mathcal{N}}$ is cofinal in
${\mathcal{N}}_{\mathcal{C}}(G)$, then it follows from Proposition 2.4 that
the pro-${\mathcal{C}}$ topology determined on $G$ by ${\mathcal{N}}$ is the
full pro-${\mathcal{C}}$ topology, and it follows from Corollary 2.3 that
$\iota_{\mathcal{N}}$ is a monomorphism, i.e., $G$ is residually
${\mathcal{C}}$, hence we can apply Theorem 2.1 with
${\mathcal{K}}_{\mathcal{C}}(G)$ replaced by ${\mathcal{K}}_{\mathcal{N}}(G)$,
${\mathcal{N}}_{\mathcal{C}}(G)$ replaced with ${\mathcal{N}}$, and
$\iota_{\mathcal{C}}$ replaced with $\iota_{\mathcal{N}}$.

##  3\. The Profinite Completion of ${\mathbb{Z}}^{n}$

###  3.1. The Abelian Group ${\mathbb{Z}}^{n}$

We set $G$ to be free abelian group ${\mathbb{Z}}^{n}$ of rank $n\geq 2$. We
represent the elements of ${\mathbb{Z}}^{n}$ as row vectors in terms of the
standard basis $\\{e_{1},e_{2},\dots,e_{n}\\}$ where $e_{1}=(1,0,\dots,0)$,
$e_{2}=(0,1,\dots,0)$, and $e_{n}=(0,0,\dots,1)$, so that each
$x\in{\mathbb{Z}}^{n}$ is expressed as

$x=x_{1}e_{1}+x_{2}e_{2}+\cdots+x_{n}e_{n}$

for $x_{1},x_{2},\dots,x_{n}\in{\mathbb{Z}}$.

###  3.2. The Full Pro-${\mathcal{C}}$ Topology

For ${\mathcal{C}}$ fixed as the formation of all finite abelian groups, the
full pro-${\mathcal{C}}$ topology on ${\mathbb{Z}}^{n}$ is determined by the
nonempty collection of all finite index subgroups of ${\mathbb{Z}}^{n}$,

${\mathcal{N}}_{\mathcal{C}}({\mathbb{Z}}^{n})=\\{N\leq_{f}{\mathbb{Z}}^{n}:{\mathbb{Z}}^{n}/N\in{\mathcal{C}}\\}=\\{N:N\leq_{f}{\mathbb{Z}}^{n}\\}.$

To simplify notation we denote ${\mathcal{N}}_{\mathcal{C}}({\mathbb{Z}}^{n})$
by ${\mathcal{F}}$. The profinite completion of ${\mathbb{Z}}^{n}$ induced by
${\mathcal{F}}$ will be denoted by
${\mathcal{K}}_{\mathcal{F}}({\mathbb{Z}}^{n})$ and the continuous
homomorphism
${\mathbb{Z}}^{n}\to{\mathcal{K}}_{\mathcal{F}}({\mathbb{Z}}^{n})$ will be
denoted by $\iota_{\mathcal{F}}$.

To show that ${\mathbb{Z}}^{n}$ is residually ${\mathcal{C}}$ we use the
collection

${\mathcal{D}}=\\{d{\mathbb{Z}}^{n}:d\in{\mathbb{N}}\\}$

and Corollary 2.3. For each $d\in{\mathbb{N}}$ the subgroup
$d{\mathbb{Z}}^{n}$ has the finite index
$[{\mathbb{Z}}^{n}:d{\mathbb{Z}}^{n}]=d^{n}$. The collection ${\mathcal{D}}$
is filtered from below because for $d_{1},d_{2}\in{\mathbb{Z}}$ the choice of
$d_{3}={\rm lcm}(d_{1},d_{2})$ satisfies

$d_{3}{\mathbb{Z}}^{n}\subset d_{1}{\mathbb{Z}}^{n}\cap
d_{2}{\mathbb{Z}}^{n},$

i.e., for $(d_{3}m_{1},d_{3}m_{2},\dots,d_{3}m_{n})\in d_{3}{\mathbb{Z}}^{n}$
the relations $d_{1}\mid d_{3}$ and $d_{2}\mid d_{3}$ imply that $d_{1}\mid
d_{3}m_{i}$ and $d_{2}\mid d_{3}m_{i}$ for all $i=1,2,\dots,n$. In proving
that ${\mathcal{D}}$ is cofinal in ${\mathcal{F}}$ we make use of the result
that states for a free abelian group $F$ of finite rank $n$ and a subgroup $H$
of rank $n$ there exists a basis $\\{x_{1},x_{2},\dots,x_{n}\\}$ of $F$ and
positive integers $d_{1},d_{2},\dots,d_{n}$ satisfying $d_{1}\mid
d_{2}\mid\cdots\mid d_{n}$ such that
$\\{d_{1}x_{1},d_{2}x_{2},\dots,d_{n}x_{n}\\}$ is a basis for the subgroup $H$
of $F$ (see Theorem 1.6 in [20]).

######  Lemma 3.1.

For each $n\geq 2$ the collection ${\mathcal{D}}$ is cofinal in
${\mathcal{F}}$.

###### Proof.

Clearly ${\mathcal{D}}\subset{\mathcal{F}}$. Let $N\in{\mathcal{F}}$. Since
$N\leq_{f}{\mathbb{Z}}^{n}$, there exists a basis
$\xi_{1},\xi_{2},\dots,\xi_{n}$ of ${\mathbb{Z}}^{n}$ and positive integers
$d_{1},d_{2},\dots,d_{n}$ such that $d_{1}\mid d_{2}\mid\cdots\mid d_{n}$ and

$d_{1}\xi_{1},d_{2}\xi_{2},\dots,d_{n}\xi_{n}$

is a basis of $N$ . Since $d_{1}\mid d_{2}\mid\cdots\mid d_{n}$, all of the
elements $d_{n}\xi_{1},d_{n}\xi_{2},\dots,d_{n}\xi_{n}$ belong to $N$. Set
$d=d_{n}$. Since $\xi_{1},\xi_{2},\dots,\xi_{n}$ is a basis for
${\mathbb{Z}}^{n}$, we have

$d(m_{1}\xi_{1}+m_{2}\xi_{2}+\cdots+m_{n}\xi_{n})\in N$

for all $m_{1},m_{2},\dots,m_{n}\in{\mathbb{Z}}$. This implies that
$d{\mathbb{Z}}^{n}\subset N$ where $d{\mathbb{Z}}^{n}\in{\mathcal{D}}$. Thus
for every $N\in{\mathcal{F}}$ there exists $d{\mathbb{Z}}^{n}\in{\mathcal{D}}$
such that $N\preceq d{\mathbb{Z}}^{n}$, making ${\mathcal{D}}$ cofinal in
${\mathcal{F}}$. ∎

By Lemma 3.1 and Proposition 2.4, the cofinal collection ${\mathcal{D}}$
induces the full pro-${\mathcal{C}}$ topology on ${\mathbb{Z}}^{n}$.
Furthermore, by Proposition 2.2 the pro-${\mathcal{C}}$ completion
${\mathcal{K}}_{\mathcal{D}}({\mathbb{Z}}^{n})$ is topologically isomorphic to
${\mathcal{K}}_{\mathcal{C}}({\mathbb{Z}}^{n})$.

######  Lemma 3.2.

For each $n\geq 2$, the map
$\iota_{{\mathcal{D}}}:{\mathbb{Z}}^{n}\to{\mathcal{K}}_{{\mathcal{D}}}({\mathbb{Z}}^{n})$
is a monomorphism.

###### Proof.

Suppose

$m\in\bigcap_{d\in{\mathbb{N}}}d{\mathbb{Z}}^{n}.$

Then $m=(m_{1},\dots,m_{n})\in d{\mathbb{Z}}^{n}$ for all $d\in{\mathbb{N}}$.
In particular each $m_{i}$ is divisible by every $d\in{\mathbb{N}}$. If
$m_{i}\neq 0$, there is $d>|m_{i}|+1$ for which $m_{i}$ is not divisible by
$d$. This means $m_{i}=0$, implying that $m=0$. ∎

######  Proposition 3.3.

For each $n\geq 2$, the map
$\iota_{{\mathcal{F}}}:{\mathbb{Z}}^{n}\to{\mathcal{K}}_{{\mathcal{F}}}({\mathbb{Z}}^{n})$
is a monomorphism.

###### Proof.

Since ${\mathcal{D}}$ is cofinal in ${\mathcal{F}}$ by Lemma 3.1 and
$\iota_{\mathcal{D}}$ is a monomorphism by Lemma 3.2, it follows by Corollary
2.3 that $\iota_{\mathcal{F}}$ is a monomorphism. ∎

######  Corollary 3.4.

The pro-${\mathcal{C}}$ group ${\mathcal{K}}_{\mathcal{F}}({\mathbb{Z}}^{n})$
is residually ${\mathcal{C}}$.

##  4\. Strong BF-Equivalence

###  4.1. Dynamically Defined Pro-${\mathcal{C}}$ Topologies

For similar hyperbolic $A,B\in{\rm GL}_{n}({\mathbb{Z}})$, all of the
generalized Bowen-Franks $R$-modules, viewed as groups, naturally correspond
to a pair of pro-${\mathcal{C}}$ topologies on ${\mathbb{Z}}^{n}$. This pair
of pro-${\mathcal{C}}$ topologies is parameterized by the collection of
polynomials

${\mathcal{Q}}^{\rm a}=\\{g\in{\mathbb{Z}}[t]:{\rm det}(g(A))\neq 0\\}.$

The invertibility of $g(B)$ follows from the invertibility of $g(A)$ and the
similarity of $A$ with $B$, and so the invertibility of $g(B)$ is not
explicitly stated in the definition of ${\mathcal{Q}}^{\rm a}$. Set
$N_{g,A}={\mathbb{Z}}^{n}g(A)$, $N_{g,B}={\mathbb{Z}}^{n}g(B)$,
${\mathcal{N}}_{{\mathcal{Q}}^{\rm a},A}=\\{N_{g,A}:g\in{\mathcal{Q}}^{\rm
a}\\}$, and ${\mathcal{N}}_{{\mathcal{Q}}^{\rm
a},B}=\\{N_{g,B}:g\in{\mathcal{Q}}^{\rm a}\\}$.

