#  Complete affine manifolds with Anosov holonomy groups I: hyperbolic bundles
and Anosov representations

Suhyoung Choi  Department of Mathematical Sciences
KAIST
Daejeon 34141, South Korea<EMAIL_ADDRESS>
###### Abstract.

Let $N$ be a complete affine manifold $\mathds{A}^{n}/\Gamma$ of dimension $n$
where $\Gamma$ is an affine transformation group and $K(\Gamma,1)$ is realized
as a finite CW-complex. $N$ has a $k$-partially hyperbolic holonomy group if
the tangent bundle pulled over the unit tangent bundle over a sufficiently
large compact part splits into expanding, neutral, and contracting subbundles
along the geodesic flow where the expanding and contracting subbundles have
the dimension $k$, $k<n/2$. We will show that the complete affine $n$-manifold
has a $P$-Anosov linear holonomy group for a parabolic subgroup $P$ of
${\mathsf{GL}}(n,{\mathbb{R}})$ if and only if it has a partially hyperbolic
linear holonomy group. This will be the first part where the techniques are
mostly representation theoretical.

###### Key words and phrases:

affine manifold, Anosov representation, word-hyperbolic group, Gromov
hyperbolic space

###### 2010 Mathematics Subject Classification:

Primary 57M50; Secondary 53A20, 53C15

This research was supported by Basic Science Research Program through the
National Research Foundation of Korea(NRF) funded by the Ministry of
Education(2019R1A2C108454412), Author orcid number: 0000-0002-9201-8210

###### Contents

  1. 1 Introduction
    1. 1.1 Notation
    2. 1.2 Main result
    3. 1.3 Outline
    4. 1.4 Acknowledgments
  2. 2  Preliminary
    1. 2.1 Convergences of geodesics
    2. 2.2 $(G,X)$-structures and affine manifolds
    3. 2.3 Gromov hyperbolic spaces
    4. 2.4 Flow space $G\hat{M}$ quasi-isometric to $\partial_{\infty}^{(2)}\hat{M}\times{\mathbb{R}}$
  3. 3 Partial hyperbolicity and $P$-Anosov properties
    1. 3.1 Some background
    2. 3.2 $P$-Anosov representations
    3. 3.3 $P$-Anosov representations and dominations
  4. 4 The proof of Theorem 1.1
    1. 4.1 $k$-convexity
    2. 4.2 Promoting $P$-Anosov representations to partially hyperbolic ones

##  1\. Introduction

###  1.1. Notation

Let ${\mathbf{Aff}}(\mathds{A}^{n})$ denote the group of affine
transformations of an affine space $\mathds{A}^{n}$ whose elements are of the
form:

$x\mapsto Ax+{\mathbf{v}}$

for a vector ${\mathbf{v}}\in{\mathbb{R}}^{n}$ and
$A\in{\mathsf{GL}}(n,{\mathbb{R}})$. Let
$\mathcal{L}:{\mathbf{Aff}}(\mathds{A}^{n})\rightarrow{\mathsf{GL}}(n,{\mathbb{R}})$
denote the map sending the affine transformations to their linear parts.

Let $\Gamma$ be an affine group acting properly and freely on
$\mathds{A}^{n}$. We assume that $K(\Gamma,1)$ is realized as a finite
complex, which is a fairly mild assumption in our field. Let
$\mathds{A}^{n}/\Gamma$ be a manifold with a complete Riemannian metric
structure.

Let $M$ be a compact submanifold (with or without boundary) in
$\mathds{A}^{n}/\Gamma$ where $\Gamma$ acts properly discontinuously and
freely on $\mathds{A}^{n}$. We suppose that the inclusion map
$M\hookrightarrow\mathds{A}^{n}/\Gamma$ induces a surjection of the
fundamental groups. The inverse image $\hat{M}$ of $M$ in $\mathds{A}^{n}$ is
connected and the deck transformation group $\Gamma_{M}$ of $\hat{M}$ is
isomorphic to $\Gamma$. In these cases, we shall say

$M,M\hookrightarrow\mathds{A}^{n}/\Gamma,\hat{M},p_{\hat{M}}:\hat{M}\rightarrow
M,\hbox{ and }\Gamma_{M}$

a fundamental group surjective (FS) submanifold, a FS map, a FS cover, a FS
covering map and a FS deck transformation group respectively.

A compact core of a manifold is a compact submanifold homotopy equivalent to
the ambient manifold. When a manifold is homotopy equivalent to a finite
complex $K$, for $n=3$, Scott and Tucker [25] proved the existence of the
compact core while for $n-\dim K\geq 3$, Stallings [26] established this fact.
Unfortunately, sometimes they don’t exist (See Venema [29]). Although it is
better for our arguments when we have compact cores, we need to use FS
submanifolds.

Now, $\mathds{A}^{n}$ has an induced complete $\Gamma$-equivariant Riemannian
metric from $\mathds{A}^{n}/\Gamma$ to be denoted by $d_{\mathds{A}^{n}}$. Let
$d_{E}$ denote a chosen standard Euclidean metric of $\mathds{A}^{n}$ fixed
for this paper. We will assume that $\partial M$ is convex in this paper. Let
$d_{M}$ denote the path metric induced from a Riemannian metric on
$\mathds{A}^{n}/\Gamma$, and we let $d_{\hat{M}}$ denote the path metric on
$\hat{M}$ induced from it.

  * •

Let $\mathbf{U}M$ denote the space of direction vectors on $M$, i.e., the
space of elements of the form $(x,\vec{v})$ for $x\in M$ and a unit vector
$\vec{v}\in T_{x}M-\\{O\\}$ with the projection
$\pi_{\mathbf{U}M}:\mathbf{U}M\rightarrow M$.

  * •

Let $\mathbf{U}\hat{M}$ denote the space of direction vectors on $\hat{M}$
covering $\mathbf{U}M$ with the deck transformation group $\Gamma_{M}$. Let
$\pi_{\mathbf{U}\hat{M}}:\mathbf{U}\hat{M}\rightarrow\hat{M}$ denote the
projection.

  * •

Let $d_{\mathbf{U}M}$ denote the path-metric on $\mathbf{U}M$ obtained from
the natural Riemannian metric on the unit tangent bundle $\mathbf{U}M$ of $M$
obtained from the Riemannian metric of $M$ as a sphere bundle. We will use
$d_{\mathbf{U}\hat{M}}$ to denote the induced path-metric on
$\mathbf{U}\hat{M}$.

A complete isometric geodesic in $\hat{M}$ is a geodesic that is an isometry
of ${\mathbb{R}}$ into $\hat{M}$ equipped with $M$. Actually, in the geodesic
metric spaces, this is equivalent notion to geodesic. Some but not all
Riemannian geodesics are isometries. A complete isometric geodesic in $M$ is a
geodesic that lifts to a complete isometric geodesic in $\hat{M}$. A ray in
$\hat{M}$ is an isometric geodesic defined on $[0,\infty)$.