######  Lemma 4.1.

For similar hyperbolic $A,B\in{\rm GL}_{n}({\mathbb{Z}})$, the collections
${\mathcal{N}}_{{\mathcal{Q}}^{\rm a},A}$ and
${\mathcal{N}}_{{\mathcal{Q}}^{\rm a},B}$ are pro-${\mathcal{C}}$ topologies
on ${\mathbb{Z}}^{n}$.

###### Proof.

For each $g\in{\mathcal{Q}}^{\rm a}$, the invertibility of $g(A),g(B)\in
M_{n}({\mathbb{Z}})$ implies by their Smith normal form that $N_{g,A}$ and
$N_{g,B}$ are finite index subgroups of ${\mathbb{Z}}^{n}$. The filtered from
below property follows from ${\mathcal{Q}}^{\rm a}$ being a multiplicative
set, i.e., closed under multiplication. For $g_{1},g_{2}\in{\mathcal{Q}}^{\rm
a}$ there holds $g_{1},g_{2}\in{\mathbb{Z}}[t]$ with $g_{1}(A)$ and $g_{2}(A)$
invertible, so that setting $g_{3}=g_{1}g_{2}$ there holds
$g_{3}\in{\mathbb{Z}}[t]$ with $g_{3}(A)$ invertible, hence
$g_{3}\in{\mathcal{Q}}^{\rm a}$. If $g_{1},g_{2}\in{\mathcal{Q}}^{\rm a}$ then
for $g_{3}=g_{1}g_{2}\in{\mathcal{Q}}^{\rm a}$ and
$m\in{\mathbb{Z}}^{n}g_{3}(A)$ there is $z\in{\mathbb{Z}}^{n}$ such that
$m=zg_{3}(A)$, hence there holds $m=(zg_{2}(A))g_{1}(A)$ and
$m=(zg_{1}(A))g_{2}(A)$, implying that $N_{g_{3},A}\subset N_{g_{1},A}\cap
N_{g_{2},A}$. A similar argument gives $N_{g_{3},B}\subset N_{g_{1},B}\cap
N_{g_{2},B}$. ∎

For similar hyperbolic $A,B\in{\rm GL}_{n}({\mathbb{Z}})$, the principal
generalized Bowen-Franks $R$-modules, viewed as groups, naturally correspond
to another pair of pro-${\mathcal{C}}$ topologies on ${\mathbb{Z}}^{n}$. This
pair of pro-${\mathcal{C}}$ topologies is parameterized by the collection of
polynomials

${\mathcal{Q}}^{\rm p}=\\{g_{k}(t)=t^{k}-1:k\in{\mathbb{N}}\\}.$

To simplify notation, set $N_{k,A}=N_{g_{k},A}$ and $N_{k,B}=N_{g_{k},B}$.
Then ${\mathcal{N}}_{{\mathcal{Q}}^{\rm p},A}=\\{N_{k,A}:k\in{\mathbb{N}}\\}$
and ${\mathcal{N}}_{{\mathcal{Q}}^{\rm p},B}=\\{N_{k,B}:k\in{\mathbb{N}}\\}$.

######  Lemma 4.2.

For similar hyperbolic $A,B\in{\rm GL}_{n}({\mathbb{Z}})$, there holds
${\mathcal{Q}}^{\rm p}\subset{\mathcal{Q}}^{\rm a}$, and the collections
${\mathcal{N}}_{{\mathcal{Q}}^{\rm p},A}$ and
${\mathcal{N}}_{{\mathcal{Q}}^{\rm p},B}$ are pro-${\mathcal{C}}$ topologies
on ${\mathbb{Z}}^{n}$.

###### Proof.

For each $k\in{\mathbb{N}}$ there holds $g_{k}\in{\mathbb{Z}}[t]$ and the
hyperbolicity of $A$ implies that $g_{k}(A)$ is invertible. Thus
${\mathcal{Q}}^{\rm p}\subset{\mathcal{Q}}^{\rm a}$. For each
$k\in{\mathbb{N}}$, the similarity of $A$ and $B$ implies the invertibility of
$g_{k}(B)$. By the Smith normal forms of $g(A)$ and $g(B)$, the subgroups
$N_{k,A}$ and $N_{k,B}$ have finite index in ${\mathbb{Z}}^{n}$. The filtered
from below property for ${\mathcal{N}}_{{\mathcal{Q}}^{\rm p},A}$ follows by
(Lemma 2.2 in [12]) wherein we showed that for $k_{1},k_{2}\in{\mathbb{N}}$,
the choice of $k_{3}=k_{1}k_{2}$ leads to $g_{k_{3}}=q_{1}g_{k_{1}}$ and
$g_{k_{3}}=q_{2}g_{k_{2}}$ for $q_{1},q_{2}\in{\mathbb{Z}}[t]$, so that for
$m\in{\mathbb{Z}}^{n}g_{k_{3}}(A)$ there exists $z\in{\mathbb{Z}}^{n}$ such
that $m=zg_{k_{3}}(A)$, whence
$m=zq_{1}(A)g_{k_{1}}(A)\in{\mathbb{Z}}^{n}g_{k_{1}}(A)$ and
$m=zq_{2}(A)g_{k_{2}}(A)\in{\mathbb{Z}}^{n}g_{k_{2}}(A)$, implying that
$N_{k_{3},A}\subset N_{k_{1},A}\cap N_{k_{2},A}$. A similar argument gives
$N_{k_{3},B}\subset N_{k_{1},B}\cap N_{k_{2},B}$. ∎

For similar hyperbolic $A,B\in{\rm GL}_{n}({\mathbb{Z}})$, we show that
${\mathcal{N}}_{{\mathcal{Q}}^{\rm p},A}$ and
${\mathcal{N}}_{{\mathcal{Q}}^{\rm p},B}$ are each cofinal in ${\mathcal{F}}$,
that ${\mathcal{N}}_{{\mathcal{Q}}^{\rm p},A}$ is cofinal in
${\mathcal{N}}_{{\mathcal{Q}}^{\rm a},A}$ and
${\mathcal{N}}_{{\mathcal{Q}}^{\rm p},B}$ is cofinal in
${\mathcal{N}}_{{\mathcal{Q}}^{\rm a},B}$. We only state the results for $A$,
as those for $B$ are obtained by replacing $A$ with $B$.

######  Lemma 4.3.

Suppose $A\in{\rm GL}_{n}({\mathbb{Z}})$ is hyperbolic. For every
$d\in{\mathbb{N}}$ there exists $k\in{\mathbb{N}}$ such that $N_{k,A}\subset
d{\mathbb{Z}}^{n}$.

###### Proof.

For fixed but arbitrary $d\in{\mathbb{N}}$, consider the finite subset of
${\mathbb{T}}^{n}$ given by

$T_{d}=\\{x\in{\mathbb{T}}^{n}:dx=0\\}.$

This finite set consists entirely of periodic points of the left action of the
hyperbolic toral automorphism $A$ of ${\mathbb{T}}^{n}$. Let $k$ be the least
common multiple of the periods of $x\in T_{d}$ under the left action of $A$.
Then $(A^{k}-I)x=0$ for all $x\in T_{d}$, so that $T_{d}\subset{\rm
Per}_{k}(A)$. Each rational point of the form $(0,\dots,0,1/d,0,\dots,0)^{\rm
T}$ belongs to $T_{d}$ and satisfies

$(A^{k}-I)(0,\dots,0,1/d,0,\dots,0)^{\rm T}=0$

because $T_{d}\subset{\rm Per}_{k}(A)$. This implies that each entry of
$A^{k}-I$ is divisible by $d$. Thus ${\mathbb{Z}}^{n}(A^{k}-I)\subset
d{\mathbb{Z}}^{n}$, i.e., $N_{k,A}\subset d{\mathbb{Z}}^{n}$. ∎

######  Proposition 4.4.

Suppose $A\in{\rm GL}_{n}({\mathbb{Z}})$ is hyperbolic. Then
${\mathcal{N}}_{{\mathcal{Q}}^{\rm p},A}$ is cofinal in
${\mathcal{N}}_{{\mathcal{Q}}^{\rm a},A}$, and
${\mathcal{N}}_{{\mathcal{Q}}^{\rm p},A}$ is cofinal in ${\mathcal{F}}$.

###### Proof.

By Lemma 4.2 there holds ${\mathcal{Q}}^{\rm p}\subset{\mathcal{Q}}^{\rm a}$
which implies ${\mathcal{N}}_{{\mathcal{Q}}^{\rm
p},A}\subset{\mathcal{N}}_{{\mathcal{Q}}^{\rm a},A}$. Furthermore, by Lemma
4.2 there holds ${\mathcal{N}}_{{\mathcal{Q}}^{\rm p},A}\subset{\mathcal{F}}$.
For $N\in{\mathcal{N}}_{{\mathcal{Q}}^{\rm a},A}$ there holds
$N\leq_{f}{\mathbb{Z}}^{n}$ by Lemma 4.1. For $N\in{\mathcal{F}}$ there holds
$N\leq_{f}{\mathbb{Z}}^{n}$. In either case we have
$N\leq_{f}{\mathbb{Z}}^{n}$. By Lemma 3.1 there is a $d\in{\mathbb{N}}$ such
that $d{\mathbb{Z}}^{n}\subset N$. Associated to this $d\in{\mathbb{N}}$ there
is by Lemma 4.3 a $k\in{\mathbb{N}}$ such that $N_{k,A}\subset
d{\mathbb{Z}}^{n}$. Hence $N_{k,A}\subset d{\mathbb{Z}}^{n}\subset N$, i.e.,
$N\preceq N_{k,A}$. ∎

######  Corollary 4.5.

If $A\in{\rm GL}_{n}({\mathbb{Z}})$ is hyperbolic, then

${\mathcal{K}}_{{\mathcal{N}}_{{\mathcal{Q}}^{\rm
p},A}}({\mathbb{Z}}^{n})=\varprojlim_{k\in{\mathbb{N}}}{\mathbb{Z}}^{n}/N_{k,A}$

is topologically isomorphic to ${\mathcal{K}}_{\mathcal{F}}({\mathbb{Z}}^{n})$
and to