We consider the subspace of $\mathbf{U}M$ where complete isometric geodesics
pass. We denote this subspace by $\mathbf{UC}M$, and call it the complete
isometric geodesic unit-tangent bundle. The inverse image in
$\mathbf{U}\hat{M}$ is a closed set denoted by $\mathbf{UC}\hat{M}$. Since a
limit of a sequence of isometric geodesics is an isometric geodesic,
$\mathbf{UC}M$ is compact and $\mathbf{UC}\hat{M}$ is closed and locally
compact. However, $\pi_{\mathbf{U}M}(\mathbf{UC}M)$ may be a proper subset of
$M$ for the projection $\pi_{\mathbf{U}M}:\mathbf{U}M\rightarrow M$, and
$\pi_{\mathbf{U}\hat{M}}(\mathbf{UC}\hat{M})$ may be one of $\hat{M}$.

Let $\rho:\Gamma_{M}\rightarrow{\mathsf{GL}}(n,{\mathbb{R}})$ be a
homomorphism. There is an action of $\Gamma_{M}$ on
$\mathbf{UC}\hat{M}\times{\mathbb{R}}^{n}$ given by

$\gamma\cdot(x,\vec{v})=(\gamma(x),\rho(\gamma)(\vec{v}))\hbox{ for
}x\in\mathbf{UC}\hat{M},\vec{v}\in{\mathbb{R}}^{n}.$

Let ${\mathbb{R}}^{n}_{\rho}$ denote the bundle over $\mathbf{UC}M$ given by
taking the quotient $\mathbf{UC}\hat{M}\times{\mathbb{R}}^{n}$ by
$\Gamma_{M}$. Let
$\left|\mkern-2.0mu\left|\cdot\right|\mkern-2.0mu\right|_{{\mathbb{R}}^{n}_{\rho}}$
denote the fiberwise quadratic norm on ${\mathbb{R}}^{n}_{\rho}$ obtained by a
partition of unity for example.

Recall that there is a geodesic flow $\phi$ on $\mathbf{U}M$ restricting to
one on $\mathbf{UC}M$, again denoted by $\phi$. This lifts to a flow denoted
by $\phi$ of $\mathbf{UC}\hat{M}$. Hence, there is a flow $\Phi$ on
$\mathbf{UC}\hat{M}\times{\mathbb{R}}^{n}$ again descending to a flow to be
denoted by $\Phi$ on ${\mathbb{R}}^{n}_{\rho}$.

###  1.2. Main result

In this paper, we connect the partial hyperbolicity in the sense of the
dynamical system and the P-Anosov properties. The significance and the
motivation of the main result are written in [7].

We need the following objects since we need to work on manifolds for the next
article [7].

######  Definition 1.1 (Partial hyperbolicity).

Suppose that $M$ is a compact Riemannian manifold with convex boundary. Let
$\hat{M}$ be a regular cover of $M$ with the deck transformation group
$G_{M}$. A representation $\rho:G_{M}\rightarrow{\mathsf{GL}}(n,{\mathbb{R}})$
is partially hyperbolic in a bundle sense if the following hold for a
Riemannian metric on $M$:

  1. (i)

There exist nontrivial $C^{0}$-subbundles ${\mathbb{V}}_{+},{\mathbb{V}}_{0}$,
and ${\mathbb{V}}_{-}$ in ${\mathbb{R}}^{n}_{\rho}$ with fibers

${\mathbb{V}}_{+}(x),{\mathbb{V}}_{0}(x),{\mathbb{V}}_{-}(x)\subset{\mathbb{V}}(x)$

for the fiber ${\mathbb{V}}(x)$ of ${\mathbb{V}}$ over each point $x$ of
$\mathbf{UC}M$ invariant under the flow $\Phi_{t}$ of ${\mathbb{V}}$ over the
geodesic flow $\phi_{t}$ of $\mathbf{UC}M$.

  2. (ii)

${\mathbb{V}}_{+}(x),{\mathbb{V}}_{0}(x)$ and ${\mathbb{V}}_{-}(x)$ for each
$x\in\mathbf{UC}M$ are independent subspaces, and their sum equals
${\mathbb{V}}(x)$.

  3. (iii)

For any fiberwise metric on ${\mathbb{R}}^{n}_{\rho}$ over $\mathbf{UC}M$, the
lifted action of $\Phi_{t}$ on ${\mathbb{V}}_{+}$ (resp. ${\mathbb{V}}_{-}$)
is dilating (resp. contracting). That is, for coefficients $A>0,a>0$,
$A^{\prime}>0,a^{\prime}>0$:

    1. (a)

$\left|\mkern-2.0mu\left|\Phi_{-t}(\mathbf{v})\right|\mkern-2.0mu\right|_{{\mathbb{R}}^{n}_{\rho},\phi_{-t}(m)}\leq
A\exp(-at)\left|\mkern-2.0mu\left|\mathbf{v}\right|\mkern-2.0mu\right|_{{\mathbb{R}}^{n}_{\rho},m}$
for $\mathbf{v}\in{\mathbb{V}}_{+}(m)$ as $t\rightarrow\infty$.

    2. (b)

$\left|\mkern-2.0mu\left|\Phi_{t}(\mathbf{v})\right|\mkern-2.0mu\right|_{{\mathbb{R}}^{n}_{\rho},{\phi_{t}(m)}}\leq
A\exp(-at)\left|\mkern-2.0mu\left|\mathbf{v}\right|\mkern-2.0mu\right|_{{\mathbb{R}}^{n}_{\rho},m}$
for $\mathbf{v}\in{\mathbb{V}}_{-}(m))$ as $t\rightarrow\infty$.

    3. (c)

(Dominance properties)

(1)
$\frac{\left|\mkern-2.0mu\left|\Phi_{t}(\mathbf{w})\right|\mkern-2.0mu\right|_{{\mathbb{R}}^{n}_{\rho},\phi_{t}(m)}}{\left|\mkern-2.0mu\left|\Phi_{t}(\mathbf{v})\right|\mkern-2.0mu\right|_{{\mathbb{R}}^{n}_{\rho},\phi_{t}(m)}}\leq
A^{\prime}\exp(-a^{\prime}t)\frac{\left|\mkern-2.0mu\left|\mathbf{w}\right|\mkern-2.0mu\right|_{{\mathbb{R}}^{n}_{\rho},m}}{\left|\mkern-2.0mu\left|\mathbf{v}\right|\mkern-2.0mu\right|_{{\mathbb{R}}^{n}_{\rho},m}}\\\
\hbox{ for
}\mathbf{v}\in{\mathbb{V}}_{+}(m),\mathbf{w}\in{\mathbb{V}}_{0}(m)\hbox{ as
}t\rightarrow\infty,\\\ \hbox{ or for
}\mathbf{v}\in{\mathbb{V}}_{0}(m),\mathbf{w}\in{\mathbb{V}}_{-}(m)\hbox{ as
}t\rightarrow\infty.$

We require additionally that $\dim{\mathbb{V}}_{+}=\dim{\mathbb{V}}_{-}\geq
1$. Here $\dim{\mathbb{V}}_{+}$ is the partial hyperbolicity index of $\rho$
for the associated bundle decomposition
${\mathbb{V}}_{+},{\mathbb{V}}_{0},{\mathbb{V}}_{-}$ where
$\dim{\mathbb{V}}_{+}$ is maximal and
$\dim{\mathbb{V}}_{+}=\dim{\mathbb{V}}_{-}<n/2$, so that
$\dim{\mathbb{V}}_{0}\geq 1$. Furthermore, ${\mathbb{V}}_{0}$ is said to be
the neutral subbundle of ${\mathbb{V}}$. (See Definition 1.5 of [9] or [4].)

The definition does depend on the Riemannian metric but not on
$\left|\mkern-2.0mu\left|\cdot\right|\mkern-2.0mu\right|_{{\mathbb{R}}^{n}_{\rho}}$.
However, if $\Gamma_{M}$ is word hyperbolic, then we can replace
$\mathbf{UC}\hat{M}$ by a Gromov flow space which depends only on
$\Gamma_{M}$. (See Theorem 2.6 and Gromov [13] or Mineyev [24].)

We say that a complete affine manifold $N$ has a partially hyperbolic linear
holonomy group if

  * •

there exists a FS submanifold $M$ in $N$ with convex boundary for a Riemannian
metric of $N$, and

  * •

the linear part
$\rho:\pi_{1}(N)=\Gamma_{\hat{M}}\rightarrow{\mathsf{GL}}(n,{\mathbb{R}})$ of
an affine holonomy homomorphism $\rho^{\prime}$ is a partially hyperbolic
representation in a bundle sense.

Since partial hyperbolicity is equivalent to the Anosov property by Theorem
1.1, $\pi_{1}(N)$ always is hyperbolic. By Theorem 2.6 , there is a quasi-
isometry from $\mathbf{UC}\hat{M}$ to the Gromov flow space
$\partial^{(2)}_{\infty}M\times{\mathbb{R}}$ preserving the geodesic length
parameter. Hence, the discussion reduces to one on the Gromov flow space,
which is independent of the choice of FS submanifolds.

Let $\log\lambda_{1},\dots,\log\lambda_{n}$ denote the logarithmic eigenvalue
functions defined on the maximal abelian Lie algebra of
$\mathfrak{gl}(n,{\mathbb{R}})$. For roots

$\theta=\left\\{\log\lambda_{i_{1}}-\log\lambda_{i_{1}+1},\ldots,\log\lambda_{i_{m}}-\log\lambda_{i_{m}+1}\right\\},1\leq
i_{1}<\cdots<i_{m}\leq n-1,$

of $\mathrm{GL}(n,{\mathbb{R}})$, the parabolic subgroup $P_{\theta}$ (resp.
$\left.P_{\theta}^{-}\right)$ is the subgroup of block upper (resp. lower)
triangular matrices in $\mathrm{GL}(n,{\mathbb{R}})$ with square diagonal
blocks of sizes $i_{1},i_{2}-i_{1},\ldots,i_{m}-i_{m-1},n-i_{m}$ respectively.

######  Theorem 1.1 (Choi-Kapovich-Stecker).

Let $N$ be a complete affine $n$-manifold where $K(\pi_{1}(N),1)$ is realized
by a finite complex. Let
$\rho:\pi_{1}(N)\rightarrow{\mathsf{GL}}(n,{\mathbb{R}})$ is a linear part of
an affine holonomy homomorphism $\rho^{\prime}$.

Then $\rho$ is $P$-Anosov for any parabolic subgroup $P$ of
${\mathsf{GL}}(n,{\mathbb{R}})$ with $P=P_{\theta}$ for $\theta$ containing
$\log\lambda_{k}-\log\lambda_{k+1},k\leq n/2$ if and only if $\rho$ is
partially hyperbolic of index $k$. In these cases $k<n/2$.

###  1.3. Outline

In Section 2, we will introduce affine structures and some basic facts on
$\delta$-hyperbolic metric spaces. Suppose that $\Gamma_{M}$ is word-
hyperbolic. Then $\hat{M}$ has a Gromov hyperbolic Riemannian metric by the
Svarc-Milnor theorem. We discuss the ideal boundary of $\hat{M}$, identifiable
with the Gromov boundary, and about the complete isometric geodesics in
$\hat{M}$. We relate the space of complete isometric geodesics to the Gromov
flow space.

In Section 3, we provide the review of the definition of the $P$-Anosov
property of Guichard-Wienhard [15]. For convenience for reductive groups, we
will use the approaches of Gúeritaud-Guichard-Kassel-Wienhard [14] and Bochi-
Potrie-Sambarino [3] using singular values of the linear holonomy group. (The
theory of Kapovich, Leeb, and Porti [17], [19] is equivalent to these but it
is not so explicit for our purposes here.) We give various associated
definitions.

In Section 4, we have the main arguments. In Section 4.1, we will relate the
partial hyperbolic property in the singular value sense to that in the bundle
sense. In Section 4.2, we show that the linear part of the holonomy
representation of a complete affine manifold is $P$-Anosov if and only if it
is partially hyperbolic. First, we do this when the linear holonomy group has
a reductive Zariski closure. This involves the fact that $1$ has to be an
eigenvalue of each element of the linear holonomy groups of complete affine
manifolds. We relate it to singular values using the clever ideas of Danciger
and Stecker, the spectrum theory of Breulliard-Sert [5], Benoist [1], [2], and
the work of Potrie-Kassel [20]. We could generalize to the cases of
nonreductive linear holonomy groups using the stability of the Anosov and
partial hyperbolic properties. This will prove Theorem 1.1.

###  1.4. Acknowledgments

We thank Michael Kapovich for various help with geometric group theory and
coarse geometry. This article began with some discussions with Michael
Kapovich during the conference honoring the 60th birthday of William Goldman
at the University of Maryland, College Park, in 2016. We thank Jeffrey
Danciger and Florian Stecker for help with $P$-Anosov properties which yielded
the main corollary of the paper. We also thank BIRS, Banff, Canada, and KIAS,
Seoul, where some of this work was done.

##  2\.  Preliminary

###  2.1. Convergences of geodesics

Let $M$ be a manifold of dimension $n$ with a regular cover $\hat{M}$ and the
deck transformation group $\Gamma_{M}$. We suppose that $M$ has a Riemannian
metric with convex boundary. We say that a sequence of rays (resp. complete
isometric geodesics) $l_{i}$ converges to a ray (resp. complete geodesic) $l$
(that is, $l_{i}\rightarrow l$) if $l_{i}(t)\rightarrow l(t)$ for each
$t\in[0,\infty)$ (resp. $t\in{\mathbb{R}}$).

By the Arzelà-Ascoli theorem and the properness of $\hat{M}$, any sequence of
rays (resp. complete isometric geodesics) converges to a ray (resp. complete
isometric geodesic) if the sequence of the pairs of their $0$-points and
directions at the $0$-points converges in $\mathbf{U}\hat{M}$. (See Section
11.11 of [10].)