${\mathcal{K}}_{{\mathcal{N}}_{{\mathcal{Q}}^{\rm
a},A}}({\mathbb{Z}}^{n})=\varprojlim_{g\in{\mathcal{Q}}^{\rm
a}}{\mathbb{Z}}^{n}/N_{g,A}.$

###### Proof.

Apply Proposition 4.4 and Proposition 2.2. ∎

By Corollary 4.5, applied to similar hyperbolic $A,B\in{\rm
GL}_{n}({\mathbb{Z}})$, there are topological isomorphisms

${\mathcal{K}}_{{\mathcal{N}}_{{\mathcal{Q}}^{\rm
a},A}}({\mathbb{Z}}^{n})\cong{\mathcal{K}}_{{\mathcal{N}}_{{\mathcal{Q}}^{\rm
p},A}}({\mathbb{Z}}^{n})\cong{\mathcal{K}}_{\mathcal{F}}({\mathbb{Z}}^{n})\cong{\mathcal{K}}_{{\mathcal{N}}_{{\mathcal{Q}}^{\rm
p},B}}({\mathbb{Z}}^{n})\cong{\mathcal{K}}_{{\mathcal{N}}_{{\mathcal{Q}}^{\rm
a},B}}({\mathbb{Z}}^{n}).$

Thus, for similar hyperbolic $A,B\in{\rm GL}_{n}({\mathbb{Z}})$, the
pro-${\mathcal{C}}$ completions
${\mathcal{K}}_{{\mathcal{N}}_{{\mathcal{Q}}^{\rm a},A}}({\mathbb{Z}}^{n})$,
${\mathcal{K}}_{{\mathcal{N}}_{{\mathcal{Q}}^{\rm p},A}}({\mathbb{Z}}^{n})$,
${\mathcal{K}}_{{\mathcal{N}}_{{\mathcal{Q}}^{\rm p},B}}({\mathbb{Z}}^{n})$,
and ${\mathcal{K}}_{{\mathcal{N}}_{{\mathcal{Q}}^{\rm
a},B}}({\mathbb{Z}}^{n})$, are indistinguishable as topological abelian
groups. We will show that what does distinguish
${\mathcal{K}}_{{\mathcal{N}}_{{\mathcal{Q}}^{\rm a},A}}({\mathbb{Z}}^{n})$
and ${\mathcal{K}}_{{\mathcal{N}}_{{\mathcal{Q}}^{\rm
p},A}}({\mathbb{Z}}^{n})$ from
${\mathcal{K}}_{{\mathcal{N}}_{{\mathcal{Q}}^{\rm p},B}}({\mathbb{Z}}^{n})$
and ${\mathcal{K}}_{{\mathcal{N}}_{{\mathcal{Q}}^{\rm
a},B}}({\mathbb{Z}}^{n})$ is an $R$-module structure on these
pro-${\mathcal{C}}$ completions.

###  4.2. Module Structure

For similar hyperbolic $A,B\in{\rm GL}_{n}({\mathbb{Z}})$, there is a
commutative ring $R$ with identity to which both matrix rings
${\mathbb{Z}}[A]$ and ${\mathbb{Z}}[B]$ are isomorphic. When the common
characteristic polynomial $f(t)$ of $A$ and $B$ is irreducible the ring $R$ is
the integral domain given by quotient

$R={\mathbb{Z}}[t]/\big{(}f(t)\big{)}$

where $\big{(}f(t)\big{)}=f(t){\mathbb{Z}}[t]$ is the principal ideal
generated by $f(t)$. The map
$\vartheta_{A}:{\mathbb{Z}}[t]/\big{(}f(t)\big{)}\to{\mathbb{Z}}[A]$ given by
$q(t)+\big{(}f(t)\big{)}\to q(A)$, where

${\mathbb{Z}}[A]=\left\\{\sum_{i=0}^{n-1}c_{i}A^{i}:c_{i}\in{\mathbb{Z}}\right\\},$

is a ring isomorphism, and the similarly defined map
$\vartheta_{B}:{\mathbb{Z}}[t]/\big{(}f(t)\big{)}\to{\mathbb{Z}}[B]$ is also a
ring isomorphism. When $f(t)$ is irreducible, this gives ${\mathbb{Z}}[A]\cong
R\cong{\mathbb{Z}}[B]$ as rings. When $f(t)$ is reducible, the map
$\vartheta_{A}$ is a ring epimorphism, and so by the First Isomorphism Theorem
for Rings it induces a ring isomorphism

$\bar{\vartheta}_{A}:\big{(}{\mathbb{Z}}[t]/\big{(}f(t)\big{)}\big{)}/{\rm
ker}(\vartheta_{A})\to{\mathbb{Z}}[A].$

Similarly $\vartheta_{B}$ is a ring epimorphism, and so it induces a ring
isomorphism

$\bar{\vartheta}_{B}:\big{(}{\mathbb{Z}}[t]/\big{(}f(t)\big{)}\big{)}/{\rm
ker}\vartheta_{B}\to{\mathbb{Z}}[B].$

Similarity of $A$ and $B$ implies that ${\rm ker}(\vartheta_{A})={\rm
ker}(\vartheta_{B})$. Thus it follows that

${\mathbb{Z}}[A]\cong\big{(}{\mathbb{Z}}[t]/\big{(}f(t)\big{)}/{\rm
ker}(\vartheta_{A})=\big{(}{\mathbb{Z}}[t]/\big{(}f(t)\big{)}/{\rm
ker}(\vartheta_{B})\cong{\mathbb{Z}}[B].$

When $f(t)$ is reducible, the ring $R$ to which ${\mathbb{Z}}[A]$ and
${\mathbb{Z}}[B]$ are both isomorphic is
$\big{(}{\mathbb{Z}}[t]/\big{(}f(t)\big{)}/{\rm ker}(\vartheta_{A})$, which is
an integral domain when the minimal polynomial of the similar $A$ and $B$ is
irreducible, and is not an integral domain when the minimal polynomial of the
similar $A$ and $B$ is reducible. With ${\mathcal{Q}}={\mathcal{Q}}^{\rm p}$
or ${\mathcal{Q}}^{\rm a}$, when we refer to the $R$-module structure it is
understood that it is ${\mathbb{Z}}[A]$ on ${\mathcal{N}}_{{\mathcal{Q}},A}$,
${\mathcal{K}}_{{\mathcal{N}}_{{\mathcal{Q}},A}}({\mathbb{Z}}^{n})$, etc., and
it is ${\mathbb{Z}}[B]$ on ${\mathcal{N}}_{{\mathcal{Q}},B}$,
${\mathcal{K}}_{{{\mathcal{N}}_{\mathcal{Q}},B}}({\mathbb{Z}}^{n})$, etc.

For a hyperbolic $A\in{\rm GL}_{n}({\mathbb{Z}})$ and ${\mathcal{Q}}$ equal to
either ${\mathcal{Q}}^{\rm p}$ or ${\mathcal{Q}}^{\rm a}$, we describe the
$R$-module structure on the pro-${\mathcal{C}}$ topology
${\mathcal{N}}_{{\mathcal{Q}},A}=\\{N_{g,A}:g\in{\mathcal{Q}}\\}$, the
pro-${\mathcal{C}}$ completion
${\mathcal{K}}_{{\mathcal{N}}_{{\mathcal{Q}},A}}({\mathbb{Z}}^{n})$, and the
several natural homomorphisms associated to the pro-${\mathcal{C}}$
completion. (The $R$-module structure for hyperbolic $B$ similar to $A$ is
described in like manner.) The ring $R\cong{\mathbb{Z}}[A]$ acts on
${\mathbb{Z}}^{n}$ via the automorphism $m\to mA$ for $m\in{\mathbb{Z}}^{n}$,
making ${\mathbb{Z}}^{n}$ into a right ${\mathbb{Z}}[A]$-module. Since each
$N_{g,A}\in{\mathcal{N}}_{{\mathcal{Q}},A}$ is invariant under the right
action of $A$, the subgroup $N_{g,A}$ is a right ${\mathbb{Z}}[A]$-module.
Denoting the elements

$G_{g,A}={\mathbb{Z}}^{n}/N_{g,A}$

by $[m]_{g,A}=m+N_{g,A}$ for $m\in{\mathbb{Z}}^{n}$, the right
${\mathbb{Z}}[A]$-module structure on $G_{g,A}$ is induced by the topological
automorphism $A_{g}$ of $G_{g,A}$ defined by

$A_{g}([m]_{g,A})=[mA]_{g,A}.$

For $h\preceq g$ the transition maps are the continuous epimorphisms
$\varphi_{g,h,A}:G_{g,A}\to G_{h,A}$ defined by

$\varphi_{g,h,A}([m]_{g,A})=[m]_{h,A}$

which are ${\mathbb{Z}}[A]$-module epimorphisms. The profinite completion of
${\mathbb{Z}}^{n}$ with respect to the pro-${\mathcal{C}}$ topology determined
by ${\mathcal{N}}_{{\mathcal{Q}},A}$ we denote by

$G_{{\mathcal{Q}},A}=\varprojlim_{g\in{\mathcal{Q}}}G_{g,A}$

instead of ${\mathcal{K}}_{\mathcal{N}_{{\mathcal{Q}},A}}({\mathbb{Z}}^{n})$
for simplicity of notation. The elements of $G_{{\mathcal{Q}},A}$ consist of
those coherent tuples $\\{[m_{g}]_{g,A}\\}_{g\in{\mathcal{Q}}}$ lying in
$\prod_{g\in{\mathcal{Q}}}G_{g,A}$ (where $m_{g}\in{\mathbb{Z}}^{n}$ for each
$g\in{\mathcal{Q}}$), i.e., those tuples that satisfy

$[m_{h}]_{h,A}=\varphi_{g,h,A}([m_{g}]_{g,A})=[m_{g}]_{h,A}$

whenever $h\preceq g$. The right ${\mathbb{Z}}[A]$-module structure on
$G_{{\mathcal{Q}},A}$ is induced by the topological automorphism $\Gamma_{A}$
of $G_{{\mathcal{Q}},A}$ defined by

$\Gamma_{A}(\\{[m_{g}]_{g,A}\\}_{g\in{\mathcal{Q}}})=\\{A_{g}([m_{g}]_{g,A})\\}_{g\in{\mathcal{Q}}}.$