###  2.2. $(G,X)$-structures and affine manifolds

Let $G$ be a Lie group acting transitively and faithfully on a space $X$. A
$(G,X)$-structure on a manifold $N$ is a maximal atlas of charts to $X$ so
that transition maps are in $G$. This is equivalent to $M$ having a pair
$({\bf dev},h)$ where $\hat{M}$ is a regular cover of $M$ with a deck
transformation group $\Gamma_{M}$ such that

  * •

$h:\Gamma_{M}\rightarrow G$ is a homomorphism, called a holonomy homomorphism,
and

  * •

${\bf dev}:\hat{M}\rightarrow X$ is an immersion, called a developing map,
satisfying

${\bf dev}\circ\gamma=h(\gamma)\circ{\bf dev}$

for each deck transformation $\gamma\in\Gamma_{M}$.

We say that an $n$-manifold $M$ with a $(G,X)$-structure is complete if ${\bf
dev}:\hat{M}\rightarrow X$ is a diffeomorphism. If $M$ is complete, we have a
diffeomorphism $M\rightarrow X/h(\Gamma_{M})$, and $h(\Gamma_{M})$ acts
properly discontinuously and freely on $X$. Complete $(G,X)$-structures on $M$
are classified by the conjugacy classes of representations
$\Gamma_{M}\rightarrow G$ with properly discontinuous and free actions on $X$.
(See Section 3.4 of [28].)

Note that the completeness and compactness of $M$ have no relation for some
$(G,X)$-structures unless $G$ is in the isometry group of $X$ with a
Riemannian metric. (The Hopf-Rinow lemma may fail.)

Now, we go over to the affine structures. Let $\mathds{A}^{n}$ be a complete
affine space. Let ${\mathbf{Aff}}(\mathds{A}^{n})$ denote the group of affine
transformations of $\mathds{A}^{n}$ whose elements are of the form:

$x\mapsto Ax+{\mathbf{v}}$

for a vector ${\mathbf{v}}\in{\mathbb{R}}^{n}$ and
$A\in{\mathsf{GL}}(n,{\mathbb{R}})$. Let
$\mathcal{L}:{\mathbf{Aff}}(\mathds{A}^{n})\rightarrow{\mathsf{GL}}(n,{\mathbb{R}})$
denote map sending each element of ${\mathbf{Aff}}(\mathds{A}^{n})$ to its
linear part in ${\mathsf{GL}}(n,{\mathbb{R}})$.

An affine $n$-manifold is an $n$-manifold with an
$({\mathbf{Aff}}(\mathds{A}^{n}),\mathds{A}^{n})$-structure. A complete affine
$n$-manifold is an $n$-manifold $M$ of the form $\mathds{A}^{n}/\Gamma$ for
$\Gamma\subset{\mathbf{Aff}}(\mathds{A}^{n})$ acting properly discontinuously
and freely on $\mathds{A}^{n}$.

###  2.3. Gromov hyperbolic spaces

Let us recall some definitions: A metric space is geodesic if every pair of
points are connected by a geodesic, i.e., a path that is an isometry from an
interval to the metric space. We will only work with geodesic metric spaces.
Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces. A map $f:X\rightarrow Y$ is
called $(L,C^{\prime})$-coarse Lipschitz for $L,C>0$ if

$d_{Y}(f(x),f(x^{\prime})\leq Ld_{X}(x,x^{\prime})+C^{\prime}\hbox{ for all
}x,x^{\prime}\in X.$

A map $f:X\rightarrow Y$ is an $(L,C)$-quasi-isomeric embedding if $f$
satisfies

$L^{-1}d_{X}(x,x^{\prime})-C\leq d_{Y}(f(x),f(x^{\prime}))\leq
Ld_{X}(x,x^{\prime})+C\hbox{ for all }x,x^{\prime}\in X.$

A map $\bar{f}:Y\rightarrow X$ is a $C$-coarse inverse of $f$ for $C>0$ if

$d_{X}(\bar{f}\circ f,{\rm I}_{X})\leq C,\hbox{ and }d_{Y}(f\circ\bar{f},{\rm
I}_{Y})\leq C.$

A map $f:X\rightarrow Y$ between metric spaces is called a quasi-isometry if
it is a coarse Lipschitz and admits a coarse Lipschitz coarse inverse map.

######  Lemma 2.1.

$\pi_{\mathbf{U}\hat{M}}$ is a quasi-isometry.

###### Proof.

Each fiber is uniformly bounded under $d_{\mathbf{U}\hat{M}}$. ∎

A metric space $(X,d)$ is proper if every metric ball has a compact closure.
In other words, $d_{p}(\cdot)=d(p,\cdot)$ is a proper function
$X\rightarrow{\mathbb{R}}$. Clearly, a complete Riemannian manifold is a
proper metric space. We denote by $B^{d}(x,R)$ the ball of radius $<R$ in $X$
with center $x,x\in X$.

Let $X$ be a geodesic metric space with metric $d$. A geodesic triangle $T$ is
a concatenation of three geodesics $\tau_{1},\tau_{2},\tau_{3}$ where the
indices are modulo $3$ ones. The thinness radius of a geodesic triangle $T$ is
the number

$\delta(T):=\max_{j=1,2,3}\left(\sup_{p\in\tau_{j}}d(p,\tau_{j+1}\cup\tau_{j+2})\right).$

A geodesic metric space $X$ is called $\delta$-hyperbolic in the sense of Rips
if every geodesic triangle $T$ in $X$ is $\delta$-thin.

By Corollary 11.29 of [10] as proved by Section 6.3C of Gromov [13], the
Gromov hyperbolicity is equivalent to the Rips hyperbolicity for geodesic
metric spaces. We will use these concepts interchangeably.

We will assume that $\Gamma_{M}$ is word-hyperbolic for our discussions below.
This is equivalent to assuming that $\hat{M}$ is Gromov hyperbolic by the
Svarc-Milnor Lemma (see [23] and [27]).

Two rays $\rho_{1}$ and $\rho_{2}$ of $X$ are equivalent if $t\mapsto
d(\rho_{1}(t),\rho_{2}(t))$ is bounded. This condition is equivalent to the
one that their Hausdorff distance under $d$ is bounded. Assume that $X$ is
$\delta$-hyperbolic. $X$ has a well-defined ideal boundary
$\partial_{\infty}X$ as in Definition 3.78 of [10], i.e., the space of
equivalence classes of geodesic rays. (See [8] and [11].) We denote by
$\partial^{(2)}_{\infty}X=\partial_{\infty}X\times\partial_{\infty}X-\Delta$
where $\Delta$ is the diagonal set.

######  Lemma 2.2.

Let $M$ be a compact manifold with the induced path metric $d_{\hat{M}}$ on
$\hat{M}$ from a Riemannian metric of $M$. Let $\Gamma_{M}$ be the deck
transformation group of $\hat{M}\rightarrow M$. Suppose that $\Gamma_{M}$ is
word-hyperbolic.

Then there exists a constant $C>0$ so that for every pair of complete
isometric geodesics $l_{1}$ and $l_{2}$ in $\hat{M}$ with the identical set of
endpoints in $\partial_{\infty}\hat{M}$, the Hausdorff distance between
$l_{1}$ and $l_{2}$ under $d$ is bounded above by $C$.

###### Proof.

Since $l_{1}$ and $l_{2}$ are both $(1,0)$-quasi-geodesics, this follows from
Proposition 3.1 of Chapter 3 of [8]. ∎

We put on $\hat{M}\cup\partial_{\infty}\hat{M}$ the shadow topology which is a
first countable Hausdorff space by Lemma 11.76 of [10]. It is also compact
according to Section 11.11 of [10]. Since it is first countable, we do not
need to consider nets on the space but only sequences to understand the
continuity of the real-valued functions.

We denote by $\partial_{G}X$ the Gromov boundary of a $\delta$-hyperbolic
geodesic metric space $X$. (See Section 11.12 of [10].) By Theorem 11.104 of
[10], there is a homeomorphism $h:\partial_{\infty}X\rightarrow\partial_{G}X$
given by sending the equivalence class $[\rho]$ of a ray $\rho$ to the
equivalence class of $\\{\rho(n)\\}_{n\in{\mathbb{Z}}_{+}}$. We will identify
these two spaces using this map.

Let $\hat{\Gamma}_{M}$ denote the Cayley graph of $\Gamma_{M}$. The Gromov
boundary $\partial_{G}\hat{M}$ of $\hat{M}$ and can be identified with the
boundary of $\hat{\Gamma}_{M}$ with the word metric by Theorem 11.108 of [10].

For any Gromov hyperbolic geodesic metric space $X$, the set of $GX$ of
complete isometric geodesics has a metric: for $g,h\in GX$, we define the
metric $d_{GX}$ given by

(2) $d_{GX}(g,h):=\int^{\infty}_{\infty}d(g(t),h(t))2^{-|t|}dt.$

(See Gromov [13].) Let $GX$ have this metric topology. $GX$ is clearly locally
closed since the limit of a sequence of isometries from ${\mathbb{R}}$ is an
isometry from ${\mathbb{R}}$. (See Section 11.11 of [10].)

######  Lemma 2.3.

Suppose that $\hat{M}$ is Gromov hyperbolic. Then there is a quasi-isometric
homeomorphism $\mathcal{F}:\mathbf{UC}\hat{M}\rightarrow G\hat{M}$ by taking a
unit vector $\vec{u}$ at $x$ to a complete isometric geodesic
${\mathbb{R}}\rightarrow\hat{M}$ passing $x$ tangent to $\vec{u}$.

###### Proof.

There is a map $G\hat{M}\rightarrow\mathbf{UC}\hat{M}$ given by sending the
complete isometric geodesic $g:{\mathbb{R}}\rightarrow\hat{M}$ to
$(g(0),\vec{v}_{0})$ where $\vec{v}_{0}$ is the unit tangent vector at $g(0)$.
This map is clearly one-to-one and onto. In Section III in [22], the map
$G\hat{M}\rightarrow\hat{M}$ given by $g\rightarrow g(0)$ is shown to be
quasi-isometry. Hence, this map post-composed by a lift sending each $g(0)$ to
a vector in the fiber of $\mathbf{UC}\hat{M}$ over $g(0)$ is a quasi-isometry.
For a sequence $\\{g_{i}\\}$ of geodesics in $\hat{M}$, if
$d_{GX}(g,g_{i})\rightarrow 0$, then
$(g_{i}(0),g_{i}^{\prime}(0))\rightarrow(g(0),g^{\prime}(0))$ since otherwise
we easily obtain a positive lower bound of the integral (2).

The inverse map $\mathbf{UC}\hat{M}\rightarrow G\hat{M}$ is also continuous by
the continuity of the exponential map and considering equations of the form
(2) where $d_{X}(g(t),h(t))$ grows sublinearly. We can show the continuity of
the integral values under $g$ and $h$ by cutting off for $|t|>N$. ∎

###  2.4. Flow space $G\hat{M}$ quasi-isometric to
$\partial_{\infty}^{(2)}\hat{M}\times{\mathbb{R}}$

We assume that $\Gamma_{M}$ is word-hyperbolic and denote by $F\Gamma_{M}$ the
Gromov geodesic flow space obtained by the following proposition. For a group
$\Gamma$, we let $\hat{\Gamma}$ denote its Cayley graph. We denote
$\partial_{\infty}^{(2)}\hat{\Gamma}:=\partial_{\infty}\hat{\Gamma}\times\partial_{\infty}\hat{\Gamma}-\\{(t,t)|t\in\partial_{\infty}\hat{\Gamma}\\}$.

######  Theorem 2.4 (Theorem 8.3.C of Gromov [12], Theorem 60 of Mineyev
[24]).

Let $\Gamma$ be a finitely generated word hyperbolic group.

Then there exists a proper hyperbolic metric space $F\Gamma$ with following
properties:

  * (i)

$\Gamma\times{\mathbb{R}}\times{\mathbb{Z}}/2{\mathbb{Z}}$ acts on $F\Gamma$.

  * (ii)

The $\Gamma\times{\mathbb{Z}}/2{\mathbb{Z}}$-action is isometric.

  * (iii)

Every orbit $\Gamma\rightarrow F\Gamma$ is a quasi-isometry.

  * (iv)

The ${\mathbb{R}}$-action is free, and every orbit ${\mathbb{R}}\rightarrow
F\Gamma$ is a quasi-isometric embedding. The induced map
$F\Gamma/{\mathbb{R}}\rightarrow\partial_{\infty}^{(2)}\hat{\Gamma}$ is a
homeomorphism.

We shall say that $F\Gamma$ is the flow space of $\Gamma$. In fact, $F\Gamma$
is unique up to a $\Gamma\times{\mathbb{Z}}/2{\mathbb{Z}}$-equivariant quasi-
isometry sending ${\mathbb{R}}$-orbits to ${\mathbb{R}}$-orbits. We shall
denote by $\phi_{t}$ the ${\mathbb{R}}$-action on $F\Gamma$ and by
$(\tau_{+},\tau_{-}):F\Gamma\rightarrow
F\Gamma/{\mathbb{R}}\cong\partial_{\infty}^{(2)}\hat{\Gamma}$ the maps
associating to an element of $F\Gamma$ the endpoint of its
${\mathbb{R}}$-orbit. Gromov identifies $F\Gamma$ with
$\partial_{\infty}^{(2)}\Gamma\times{\mathbb{R}}$ with a certain metric,
called the Gromov metric.

######  Proposition 2.5.

Let $M$ be a compact manifold with a covering map $\hat{M}\rightarrow M$ with
a deck transformation group $\Gamma_{M}$. Suppose that $\Gamma_{M}$ is word-
hyperbolic.

Then there is a quasi-isometric surjective map
$\mathcal{E}:G\hat{M}\rightarrow\partial^{(2)}_{\infty}\hat{M}\times{\mathbb{R}}$
with compact fibers. The map obtained from composing with a projection to the
first factor is a map given by taking the endpoints of complete isometric
geodesics.