For each $g\in{\mathcal{Q}}$, the continuous epimorphism
$\varphi_{g,A}:G_{{\mathcal{Q}},A}\to G_{g,A}$ defined by

$\varphi_{g,A}(\\{[m_{h}]_{h,A}\\}_{h\in{\mathcal{Q}}})=[m_{g}]_{g,A}$

is a ${\mathbb{Z}}[A]$-module epimorphism. By $R$-module isomorphism theory,
each epimorphism $\varphi_{g,A}$ induces the canonical $R$-module isomorphism

$G_{{\mathcal{Q}},A}/{\rm ker}\varphi_{g,A}\cong_{R}G_{g,A}.$

The continuous homomorphism $\iota_{{\mathcal{Q}},A}:{\mathbb{Z}}^{n}\to
G_{{\mathcal{Q}},A}$ (we use $\iota_{{\mathcal{Q}},A}$ instead of
$\iota_{{\mathcal{N}}_{{\mathcal{Q}},A}}$ for simplicity of notation) defined
by

$\iota_{{\mathcal{Q}},A}(m)=\\{[m]_{g,A}\\}_{g\in{\mathcal{Q}}}$

is also a ${\mathbb{Z}}[A]$-module homomorphism. The map
$\iota_{{\mathcal{Q}},A}$ is a monomorphism for both
${\mathcal{Q}}={\mathcal{Q}}^{\rm p}$ and ${\mathcal{Q}}={\mathcal{Q}}^{\rm
a}$ by Corollary 2.3 because ${\mathcal{N}}_{{\mathcal{Q}}^{\rm p},A}$ is
cofinal in ${\mathcal{F}}$ and ${\mathcal{N}}_{{\mathcal{Q}}^{\rm p},A}$ is
cofinal in ${\mathcal{N}}_{{\mathcal{Q}}^{\rm a},A}$ by Proposition 4.4, and
$\iota_{\mathcal{F}}$ is a monomorphism by Proposition 3.3. We proved directly
that hyperbolicity of $A$ implies the intersection of $N_{k,A}$,
$k\in{\mathbb{N}}$, is $\\{0\\}$ (see Lemma 2.2 in [12]), hence, since
${\mathcal{N}}_{{\mathcal{Q}}^{\rm
p},A}\subset{\mathcal{N}}_{{\mathcal{Q}}^{\rm a},A}$ by Proposition 4.4, that

$\bigcap_{k\in{\mathbb{N}}}N_{k,A}=\\{0\\}\ \Rightarrow\
\bigcap_{g\in{\mathcal{Q}}^{\rm a}}N_{g,A}=\\{0\\},$

also showing that $\iota_{{\mathcal{Q}},A}$ is a monomorphism for both
${\mathcal{Q}}={\mathcal{Q}}^{\rm p}$ and ${\mathcal{Q}}={\mathcal{Q}}^{\rm
a}$.

###  4.3. Profinite Conjugacy

######  Definition 4.6.

For ${\mathcal{Q}}$ being either ${\mathcal{Q}}^{\rm p}$ or
${\mathcal{Q}}^{\rm a}$, two similar hyperbolic $A,B\in{\rm
GL}_{n}({\mathbb{Z}})$ are profinitely conjugate over ${\mathcal{Q}}$ if there
exists a topological $R$-module isomorphism
$\Psi_{\mathcal{Q}}:G_{{\mathcal{Q}},A}\to G_{{\mathcal{Q}},B}$.

Profinite conjugacy over ${\mathcal{Q}}$ for similar hyperbolic toral
automorphisms is an equivalence relation, as is readily shown.

######  Remark 2.

We showed that strong BF-equivalence over ${\mathcal{Q}}^{\rm p}$ implies that
$G_{{\mathcal{Q}}^{\rm p},A}$ and $G_{{\mathcal{Q}}^{\rm p},B}$ are
topologically $R$-module isomorphic (see Theorem 4.1 in [12]) where we used
the $R$-module version of Theorem 3.27 in [18] (up to $R$-module isomorphism,
$G_{{\mathcal{Q}}^{\rm p},A}$ and $G_{{\mathcal{Q}}^{\rm p},B}$ have the same
finite $R$-module quotients, namely ${\rm BF}_{g}(A)\cong_{R}{\rm BF}_{g}(B)$
for all $g\in{\mathcal{Q}}^{\rm p}$). The same argument shows that strong BF-
equivalence over ${\mathcal{Q}}^{\rm a}$ implies that $G_{{\mathcal{Q}}^{\rm
a},A}$ and $G_{{\mathcal{Q}}^{\rm a},B}$ are topologically $R$-module
isomorphic. Thus, for ${\mathcal{Q}}={\mathcal{Q}}^{\rm p}$ or
${\mathcal{Q}}^{\rm a}$, a sufficient condition for when two similar
hyperbolic $A,B\in{\rm GL}_{n}({\mathbb{Z}})$ are profinitely conjugate over
${\mathcal{Q}}$, and hence in the same equivalence class, is strong BF-
equivalence of $A$ and $B$ over ${\mathcal{Q}}$, i.e., ${\rm
BF}_{g}(A)\cong_{R}{\rm BF}_{g}(B)$ for all $g\in{\mathcal{Q}}$.

######  Lemma 4.7.

For a hyperbolic $A\in{\rm GL}_{n}({\mathbb{Z}})$ the profinite $R$-modules
$G_{{\mathcal{Q}}^{\rm p},A}$ and $G_{{\mathcal{Q}}^{\rm a},A}$ are
topologically $R$-module isomorphic.

###### Proof.

By Corollary 4.5 the profinite $R$-modules $G_{{\mathcal{Q}}^{\rm p},A}$ and
$G_{{\mathcal{Q}}^{\rm a},A}$ are topologically isomorphic as topological
abelian groups. The proof of Corollary 4.5 uses Proposition 2.2 where an
explicit topological isomorphism is given. It is straightforward to verify
that this explicitly given topological isomorphism is an $R$-module map,
making $G_{{\mathcal{Q}}^{\rm p},A}$ and $G_{{\mathcal{Q}}^{\rm a},A}$
topologically $R$-module isomorphic. ∎

For ${\mathcal{Q}}$ being either ${\mathcal{Q}}^{\rm p}$ or
${\mathcal{Q}}^{\rm a}$, the pro-${\mathcal{C}}$ completions
$G_{{\mathcal{Q}},A}$ and $G_{{\mathcal{Q}},B}$ have, of course, the closest
connection with the generalized Bowen-Franks $R$-modules ${\rm BF}_{g}(A)$ and
${\rm BF}_{g}(B)$ for $g\in{\mathcal{Q}}$. This is because

${\rm
BF}_{g}(A)={\mathbb{Z}}^{n}/{\mathbb{Z}}^{n}g(A)=G_{g,A}\cong_{R}G_{A}/{\rm
ker}\varphi_{g,A},\ g\in{\mathcal{Q}},$

with a similar statement holding with $A$ replaced by $B$. The
pro-${\mathcal{C}}$ completions $G_{{\mathcal{Q}},A}$ and
$G_{{\mathcal{Q}},B}$ are both topologically isomorphic to the full
pro-${\mathcal{C}}$ completion ${\mathcal{K}}_{\mathcal{F}}({\mathbb{Z}}^{n})$
by Corollary 4.5, but not every finite index subgroup of ${\mathbb{Z}}^{n}$
that is used in the construction of
${\mathcal{K}}_{\mathcal{F}}({\mathbb{Z}}^{n})$ is invariant by the
corresponding right action of $A$ or $B$. Hence we use the dynamically defined
$R$-modules $G_{{\mathcal{Q}},A}$ and $G_{{\mathcal{Q}},B}$ to prove that
profinite conjugacy over ${\mathcal{Q}}$ of similar hyperbolic $A$ and $B$
implies strong BF-equivalence of $A$ and $B$ over ${\mathcal{Q}}$. We make use
of the one-to-one correspondence in Theorem 2.1, via Remark 1, from
$\\{U:U\leq_{o}{\mathbb{Z}}^{n};{\mathcal{N}}_{{\mathcal{Q}},A}\\}$ to
$\\{V:V\leq_{o}G_{{\mathcal{Q}},A}\\}$ and its properties without explicit
reference in what follows.

###  4.4. The Converse

The main result is proven in this subsection. The idea of the proof of the
main result is to mimic the quotient of the generalized Bowen-Franks
$R$-modules in the profinite $R$-module setting and connect it with
generalized Bowen-Franks $R$-modules through the isomorphism theory for
$R$-modules. The first step in this direction is to show through a series of
four lemmas that, for a pro-${\mathcal{C}}$ topology
${\mathcal{N}}_{{\mathcal{Q}},A}$ on ${\mathbb{Z}}^{n}$ with ${\mathcal{Q}}$
being either ${\mathcal{Q}}^{\rm p}$ or ${\mathcal{Q}}^{\rm a}$, the closure
of the image $g(\Gamma_{A})(G_{{\mathcal{Q}},A})$ is equal to ${\rm
ker}\varphi_{g,A}$ for each $g\in{\mathcal{Q}}$.