###### Proof.

Choose a basepoint $x_{0}$ in $\hat{M}$. For each complete isometric geodesic
$g:{\mathbb{R}}\rightarrow\hat{M}$, let $g_{x_{0}}$ denote the projection
point on $g({\mathbb{R}})$ that is of the minimal distance from $x_{0}$ and
let $\partial_{+}g,\partial_{-}g\in\partial_{\infty}\hat{M}$ denote the
forward and backward endpoints, and let $t(g)\in{\mathbb{R}}$ denote $\pm
d(g(0),g_{x_{0}})$ where we use $+$ if $g(0)$ is ahead of $g_{x_{0}}$.

We define $\mathcal{E}(g)=(\partial_{+}g,\partial_{-}g,t(g))$. The
surjectivity follows from Proposition 2.1 of Chapter 2 of [8] since $t$ is an
isometry on each complete isometric geodesic. The compactness of the fiber is
implied by Lemma 2.2.

Champetier [6] constructs the space $G\hat{\Gamma}_{M}$ from
$\hat{\Gamma}_{M}$ quasi-isometric to $\Gamma_{M}$ following Gromov. Recall
$\partial_{\infty}\hat{M}=\partial_{\infty}\hat{\Gamma}_{M}$. Using the orbit
map $\Gamma_{M}\rightarrow\hat{M}$ sending $\gamma$ to $\gamma(x_{0})$, we
extend the map to $\hat{\Gamma}_{M}\rightarrow\hat{M}$ by sending each edge to
a shortest geodesic. Let $I:G\hat{\Gamma}_{M}\rightarrow G\hat{M}$ be a map
given by sending a geodesic $g$ in $\hat{\Gamma}_{M}$ to a geodesic
$g^{\prime}$ in $\hat{M}$ with the same endpoints in
$\partial_{\infty}\hat{M}$ with $g(0)$ to a nearest point to the image of
$g^{\prime}$. The map $I^{\prime}:G\hat{M}\rightarrow G\hat{\Gamma}_{M}$ can
be defined by taking the geodesic $g$ of $\hat{M}$ to a geodesic $g^{\prime}$
of $\hat{\Gamma}_{M}$ in the same manner with $g(0)$ going to one of the
element of $\Gamma_{M}(x_{0})$ nearest to it. Since a geodesic in
$\hat{\Gamma}_{M}$ is a quasi-geodesic in $\hat{M}$, and every geodesic in
$\hat{M}$ is uniformly bounded away from one in $\hat{\Gamma}_{M}$ in the
Hausdorff distance by Theorem 11.72 of [10], $G\hat{M}$ is quasi-isometric
with $G\hat{\Gamma}_{M}$ by the maps $I$ and $I^{\prime}$ using (2).

We define
$\mathcal{E}^{\prime}:G\hat{\Gamma}_{M}\rightarrow\partial_{\infty}^{(2)}\hat{\Gamma}_{M}\times{\mathbb{R}}$
as we did for $\mathcal{E}$ above. By Proposition 4.8 and (4.3) of [6],
$\mathcal{E}^{\prime}$ sends $G\hat{\Gamma}_{M}$ to
$\partial_{\infty}^{(2)}\hat{\Gamma}_{M}\times{\mathbb{R}}=\partial_{\infty}^{(2)}\hat{M}\times{\mathbb{R}}$
as a quasi-isometric homeomorphism. Since $\mathcal{E}(g,t)$ and
$\mathcal{E}^{\prime}\circ I^{\prime}(g,t)$ are uniformly bounded away from
each other, the result follows. ∎

The following shows that $\mathbf{UC}\hat{M}$ can be used as our Gromov flow
space up to some collapsing.

######  Theorem 2.6.

The map
$\mathcal{E}\circ\mathcal{F}:\mathbf{UC}\hat{M}\rightarrow\partial^{(2)}M\times{\mathbb{R}}$
is a quasi-isometry so that each complete isometric geodesic in
$\mathbf{UC}\hat{M}$ goes to $(t_{1},t_{2})\times{\mathbb{R}}$ for its
endpoint pair $(t_{1},t_{2})\in\partial^{(2)}_{\infty}\hat{M}$ as an isometry.

###### Proof.

Proposition 2.5 and Lemma 2.3 imply the result. ∎

##  3\. Partial hyperbolicity and $P$-Anosov properties

###  3.1. Some background

We have to repeat some of the standard materials from Section 2.2 of [14] and
Chapter 2 of [16]. Let us consider a real reductive Lie group $G$ with a
maximal compact subgroup $K$.

We assume that $\mathrm{Ad}(G)\subset{\mathbf{Aut}}(\mathfrak{g})$ for
simplicity. For example, this holds if $G$ is connected. We can usually reduce
everything here to the case when $G$ is connected. Here, $G$ is an almost
product of $Z(G)_{0}$ and $G_{s}$ where $Z(G)_{0}$ is the identity component
of the center $Z(G)$ of $G$ and $G_{s}=D(G)$ the derived subgroup of $G$,
which is semisimple.

A parabolic subgroup $P$ of $G$ is a subgroup $P$ of the form
$G\cap\mathbf{P}({\mathbb{R}})$ for some algebraic subgroup $\mathbf{P}$ of an
algebraic group $\mathbf{G}$ where
$\mathbf{G}({\mathbb{R}})/\mathbf{P}({\mathbb{R}})$ is compact and
$\mathbf{G}({\mathbb{R}})=G$.

Let $\mathfrak{z}(\mathfrak{g})$ denote the Lie algebra of $Z(G)$, and
$\mathfrak{g}_{s}$ the Lie algebra of $G_{s}$. Then
$\mathfrak{g}=\mathfrak{z}(\mathfrak{g})\oplus\mathfrak{g}_{s}$. Letting
$\mathfrak{t}$ to denote the Lie algebra of $K$, we have
$\mathfrak{g}=\mathfrak{t}\oplus\mathfrak{q}$ where $\mathfrak{q}$ is the
$-1$-eigenspace of an involution $\theta$ preserving $\mathfrak{t}$ as the
$1$-eigenspace. Let $\mathfrak{a}\subset\mathfrak{q}$ be the maximal abelian
subalgebra, i.e., the Cartan subalgebra of $\mathfrak{g}$. Then
$\mathfrak{a}=\mathfrak{z}(\mathfrak{g})\cap\mathfrak{q}\oplus\mathfrak{a}_{s}$
where $\mathfrak{a}_{s}=\mathfrak{a}\cap\mathfrak{g}_{s}$, a maximal abelian
subalgebra of $\mathfrak{q}\cap\mathfrak{g}_{s}$.

We have a decomposition to $ad(\mathfrak{a})$-eigenspaces
$\mathfrak{g}=\mathfrak{g}_{0}\oplus\bigoplus_{\alpha\in\Sigma}\mathfrak{g}_{\alpha}$.
Here, $\mathfrak{g}_{0}$ is the direct sum of $z(\mathfrak{g})$ and of the
centralizer of $\mathfrak{a}$ in $\mathfrak{g}_{s}$. The set
$\Sigma\subset\mathfrak{a}^{*}=\mathrm{Hom}_{\mathbb{R}}(\mathfrak{a},\mathbb{R})$
projects to a (possibly nonreduced) root system of $\mathfrak{a}_{s}^{*},$ and
each $\alpha\in\Sigma$ is called a restricted root of $\mathfrak{a}$ in
$\mathfrak{g}$. There is a subset $\Delta\subset\Sigma$ called a simple
system, i.e. a subset such that any root is expressed uniquely as a linear
combination of elements of $\Delta$ with coefficients all of the same sign;
the elements of $\Delta$ are called the simple roots. There is a subset
$\Sigma^{+}\subset\Sigma$ which is the set of positive roots, i.e. roots that
are nonnegative linear combinations of elements of $\Delta$, where
$\Sigma=\Sigma^{+}\cup\left(-\Sigma^{+}\right)$ holds additionally.

We define $P_{\theta}$ to be the parabolic subgroup with Lie algebra

$\operatorname{Lie}\left(P_{\theta}\right)=\mathfrak{g}_{0}\oplus\bigoplus_{\alpha\in\Sigma^{+}}\mathfrak{g}_{\alpha}\oplus\bigoplus_{\alpha\in\Sigma^{+}\backslash\Sigma_{\theta}^{+}}\mathfrak{g}_{-\alpha}.$

The connected components of
$\mathfrak{a}\setminus\cup_{\alpha\in\Sigma}\ker(\alpha)$ are called Weyl
chambers of $\mathfrak{a}$. The component where every $\alpha\in\Sigma_{+}$ is
positive is denoted by $\mathfrak{a}^{+}$.

For $G={\mathsf{SL}}_{\pm}(n,{\mathbb{R}})$. The maximal abelian subalgebra
$A_{n}$ of $\mathfrak{p}$ in this case is the subspace of diagonal matrices
with entries $a_{1},\dots,a_{n}$ where $a_{1}+\dots+a_{n}=0$. Let
$\log\lambda_{i}:A_{n}\rightarrow{\mathbb{R}}$ denote the projection to the
$i$-th factor. $\Sigma$ consists of
$\alpha_{ij}:=\log\lambda_{i}-\log\lambda_{j}$ for $i\neq j$, and
$\mathfrak{sl}(n,{\mathbb{R}})_{\alpha_{ij}}=\\{cE_{ij}|c\in{\mathbb{R}}\\}$
where $E_{ij}$ is a matrix with entry $1$ at $(i,j)$ and zero entries
everywhere else.

A positive Weyl chamber is given by

$A_{n}^{+}=\\{(a_{1},\dots,a_{n})|a_{1}\geq\dots\geq
a_{n},a_{1}+\cdots+a_{n}=0\\},$

and we are given

$\displaystyle\Sigma^{+}=\\{\log\lambda_{i}-\log\lambda_{j}|i<j\\}\hbox{ and
}$
$\displaystyle\Delta=\\{\alpha_{i}:=\log\lambda_{i}-\log\lambda_{i+1}|i=1,\dots,n-1\\}.$

We recall Example 2.14 of [14]: Let $G$ be $\mathrm{GL}(n,{\mathbb{R}})$ seen
as a real Lie group. Its derived group is
$G_{s}=D(G)=\mathrm{SL}(n,{\mathbb{R}}).$ We can take
$\mathfrak{a}\subset\mathfrak{gl}(n,{\mathbb{R}})$ to be the set of real
diagonal matrices of size $n\times n$. As above, for $1\leq i\leq n,$ let
$\log\lambda_{i}\in\mathfrak{a}^{*}$ be the evaluation of the $i$-th diagonal
entry. Then

$\mathfrak{a}=\mathfrak{z}(\mathfrak{g})\cap\mathfrak{a}\oplus\mathfrak{a}_{s}\hbox{
where }\mathfrak{z}(\mathfrak{g})\cap\mathfrak{a}=\bigcap_{1\leq i,j\leq
n}\operatorname{Ker}\left(\log\lambda_{i}-\log\lambda_{j}\right)$

is the set of real scalar matrices, and
$\mathfrak{a}_{s}=\operatorname{Ker}\left(\log\lambda_{1}+\cdots+\log\lambda_{n}\right)$
is the set of traceless real diagonal matrices. The set of restricted roots of
$\mathfrak{a}$ in $G$ is

$\Sigma=\left\\{\log\lambda_{i}-\log\lambda_{j}\mid 1\leq i\neq j\leq
n\right\\}.$

We can take $\Delta=\left\\{\log\lambda_{i}-\log\lambda_{i+1}\mid 1\leq i\leq
n-1\right\\}$ so that

$\Sigma^{+}=\left\\{\log\lambda_{i}-\log\lambda_{j}\mid 1\leq i<j\leq
n\right\\},$

and $\mathfrak{a}^{+}$ is the set of the elements of $\mathfrak{a}$ whose
entries are in increasing order. Let ${\rm Cl}({\mathfrak{a}}^{+})$ denote its
closure. For

(3)
$\theta=\left\\{\log\lambda_{i_{1}}-\log\lambda_{i_{1}+1},\ldots,\log\lambda_{i_{m}}-\log\lambda_{i_{m}+1}\right\\}\\\
\hbox{ with }1\leq i_{1}<\cdots<i_{m}\leq n-1,$

the parabolic subgroup $P_{\theta}$ (resp. $\left.P_{\theta}^{-}\right)$ is
the group of block upper ( resp. lower) triangular matrices in
$\mathrm{GL}(n,{\mathbb{R}})$ with square diagonal blocks of sizes
$i_{1},i_{2}-i_{1},\ldots,i_{m}-i_{m-1},n-i_{m}$. In particular, $P_{\Delta}$
is the group of upper triangular matrices in $\mathrm{GL}(n,({\mathbb{R}})$.

From now on, we will fix
$\mathfrak{a}_{n}\subset{\mathsf{GL}}(n,{\mathbb{R}})$ to be the set of real
diagonal matrices of size $n\times n$. We will also regard $\theta$ as a
subset of $\\{1,\dots,n-1\\}$ where each $i$ corresponds to
$\log\lambda_{i}-\log\lambda_{i+1}$. Also, let $\mathfrak{a}_{n}^{+}$ denote
the open positive Weyl chamber.

###  3.2. $P$-Anosov representations

Let $(P^{+},P^{-})$ be a pair of opposite parabolic subgroups of
${\mathsf{GL}}(n,{\mathbb{R}})$, and let $\mathcal{F}^{{\pm}}$ denote the flag
spaces. Let $\chi$ denote the unique open
${\mathsf{GL}}(n,{\mathbb{R}})$-orbit of the product
$\mathcal{F}^{+}\times\mathcal{F}^{-}$. The product subspace $\chi$ has two
${\mathsf{GL}}(n,{\mathbb{R}})$-invariant distributions
$E^{\pm}:=T_{x_{\pm}}\mathcal{F}^{\pm}$ for $(x^{+},x^{-})\in\chi$. Let
$\phi_{t}$ be given as in Theorem 2.4 as the ${\mathbb{R}}$-action on
$\hat{\Gamma}_{M}$ whose orbits are quasi-geodesics.