######  Lemma 4.8.

For a hyperbolic $A\in{\rm GL}_{n}({\mathbb{Z}})$ and $g\in{\mathcal{Q}}$,
there holds

$\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n}g(A))\subset
g(\Gamma_{A})(G_{{\mathcal{Q}},A})\cap\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n}).$

###### Proof.

Suppose $x\in\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n}g(A))$. Then
$x=\iota_{{\mathcal{Q}},A}(mg(A))$ for some $m\in{\mathbb{Z}}^{n}$. Hence
$x=g(\Gamma_{A})(\iota_{{\mathcal{Q}},A}(m))\in
g(\Gamma_{A})(G_{{\mathcal{Q}},A})$. Since $mg(A)\in{\mathbb{Z}}^{n}$, we have
$x\in\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n})$, and so

$x\in
g(\Gamma_{A})(G_{{\mathcal{Q}},A})\cap\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n}).$

This implies that $\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n}g(A))\subset
g(\Gamma_{A})(G_{{\mathcal{Q}},A})\cap\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n})$.
∎

######  Lemma 4.9.

For a hyperbolic $A\in{\rm GL}_{n}({\mathbb{Z}})$ and $g\in{\mathcal{Q}}$,
there holds

$g(\Gamma_{A})(G_{{\mathcal{Q}},A})\subset{\rm ker}\varphi_{g,A}.$

###### Proof.

For $\\{[m_{h}]_{h,A}\\}_{h\in{\mathcal{Q}}}\in G_{{\mathcal{Q}},A}$, we have

$g(\Gamma_{A})(\\{[m_{h}]_{h,A}\\}_{h\in{\mathcal{Q}}})=\\{[m_{h}g(A)]_{h,A}\\}_{h\in{\mathcal{Q}}}.$

Then

$\varphi_{g,A}(g(\Gamma_{A})(\\{[m_{h}]_{h,A}\\}_{h\in{\mathcal{Q}}}))=[m_{g}g(A)]_{g,A}=0$

because $m_{g}g(A)\in{\mathbb{Z}}^{n}g(A)$. Thus
$g(\Gamma_{A})(G_{{\mathcal{Q}},A})\subset{\rm ker}\varphi_{g,A}$. ∎

######  Lemma 4.10.

For a hyperbolic $A\in{\rm GL}_{n}({\mathbb{Z}})$ and $g\in{\mathcal{Q}}$,
there holds

${\rm
ker}\varphi_{g,A}\cap\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n})=\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n}g(A)).$

###### Proof.

Suppose $x\in\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n}g(A))$. Then
$x=\iota_{{\mathcal{Q}},A}(mg(A))$ for some $m\in{\mathbb{Z}}^{n}$. Hence
$\iota_{{\mathcal{Q}},A}(mg(A))\in\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n})$,
and

$\varphi_{g,A}(x)=[mg(A)]_{g,A}=0$

because $mg(A)\in{\mathbb{Z}}^{n}g(A)$. Thus
$\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n}g(A))\subset{\rm
ker}\varphi_{g,A}\cap\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n})$.

Now suppose that $x\in{\rm
ker}\varphi_{g,A}\cap\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n})$. Then
$x=\iota_{{\mathcal{Q}},A}(m)$ for some $m\in{\mathbb{Z}}^{n}$ and

$0=\varphi_{g,A}(\iota_{{\mathcal{Q}},A}(m))=[m]_{g,A}.$

This says that $m\in{\mathbb{Z}}^{n}g(A)$, so that
$x=\iota_{{\mathcal{Q}},A}(m)\in\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n}g(A))$.
This implies that ${\rm
ker}\varphi_{g,A}\cap\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n})\subset\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n}g(A))$.

Therefore ${\rm
ker}\varphi_{g,A}\cap\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n})=\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n}g(A))$.
∎

######  Lemma 4.11.

For a hyperbolic $A\in{\rm GL}_{n}({\mathbb{Z}})$ and $g\in{\mathcal{Q}}$,
there holds

$\overline{g(\Gamma_{A})(G_{{\mathcal{Q}},A})}={\rm ker}\varphi_{g,A}.$

###### Proof.

By Lemma 4.9, we have
$\overline{g(\Gamma_{A})(G_{{\mathcal{Q}},A})}\subset{\rm ker}\varphi_{g,A}$
because ${\rm ker}\varphi_{g,A}$ is closed.

On the other hand, by Lemma 4.10, we have ${\rm
ker}\varphi_{g,A}\cap\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n})=\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n}g(A))$.
Since ${\rm ker}\varphi_{g,A}\leq_{o}G_{{\mathcal{Q}},A}$, we have that

$\overline{\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n}g(A))}=\overline{{\rm
ker}\varphi_{g,A}\cap\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n})}={\rm
ker}\varphi_{g,A}.$

By Lemma 4.8, we have $\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n}g(A))\subset
g(\Gamma_{A})(G_{{\mathcal{Q}},A})\cap\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n})$,
so that

$\displaystyle\overline{\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n}g(A))}$
$\displaystyle\subset\overline{g(\Gamma_{A})(G_{{\mathcal{Q}},A})\cap\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n})}$
$\displaystyle\subset\overline{g(\Gamma_{A})(G_{{\mathcal{Q}},A})}\cap\overline{\iota_{{\mathcal{Q}},A}({\mathbb{Z}}^{n})}$
$\displaystyle=\overline{g(\Gamma_{A})(G_{{\mathcal{Q}},A})}\cap
G_{{\mathcal{Q}},A}$
$\displaystyle=\overline{g(\Gamma_{A})(G_{{\mathcal{Q}},A})}.$

This gives ${\rm
ker}\varphi_{g,A}\subset\overline{g(\Gamma_{A})(G_{{\mathcal{Q}},A})}$.

Therefore $\overline{g(\Gamma_{A})(G_{{\mathcal{Q}},A})}={\rm
ker}\varphi_{g,A}$. ∎

For any profinite conjugacy $\Psi_{\mathcal{Q}}:G_{{\mathcal{Q}},A}\to
G_{{\mathcal{Q}},B}$ over ${\mathcal{Q}}$ there holds

$g(\Gamma_{B})\Psi_{\mathcal{Q}}=\Psi_{\mathcal{Q}}g(\Gamma_{A}){\rm\ for\
all\ }g\in{\mathcal{Q}},$

because $\Psi_{\mathcal{Q}}$ is an $R$-module isomorphism, i.e.,
$\Gamma_{B}\Psi_{\mathcal{Q}}=\Psi_{\mathcal{Q}}\Gamma_{A}$.

######  Lemma 4.12.

For similar hyperbolic $A,B\in{\rm GL}_{n}({\mathbb{Z}})$, if
$\Psi_{\mathcal{Q}}:G_{{\mathcal{Q}},A}\to G_{{\mathcal{Q}},B}$ is a profinite
conjugacy over ${\mathcal{Q}}$, then for all $g\in{\mathcal{Q}}$ there holds

$\Psi_{\mathcal{Q}}\big{(}g(\Gamma_{A})(G_{{\mathcal{Q}},A})\big{)}=g(\Gamma_{B})(G_{{\mathcal{Q}},B}).$

###### Proof.

Let $\Psi_{\mathcal{Q}}:G_{{\mathcal{Q}},A}\to G_{{\mathcal{Q}},B}$ be a
topological $R$-module isomorphism.

Suppose $x\in g(\Gamma_{A})(G_{{\mathcal{Q}},A})$. Then there exists $y\in
G_{{\mathcal{Q}},A}$ such that $x=g(\Gamma_{A})(y)$. Set
$z=\Psi_{\mathcal{Q}}(y)\in G_{{\mathcal{Q}},B}$. Then we have

$\Psi_{\mathcal{Q}}(x)=\Psi_{\mathcal{Q}}g(\Gamma_{A})(y)=g(\Gamma_{B})\Psi_{\mathcal{Q}}(y)=g(\Gamma_{B})(z).$

Thus $\Psi_{\mathcal{Q}}(x)\in g(\Gamma_{B})(G_{{\mathcal{Q}},B})$, and so
$\Psi_{\mathcal{Q}}(g(\Gamma_{A})(G_{{\mathcal{Q}},A}))\subset
g(\Gamma_{B})(G_{{\mathcal{Q}},B})$.

Now let $x\in g(\Gamma_{B})(G_{{\mathcal{Q}},B})$. Then there is $y\in
G_{{\mathcal{Q}},B}$ such that $x=g(\Gamma_{B})(y)$. Because
$\Psi_{\mathcal{Q}}$ is a bijection, there exists a unique $z\in
G_{{\mathcal{Q}},A}$ such that $\Psi_{\mathcal{Q}}(z)=y$. Set
$w=g(\Gamma_{A})(z)$. Then we have

$\Psi_{\mathcal{Q}}(w)=\Psi_{\mathcal{Q}}\big{(}g(\Gamma_{A})(z)\big{)}=g(\Gamma_{B})\Psi_{\mathcal{Q}}(z)=g(\Gamma_{B})(y)=x.$

This gives $x\in\Psi_{\mathcal{Q}}(g(\Gamma_{A})(G_{{\mathcal{Q}},A}))$, and
so
$g(\Gamma_{B})(G_{{\mathcal{Q}},B})\subset\Psi_{\mathcal{Q}}(g(\Gamma_{A})(G_{{\mathcal{Q}},A}))$.

Therefore
$\Psi_{\mathcal{Q}}(g(\Gamma_{A})(G_{{\mathcal{Q}},A}))=g(\Gamma_{B})(G_{{\mathcal{Q}},B})$.
∎

######  Theorem 4.13.

Suppose that $A,B\in{\rm GL}_{n}({\mathbb{Z}})$ are similar and hyperbolic. If
$A$ and $B$ are profinitely conjugate over ${\mathcal{Q}}$, then $A$ and $B$
are strongly BF-equivalent over ${\mathcal{Q}}$.

###### Proof.

Suppose $\Psi_{\mathcal{Q}}:G_{{\mathcal{Q}},A}\to G_{{\mathcal{Q}},B}$ is a
profinite conjugacy over ${\mathcal{Q}}$. By Lemma 4.12, we have for each
$g\in{\mathcal{Q}}$ that
$\Psi_{\mathcal{Q}}(g(\Gamma_{A})(G_{{\mathcal{Q}},A}))=g(\Gamma_{B})(G_{{\mathcal{Q}},B})$.
Since $\Psi_{\mathcal{Q}}$ is a homeomorphism, we have for each
$g\in{\mathcal{Q}}$ that

$\Psi_{\mathcal{Q}}\left(\overline{g(\Gamma_{A})(G_{{\mathcal{Q}},A})}\right)=\overline{g(\Gamma_{B})(G_{{\mathcal{Q}},B})}.$

By Lemma 4.11 for each $g\in{\mathcal{Q}}$ there holds
$\overline{g(\Gamma_{A})(G_{{\mathcal{Q}},A})}={\rm ker}\varphi_{g,A}$ and
$\overline{g(\Gamma_{B})(G_{{\mathcal{Q}},B})}={\rm ker}\varphi_{g,B}$, and so

$\Psi_{\mathcal{Q}}({\rm ker}\varphi_{g,A})={\rm ker}\varphi_{g,B}.$

Thus, by the isomorphism theory for $R$-modules, $\Psi_{\mathcal{Q}}$ induces
an $R$-module isomorphism $G_{{\mathcal{Q}},A}/{\rm
ker}\varphi_{g,A}\cong_{R}G_{{\mathcal{Q}},B}/{\rm ker}\varphi_{g,B}$ for each
$g\in{\mathcal{Q}}$. Since $G_{g,A}\cong_{R}G_{{\mathcal{Q}},A}/{\rm
ker}\varphi_{g,A}$ and $G_{g,B}\cong_{R}G_{{\mathcal{Q}},B}/{\rm
ker}\varphi_{g,B}$, we obtain $G_{g,A}\cong_{R}G_{g,B}$ for all
$g\in{\mathcal{Q}}$. Thus $A$ and $B$ are strongly BF-equivalent over
${\mathcal{Q}}$. ∎

######  Corollary 4.14.

Suppose $A,B\in{\rm GL}_{n}({\mathbb{Z}})$ are similar and hyperbolic. Then
$A$ and $B$ are profinitely conjugate over ${\mathcal{Q}}$ if and only if $A$
and $B$ are strongly BF-equivalent over ${\mathcal{Q}}$.

###### Proof.

Combine Theorem 4.13 and Remark 2. ∎

######  Corollary 4.15.

Suppose $A,B\in{\rm GL}_{n}({\mathbb{Z}})$ are similar and hyperbolic. The
following are equivalent.

  1. (a)

$A$ and $B$ are strongly BF-equivalent over ${\mathcal{Q}}^{\rm p}$.

  2. (b)

$A$ and $B$ are profinitely conjugate over ${\mathcal{Q}}^{\rm p}$.

  3. (c)

$A$ and $B$ are profinitely conjugate over ${\mathcal{Q}}^{\rm a}$.

  4. (d)

$A$ and $B$ are strongly BF-equivalent over ${\mathcal{Q}}^{\rm a}$.

###### Proof.

Parts (a) and (b) are equivalent by Corollary 4.14 applied when
${\mathcal{Q}}={\mathcal{Q}}^{\rm p}$ and parts (c) and (d) are equivalent by
Corollary 4.14 applied when ${\mathcal{Q}}={\mathcal{Q}}^{\rm a}$. By Lemma
4.7 the profinite $R$-modules $G_{{\mathcal{Q}}^{\rm p},A}$ and
$G_{{\mathcal{Q}}^{\rm a},A}$ are topologically $R$-module isomorphic and the
profinite $R$-modules $G_{{\mathcal{Q}}^{\rm p},B}$ and $G_{{\mathcal{Q}}^{\rm
a},B}$ are topologically $R$-module isomorphic. Hence $G_{{\mathcal{Q}}^{\rm
p},A}$ and $G_{{\mathcal{Q}}^{\rm p},B}$ are profinite conjugate over
${\mathcal{Q}}^{\rm p}$ if and only if $G_{{\mathcal{Q}}^{\rm a},A}$ and
$G_{{\mathcal{Q}}^{\rm a},B}$ are profinitely conjugate over
${\mathcal{Q}}^{\rm a}$, which is the equivalence of parts (b) and (c). ∎

The implications of Corollary 4.15 are that profinite conjugacy over
${\mathcal{Q}}^{\rm p}$ and over ${\mathcal{Q}}^{\rm a}$ are equivalent and
that the principal generalized Bowen-Franks $R$-modules are indeed the
principal invariants among the generalized Bowen-Franks $R$-modules
parameterized by ${\mathcal{Q}}^{\rm a}$. Going forward we need only be
concerned with profinite conjugacy of similar hyperbolic $A,B\in{\rm
GL}_{n}({\mathbb{Z}})$ over ${\mathcal{Q}}^{p}$. To aid with this, we simplify
the notation by writing ${\mathcal{N}}_{A}={\mathcal{N}}_{{\mathcal{Q}}^{\rm
p},A}$, $G_{A}=G_{{\mathcal{Q}}^{\rm p},A}$,
$\iota_{A}=\iota_{{\mathcal{Q}}^{\rm P},A}$, an element of $G_{A}$ as
$\\{[m_{k}]_{k,A}\\}_{k\in{\mathbb{N}}}$,
${\mathcal{N}}_{B}={\mathcal{N}}_{{\mathcal{Q}}^{\rm p},B}$
$G_{B}=G_{{\mathcal{Q}}^{\rm p},B}$, $\iota_{B}=\iota_{{\mathcal{Q}}^{\rm
p},B}$, and an element of $G_{B}$ as $\\{[m_{k}]_{k,B}\\}_{k\in{\mathbb{N}}}$.
We also refer to a profinite conjugacy over ${\mathcal{Q}}^{\rm p}$ between
$A$ and $B$ simply as a profinite conjugacy $\Psi:G_{A}\to G_{B}$ between $A$
and $B$, and to strong BF-equivalence of $A$ and $B$ over ${\mathcal{Q}}^{\rm
p}$ simply as strong BF-equivalence of $A$ and $B$.

##  5\. Properties of Profinite Conjugacies

###  5.1. Two Characterizations of Conjugacy

The existence of a profinite conjugacy between two similar hyperbolic toral
automorphisms is insufficient to imply their conjugacy, as it is known that
strong ${\rm BF}$-equivalence is insufficient to imply conjugacy (see [17,
9]). However, for similar hyperbolic toral automorphisms $A,B\in{\rm
GL}_{n}({\mathbb{Z}})$, there are two characterizations of when profinite
conjugacies between $A$ and $B$ gives conjugacy of $A$ and $B$ in terms of the
embedded copies $\iota_{A}({\mathbb{Z}}^{n})$ and
$\iota_{B}({\mathbb{Z}}^{n})$ of ${\mathbb{Z}}^{n}$, as we showed in (Theorem
5.1 in [12]):

  1. (a)

$A$ and $B$ are conjugate,

  2. (b)

there is a profinite conjugacy $\Psi:G_{A}\to G_{B}$ such that
$\Psi(\iota_{A}({\mathbb{Z}}^{n}))=\iota_{B}({\mathbb{Z}}^{n})$, and

  3. (c)

there is a profinite conjugacy $\Psi:G_{A}\to G_{B}$ such that
$\Psi(\iota_{A}({\mathbb{Z}}^{n}))\cap\iota_{B}({\mathbb{Z}}^{n})$ is
$R$-module isomorphic to the right ${\mathbb{Z}}[B]$-module
${\mathbb{Z}}^{n}$.

We structured the proof of this in the cyclic format (a) $\Rightarrow$ (b),
(b) $\Rightarrow$ (c), and (c) $\Rightarrow$ (a). The proof that (a)
$\Leftrightarrow$ (b) follows from the proof of (a) $\Rightarrow$ (b) and an
extra independent proof of (b) $\Rightarrow$ (a) included in [12], the latter
highlighting the pro-${\mathcal{C}}$ topology ${\mathcal{N}}_{B}$ on
${\mathbb{Z}}^{n}$. But there was an error in the proof of (c) $\Rightarrow$
(a) in [12], which, using the notation

$\Delta_{\Psi}=\Psi(\iota_{A}({\mathbb{Z}}^{n}))\cap\iota_{B}({\mathbb{Z}}^{n}),$

leads to following correction of part (c):

  1. (c)

there is a profinite conjugacy $\Psi:G_{A}\to G_{B}$ such that
$\Psi^{-1}(\Delta_{\Psi})$ is $R$-module isomorphic to the right
${\mathbb{Z}}[A]$-module ${\mathbb{Z}}^{n}$ and $\Delta_{\Psi}$ is $R$-module
isomorphic to the right ${\mathbb{Z}}[B]$-module ${\mathbb{Z}}^{n}$.

With the inclusion of the additional condition of the correction in part (c),
here are the corrected parts of the proof of the two characterization of the
conjugacy.

(b) $\Rightarrow$ (c). Suppose there is a profinite conjugacy $\Psi:G_{A}\to
G_{B}$ such that
$\Psi(\iota_{A}({\mathbb{Z}}^{n}))=\iota_{B}({\mathbb{Z}}^{n})$. Then
$\Delta_{\Psi}=\iota_{B}({\mathbb{Z}}^{n})$. The map
$\iota_{B}:{\mathbb{Z}}^{n}\to G_{B}$ from the right ${\mathbb{Z}}[B]$-module
to the right ${\mathbb{Z}}[B]$-module $G_{B}$ is an $R$-module monomorphism.
So the map $\iota_{B}^{-1}:\iota_{B}({\mathbb{Z}}^{n})\to{\mathbb{Z}}^{n}$ is
an $R$-module isomorphism from $\iota_{B}({\mathbb{Z}}^{n})=\Delta_{\Psi}$ to
the right ${\mathbb{Z}}[B]$-module ${\mathbb{Z}}^{n}$. On the other hand, from
$\Delta_{\Psi}=\Psi(\iota_{A}({\mathbb{Z}}^{n}))$ we have
$\Psi^{-1}(\Delta_{\Psi})=\iota_{A}({\mathbb{Z}}^{n})$. The map
$\iota_{A}:{\mathbb{Z}}^{n}\to G_{A}$ is an $R$-module monomorphism from the
right ${\mathbb{Z}}[A]$-module ${\mathbb{Z}}^{n}$ to the right
${\mathbb{Z}}[A]$-module $G_{A}$. So the map
$\iota_{A}^{-1}:\iota_{A}({\mathbb{Z}}^{n})\to{\mathbb{Z}}^{n}$ is an
$R$-module isomorphism from the right ${\mathbb{Z}}[A]$-module
$\iota_{A}({\mathbb{Z}}^{n})=\Psi^{-1}(\Delta_{\Psi})$ to the right
${\mathbb{Z}}[A]$-module ${\mathbb{Z}}^{n}$. This gives (c).

(c) $\Rightarrow$ (a). The assumptions of part (c) imply the existence of
$R$-module isomorphisms $h:{\mathbb{Z}}^{n}\to\Delta_{\Psi}$ and
$g:{\mathbb{Z}}^{n}\to\Psi^{-1}(\Delta_{\Psi})$ that satisfy

$Bh^{-1}=h^{-1}\Gamma_{B},\ \ gA=\Gamma_{A}g.$

[The additional condition in the corrected version of part (c) gives the
existence of $g$, while the original condition in part (c) gives the existence
of $h$.] The mapping $h^{-1}\Psi g:{\mathbb{Z}}^{n}\to{\mathbb{Z}}^{n}$ is an
$R$-module isomorphism from the right ${\mathbb{Z}}[A]$-module
${\mathbb{Z}}^{n}$ to the right ${\mathbb{Z}}[B]$-module ${\mathbb{Z}}^{n}$.
[The original incorrect proof of (c) $\Rightarrow$ (a) used the $R$-module
mapping $h^{-1}\Psi\iota_{A}$ which is where the error occurred.] This implies
that $h^{-1}\Psi g$ is an automorphism of ${\mathbb{Z}}^{n}$, and gives the
existence of $D\in{\rm GL}_{n}({\mathbb{Z}})$ such that $D=h^{-1}\Psi g$. This
is illustrated in the following diagram.

$\begin{CD}\Psi^{-1}(\Delta_{\Psi})@>{\Psi}>{}>\Delta_{\Psi}\\\ @A{g}A{}A@
AAhA\\\ {\mathbb{Z}}^{n}@>{}>{D}>{\mathbb{Z}}^{n}\end{CD}$

Using the right $R$-module structures of $h^{-1}$, $\Psi$, and $g$, there
holds

$DA=h^{-1}\Psi gA=h^{-1}\Psi\Gamma_{A}g=h^{-1}\Gamma_{B}\Psi g=Bh^{-1}\Psi
g=BD.$

Setting $C=D^{-1}$ gives $AC=CB$, hence (a).

We state the corrected version of the two characterizations of conjugacy for
similar hyperbolic toral automorphisms.

######  Theorem 5.1.

For similar hyperbolic $A,B\in{\rm GL}_{n}({\mathbb{Z}})$, the following are
equivalent.

  1. (a)

$A$ and $B$ are conjugate in ${\rm GL}_{n}({\mathbb{Z}})$, i.e., there is
$C\in{\rm GL}_{n}({\mathbb{Z}})$ such that $AC=CB$,

  2. (b)

there is a profinite conjugacy $\Psi:G_{A}\to G_{B}$ such that
$\Psi(\iota_{A}({\mathbb{Z}}^{n}))=\iota_{B}({\mathbb{Z}}^{n})$, and

  3. (c)

there is a profinite conjugacy $\Psi:G_{A}\to G_{B}$ such that
$\Psi^{-1}(\Delta_{\Psi})$ is $R$-module isomorphic to the right
${\mathbb{Z}}[A]$-module ${\mathbb{Z}}^{n}$ and $\Delta_{\Psi}$ is $R$-module
isomorphic to the right ${\mathbb{Z}}[B]$-module ${\mathbb{Z}}^{n}$.

###  5.2. The Embedded Copy of ${\mathbb{Z}}^{n}$

The two characterizations of conjugacy in Theorem 5.1 involve the embedded
dense copies of ${\mathbb{Z}}^{n}$ through the $R$-module monomorphisms
$\iota_{A}$ and $\iota_{B}$. The second characterization of conjugacy in part
(c) of Theorem 5.1 has the presence of the $R$-submodule
$\Delta_{\Psi}=\Psi(\iota_{A}({\mathbb{Z}}^{n}))\cap\iota_{B}({\mathbb{Z}}^{n})$.
When similar hyperbolic $A$ and $B$ are conjugate, there exists a profinite
conjugacy by Theorem 5.1 for which
$\Psi(\iota_{A}({\mathbb{Z}}^{n}))=\iota_{B}({\mathbb{Z}}^{n})$ so that
$\Delta_{\Psi}=\iota_{B}({\mathbb{Z}}^{n})$. In general, however, part (c) of
Theorem 5.1 poses the question of if it is possible for a profinite conjugacy
$\Psi$ to send $\iota_{A}({\mathbb{Z}}^{n})$ into a proper $R$-submodule of
$\iota_{B}({\mathbb{Z}}^{n})$. We answer this in the negative.

######  Lemma 5.2.

For similar hyperbolic $A,B\in{\rm GL}_{n}({\mathbb{Z}})$, if $\Psi:G_{A}\to
G_{B}$ is a profinite conjugacy between $A$ and $B$, then
$\Psi(\iota_{A}({\mathbb{Z}}^{n}))$ is not a proper $R$-submodule of
$\iota_{B}({\mathbb{Z}}^{n})$.

###### Proof.

Suppose $\Psi:G_{A}\to G_{B}$ is profinite conjugacy for which
$\Psi(\iota_{A}({\mathbb{Z}}^{n}))$ is a proper $R$-submodule of
$\iota_{B}({\mathbb{Z}}^{n})$. Since
$\iota_{A}({\mathbb{Z}}^{n})\leq_{o}{\mathbb{Z}}^{n}$ (with respect to the
pro-${\mathcal{C}}$ topology on ${\mathbb{Z}}^{n}$ determined by
${\mathcal{N}}_{A}$ and the identification of $\iota_{A}({\mathbb{Z}}^{n})$
with ${\mathbb{Z}}^{n}$), we have
$\overline{\iota_{A}({\mathbb{Z}}^{n})}\leq_{o}G_{A}$. Then
$\Psi\left(\overline{\iota_{A}({\mathbb{Z}}^{n})}\right)\leq_{o}G_{B}$, and so
by the identification of ${\mathbb{Z}}^{n}$ with $\iota_{B}({\mathbb{Z}}^{n})$
we have

$\Psi\left(\overline{\iota_{A}({\mathbb{Z}}^{n})}\right)\cap\iota_{B}({\mathbb{Z}}^{n})\leq_{o}{\mathbb{Z}}^{n}$

(with respect to the pro-${\mathcal{C}}$ topology on ${\mathbb{Z}}^{n}$
determined by ${\mathcal{N}}_{B}$). Let $H\leq_{o}{\mathbb{Z}}^{n}$ be the set
defined by

$\iota_{B}(H)=\Psi\left(\overline{\iota_{A}({\mathbb{Z}}^{n})}\right)\cap\iota_{B}({\mathbb{Z}}^{n}).$

Since $\Psi$ is a homeomorphism we have

$\iota_{B}(H)=\overline{\Psi(\iota_{A}({\mathbb{Z}}^{n}))}\cap\iota_{B}({\mathbb{Z}}^{n}).$

By hypothesis, $\Psi(\iota_{A}({\mathbb{Z}}^{n}))$ is a proper $R$-submodule
of $\iota_{B}({\mathbb{Z}}^{n})$. By the $1-1$ correspondence from
$\\{U:U\leq_{o}{\mathbb{Z}}^{n};{\mathcal{N}}_{B}\\}$ to
$\\{V:V\leq_{o}G_{B}\\}$ there is only one open subgroup of ${\mathbb{Z}}^{n}$
(with respect to pro-${\mathcal{C}}$ topology determined by
${\mathcal{N}}_{B}$) whose closure is $G_{B}$, and this open subgroup is
${\mathbb{Z}}^{n}$. Since
$\Psi(\iota_{A}({\mathbb{Z}}^{n}))\neq\iota_{B}({\mathbb{Z}}^{n})$ it follows
that $H$ is a proper open subgroup of ${\mathbb{Z}}^{n}$, for if
$H={\mathbb{Z}}^{n}$, then

$\overline{\Psi(\iota_{A}({\mathbb{Z}}^{n}))}\cap\iota_{B}({\mathbb{Z}}^{n})=\iota_{B}({\mathbb{Z}}^{n})$

which would imply that $\overline{\Psi(\iota_{A}({\mathbb{Z}}^{n}))}=G_{B}$,
giving $\Psi(\iota_{A}({\mathbb{Z}}^{n}))=\iota_{B}({\mathbb{Z}}^{n})$, a
contradiction. Hence $[{\mathbb{Z}}^{n}:H]>1$. Since
$\overline{{\mathbb{Z}}^{n}}=G_{B}$, we have

$[G_{B}:\overline{H}]=[{\mathbb{Z}}^{n}:H]>1.$

But since $\iota_{A}({\mathbb{Z}}^{n})$ is dense in $G_{A}$ and since $\Psi$
is a homeomorphism, the image $\Psi(\iota_{A}({\mathbb{Z}}^{n}))$ is dense in
$G_{B}$. Hence

$\overline{\Psi(\iota_{A}({\mathbb{Z}}^{n}))}=G_{B},$

which implies that $H={\mathbb{Z}}^{n}$, so that $[G_{B}:\overline{H}]=1$, a
contradiction. ∎

###  5.3. Cyclic Automorphisms

For profinitely conjugate similar hyperbolic toral automorphisms $A$ and $B$
we show how $\iota_{A}({\mathbb{Z}}^{n})$ and $\iota_{B}({\mathbb{Z}}^{n})$
within $G_{A}$ and $G_{B}$ distinguish $A$ from $B$ when $A$ is cyclic and $B$
is not cyclic. An automorphism $A\in{\rm GL}_{n}({\mathbb{Z}})$ is called
cyclic if there exists $\xi\in{\mathbb{Z}}^{n}$ such that the set $\\{\xi,\xi
A,\dots,\xi A^{n-1}\\}$ forms a basis for ${\mathbb{Z}}^{n}$, and $\xi$ is
called a generator for $A$, i.e., the right ${\mathbb{Z}}[A]$-module
${\mathbb{Z}}^{n}$ is cyclic, or equivalently, $\iota_{A}({\mathbb{Z}}^{n})$
is a cyclic $R$-submodule of $G_{A}$. As is well known, a cyclic automorphism
$A$ is conjugate to the companion matrix of its characteristic polynomial
$f(t)=a_{0}+a_{1}t+\cdots+a_{n-1}t^{n-1}+t^{n}\in{\mathbb{Z}}[t]$, i.e., to

$C_{f}=\begin{bmatrix}0&1&0&\dots&0&0\\\ 0&0&1&\dots&0&0\\\ 0&0&0&\dots&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&0&\dots&0&1\\\
-a_{0}&-a_{1}&-a_{2}&\dots&-a_{n-2}&-a_{n-1}\end{bmatrix}.$

In particular, cyclicity is an invariant of the conjugacy class of the
companion matrix. Profinite conjugacy realizes this invariant for hyperbolic
toral automorphisms in a very explicit manner.

######  Theorem 5.3.

Suppose that $A,B\in{\rm GL}_{n}({\mathbb{Z}})$ are similar and hyperbolic. If
$A$ is cyclic with generator $\xi\in{\mathbb{Z}}^{n}$, and $B$ is not cyclic,
then for any profinite conjugacy $\Psi:G_{A}\to G_{B}$ there holds

$\Psi(\iota_{A}(\xi))\not\in\iota_{B}({\mathbb{Z}}^{n}).$

###### Proof.

Suppose $A$ is cyclic with generator $\xi\in{\mathbb{Z}}^{n}$. Then the set
$\\{\xi,\xi A,\dots,\xi A^{n-1}\\}$ is a basis for ${\mathbb{Z}}^{n}$.
Identifying $\xi$ with its image $\iota_{A}(\xi)$ in $G_{A}$, the set

$\big{\\{}\\{[\xi A^{j}]_{k,A}\\}_{k\in{\mathbb{N}}}:j=0,1,\dots,n-1\big{\\}}$

is a basis for $\iota_{A}({\mathbb{Z}}^{n})$. For a topological $R$-module
isomorphism $\Psi:G_{A}\to G_{B}$, set

$\\{[\eta_{k}]_{k,B}\\}_{k\in{\mathbb{N}}}=\Psi(\\{[\xi]_{k,A}\\}_{k\in{\mathbb{N}}}.$

Since $\Psi$ is a $R$-module isomorphism, we have

$\displaystyle\Psi(\\{[\xi A]_{k,A}\\}_{k\in{\mathbb{N}}}$
$\displaystyle=\Psi(\\{A_{k}[\xi]_{k,A}\\}_{k\in{\mathbb{N}}}$
$\displaystyle=(\Psi\circ\Gamma_{A})(\\{[\xi]_{k,A}\\}_{k\in{\mathbb{N}}}$
$\displaystyle=(\Gamma_{B}\circ\Psi)(\\{[\xi]_{k,A}\\}_{k\in{\mathbb{N}}}$
$\displaystyle=\Gamma_{B}(\\{[\eta_{k}]_{k,B}\\}_{k\in{\mathbb{N}}}$
$\displaystyle=\\{B_{k}[\eta_{k}]_{k,B}\\}_{k\in{\mathbb{N}}}$
$\displaystyle=\\{[\eta_{k}B]_{k,B}\\}_{k\in{\mathbb{N}}}.$

By induction we have for $1\leq r\leq n-1$ that

$\Psi(\\{[\xi
A^{r}]_{k,A}\\}_{k\in{\mathbb{N}}})=\\{[\eta_{k}B^{r}]_{k,B}\\}_{k\in{\mathbb{N}}}.$

Suppose to the contrary that
$\Psi(\iota_{A}(\xi))\in\iota_{B}({\mathbb{Z}}^{n})$. Then
$\\{[\eta_{k}]_{k,B}\\}_{k\in{\mathbb{N}}}\in\iota_{B}({\mathbb{Z}}^{n})$.
Thus there is $m^{\prime}\in{\mathbb{Z}}^{n}$ such that
$[\eta_{k}]_{k,B}=[m^{\prime}]_{k,B}$ for all $k\in{\mathbb{N}}$. For each
$1\leq r\leq n-1$ we have

$\\{[\eta_{k}B^{r}]_{k,B}\\}_{k\in{\mathbb{N}}}=\\{B_{k}^{r}[\eta_{k}]_{k,B}\\}_{k\in{\mathbb{N}}}=\\{B_{k}^{r}[m^{\prime}]_{k,B}\\}_{k\in{\mathbb{N}}}=\\{[m^{\prime}B^{r}]_{k,B}\\}_{k\in{\mathbb{N}}}.$

This implies that
$\\{[n_{k}B^{r}]_{k,B}\\}_{k\in{\mathbb{N}}}\in\iota_{B}({\mathbb{Z}}^{n})$
for all $1\leq r\leq n-1$.

We show that
$\Psi(\iota_{A}({\mathbb{Z}}^{n}))\subset\iota_{B}({\mathbb{Z}}^{n})$. Let
$m\in{\mathbb{Z}}^{n}$. Since $\\{\xi,\xi A,\dots,\xi A^{n-1}\\}$ is a basis
for ${\mathbb{Z}}^{n}$ there exist $c_{1},c_{2},\dots,c_{n}\in{\mathbb{Z}}$
such that

$m=c_{1}\xi+c_{2}\xi A+\cdots+c_{n}\xi A^{n-1}.$

Then

$\displaystyle\Psi(\iota_{A}(m))$
$\displaystyle=\Psi(\\{[m]_{k,A}\\}_{k\in{\mathbb{N}}})$
$\displaystyle=\Psi(\\{[c_{1}\xi+c_{2}\xi A+\cdots+c_{n}\xi
A^{n-1}]_{k,A}\\}_{k\in{\mathbb{N}}})$
$\displaystyle=\Psi(\\{c_{1}[\xi]_{k,A}+c_{2}[\xi A]_{k,A}+\cdots+c_{n}[\xi
A^{n-1}]_{k,A}\\}_{k\in{\mathbb{N}}})$
$\displaystyle=\Psi(c_{1}\\{[\xi]_{k,A}\\}_{k\in{\mathbb{N}}}+c_{2}\\{[\xi
A]_{k,A}\\}_{k\in{\mathbb{N}}}+\cdots+c_{n}\\{[\xi
A^{n-1}]_{k,A}\\}_{k\in{\mathbb{N}}})$
$\displaystyle=c_{1}\\{[m^{\prime}]_{k,B}\\}_{k\in{\mathbb{N}}}+c_{2}\\{[m^{\prime}B]_{k,B}\\}_{k\in{\mathbb{N}}}+\cdots+c_{n}\\{[m^{\prime}B^{n-1}]_{k,B}\\}_{k\in{\mathbb{N}}}$
$\displaystyle\in\iota_{B}({\mathbb{Z}}^{n}).$

Since $m\in{\mathbb{Z}}^{n}$ is arbitrary, we obtain
$\Psi(\iota_{A}({\mathbb{Z}}^{n}))\subset\iota_{B}({\mathbb{Z}}^{n})$.

We show that
$\Psi(\iota_{A}({\mathbb{Z}}^{n}))\neq\iota_{B}({\mathbb{Z}}^{n})$. Using
$m^{\prime}$ from above, if every element of $\iota_{B}({\mathbb{Z}}^{n})$
could be written in the form

$c_{1}\\{[m^{\prime}]_{k,B}\\}_{k\in{\mathbb{N}}}+c_{2}\\{[m^{\prime}B]_{k,B}\\}_{k\in{\mathbb{N}}}+\cdots+c_{n}\\{[m^{\prime}B^{n-1}]_{k,B}\\}_{k\in{\mathbb{N}}}$

for $c_{i}\in{\mathbb{Z}}$, $i=1,2,\dots,n$, then $B$ would be cyclic with
generator $m^{\prime}$. But $B$ is not cyclic by hypothesis, and so
$\Psi(\iota_{A}({\mathbb{Z}}^{n}))\neq\iota_{B}({\mathbb{Z}}^{n})$.

We now have $\Psi(\iota_{A}({\mathbb{Z}}^{n}))$ is a proper $R$-submodule of
$\iota_{B}({\mathbb{Z}}^{n})$. This contradicts Lemma 5.2. Therefore,
$\Psi(\iota_{A}(\xi))\not\in\iota_{B}({\mathbb{Z}}^{n})$. ∎

###  5.4. Distinguishing Conjugacy Classes

Profinite conjugacy distinguishes between the conjugacy classes of similar
hyperbolic toral automorphisms according to the contrapositive of Theorem 5.1
and the assumption of strong BF-equivalence which by Corollary 4.14 gives the
existence of profinite conjugacies. For similar, hyperbolic, strongly BF-
equivalent $A,B\in{\rm GL}_{n}({\mathbb{Z}})$ the following are equivalent:

  1. (a)

$A$ and $B$ are not conjugate,

  2. (b)

for all profinite conjugacies $\Psi:G_{A}\to G_{B}$ there holds
$\Psi(\iota_{A}({\mathbb{Z}}^{n}))\neq\iota_{B}({\mathbb{Z}}^{n})$, and

  3. (c)

for all profinite conjugacies $\Psi:G_{A}\to G_{B}$ either
$\Psi^{-1}(\Delta_{\Psi})$ is not $R$-module isomorphic to right
${\mathbb{Z}}[A]$-module ${\mathbb{Z}}^{n}$ or $\Delta_{\Psi}$ is not
$R$-module isomorphic to the right ${\mathbb{Z}}[B]$-module ${\mathbb{Z}}^{n}$
(where
$\Delta_{\Psi}=\Psi(\iota_{A}({\mathbb{Z}}^{n}))\cap\iota_{B}({\mathbb{Z}}^{n})$).

Parts (a), (b), and (c) of this equivalence have particular realizations for
similar, hyperbolic, strongly BF-equivalent $A,B\in{\rm GL}_{n}({\mathbb{Z}})$
when $A$ is cyclic, with generator $\xi$, and $B$ is not cyclic. The
realization of (a) follows because cyclicity is an invariant of conjugacy. The
realization of (b) follows because for all profinite conjugacies
$\Psi:G_{A}\to G_{B}$ there holds
$\Psi(\iota_{A}(\xi))\not\in\iota_{B}({\mathbb{Z}}^{n})$ by Theorem 5.3, which
implies for all profinite conjugacies $\Psi:G_{A}\to G_{B}$ that
$\Psi(\iota_{A}({\mathbb{Z}}^{n}))\neq\iota_{B}({\mathbb{Z}}^{n})$. The
realization of (c) follows because for all profinite conjugacies
$\Psi:G_{A}\to G_{B}$ the $R$-submodules $\Psi^{-1}(\Delta_{\Psi})$ and
$\Delta_{\Psi}$ are $R$-module isomorphic, while the right
${\mathbb{Z}}[A]$-module ${\mathbb{Z}}^{n}$ is cyclic and the right
${\mathbb{Z}}[B]$-module ${\mathbb{Z}}^{n}$ is not cyclic, implying for all
profinite conjugacies $\Psi:G_{A}\to G_{B}$ that either
$\Psi^{-1}(\Delta_{\Psi})$ is not $R$-module isomorphic to the right
${\mathbb{Z}}[A]$-module ${\mathbb{Z}}^{n}$ or $\Delta_{\Psi}$ is not
$R$-module isomorphic to the right ${\mathbb{Z}}[B]$-module
${\mathbb{Z}}^{n}$.

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