######  Definition 3.1 (Definition 2.10 of [15]).

A representation $\rho:\Gamma_{M}\rightarrow{\mathsf{GL}}(n,{\mathbb{R}})$ is
$(P^{+},P^{-})$-Anosov if there exist continuous $\rho$-equivariant maps
$\xi^{+}:\partial_{\infty}\hat{\Gamma}_{M}\rightarrow{\mathcal{F}}^{+}$,
$\xi^{-}:\partial_{\infty}\hat{\Gamma}_{M}\rightarrow{\mathcal{F}}^{-}$ such
that:

  1. (i)

for all $(x,y)\in\partial_{\infty}^{(2)}\hat{\Gamma}_{M}$, the pair
$(\xi^{+}(x),\xi^{-}(y))$ is transverse, and

  2. (ii)

for one (and hence any) continuous and equivariant family of norms
$(\left|\mkern-2.0mu\left|\cdot\right|\mkern-2.0mu\right|_{m})_{m\in\hat{\Gamma}_{M}}$
on

$(T_{\xi^{+}(\tau^{+}(m))}{\mathcal{F}}^{+})_{m\in\hat{\Gamma}_{M}}\Big{(}\hbox{resp.
}(T_{\xi^{-}(\tau^{-}(m))}{\mathcal{F}}^{-})_{m\in\hat{\Gamma}_{M}}\Big{)}:$
$\left|\mkern-2.0mu\left|e\right|\mkern-2.0mu\right|_{\phi_{-t}m}\leq
Ae^{-at}\left|\mkern-2.0mu\left|e\right|\mkern-2.0mu\right|_{m}\Big{(}\hbox{resp.}\left|\mkern-2.0mu\left|e\right|\mkern-2.0mu\right|_{\phi_{t}m}\leq
Ae^{-at}\left|\mkern-2.0mu\left|e\right|\mkern-2.0mu\right|_{m}\Big{)}.$

We call
$\xi^{\pm}:\partial_{\infty}\hat{\Gamma}_{M}\rightarrow{\mathcal{F}}^{\pm}$
the Anosov maps associated with $\rho:\Gamma_{M}\rightarrow G$. (See also
Section 2.5 of [14] for more details.)

It is sufficient to consider only the case where $P^{+}$ is conjugate to
$P^{-}$ by Lemma 3.18 of [15]. (See also Definition 5.62 of [18]). In this
case, $\rho$ is called $P$-Anosov for $P=P^{+}$. Note that this means that

(4) $\theta=\theta^{*}$

where $\theta^{*}$ is the image $\theta$ under the map $i\mapsto n-i$,
$i=1,\dots,n-1$. Hence, this will always be true for the set $\theta$
occurring for $P_{\theta}$-Anosov representations below. (See the paragraph
before Fact 2.34 of [14].)

###  3.3. $P$-Anosov representations and dominations

Let $\rho:\Gamma_{M}\rightarrow{\mathsf{GL}}(n,{\mathbb{R}})$ denote a
representation. We order singular values

$a_{1}(g)\geq a_{2}(g)\geq\dots\geq a_{n}(g)\hbox{ of }\rho(g)\hbox{ for
}g\in\rho(\Gamma_{M}).$

Let $w(g)$ denote the word length of $g$. Suppose that there exists an integer
$k$, $1\leq k\leq n-k+1\leq n$, so that the following hold for a constant
$A,C>0$:

  * •

$a_{k}(\rho(g))/a_{k+1}(g)\geq C\exp(Aw(g))$, and

  * •

$a_{n-k}(\rho(g))/a_{n-k+1}(g)\geq C\exp(Aw(g))$.

In this case, we say that $\rho$ is $k$-dominated for $k\leq n/2$ (see Bochi-
Potrie-Sambarino [3]).

We will use $\left|\mkern-2.0mu\left|\cdot\right|\mkern-2.0mu\right|$ to
indicate the Euclidean norm in the maximal flat in a symmetric space $X$. (See
Example 2.12 of [17].) We will use the following notation:

$\vec{a}(g):=(a_{1}(g),\dots,a_{n}(g)),\quad\log\vec{\lambda}(g):=(\log\lambda_{1}(g),\dots,\log\lambda_{n}(g))\in{\rm
Cl}(\mathfrak{a}^{+}_{n})$

for the singular values $a_{i}(g)$ and the modulus $\lambda_{i}(g)$ of the
eigenvalues of $g$ for $g\in{\mathsf{GL}}(n,{\mathbb{R}})$.

We say that two functions $f,g:\Gamma\rightarrow{\mathbb{R}}$ are compatible
if there exists a uniform constant $C>1$ so that
$C^{-1}g(\gamma)<f(\gamma)<Cg(\gamma)$ for $\gamma\in\Gamma$.

######  Lemma 3.1.

Let a finitely generated discrete affine transformation group $G$ have linear
holonomy representation $\rho:G\rightarrow{\mathsf{GL}}(n,{\mathbb{R}})$.

Then the following are equivalent:

  * •

$\rho$ is $P$-Anosov in the Gueritaud-Guichard-Kassel-Wienhard sense [14] for
some parabolic group $P=P_{\theta}$ for an index set $\theta$ containing $k$
and $n-k$.

  * •

$\rho$ is a $k$-dominated representation for $1\leq k\leq n/2$.

###### Proof.

This follows by Theorem 1.3(3) of [14] since the set $\theta$ contains $k$ and
$n-k$. ∎

##  4\. The proof of Theorem 1.1

###  4.1. $k$-convexity

We continue to assume that $\Gamma_{M}$ is hyperbolic. Let
$\rho:\Gamma_{M}\rightarrow{\mathsf{GL}}(n,{\mathbb{R}})$ be a representation.
We say that the representation $\rho$ is $k$-convex if there exist continuous
maps
$\zeta:\partial_{\infty}\hat{\Gamma}_{M}\rightarrow\mathcal{G}_{k}({\mathbb{R}}^{n})$
and
$\theta:\partial_{\infty}\hat{\Gamma}_{M}\rightarrow\mathcal{G}_{n-k}({\mathbb{R}}^{n})$
such that:

(transversality):

for every $x,y\in\partial_{\infty}\hat{\Gamma}_{M},x\neq y$, we have
$\zeta(x)\oplus\theta(y)={\mathbb{R}}^{n}$, and

(equivariance):

for every $\gamma\in\Gamma_{M}$, we have

$\zeta(\gamma x)=\rho(\gamma)\zeta(x),\theta(\gamma
x)=\rho(\gamma)\theta(x),x\in\partial_{\infty}\hat{\Gamma}_{M}.$

Using the representation $\rho$, it is possible to construct a linear flow
$\psi_{t}$ over the geodesic flow $\phi_{t}$ of the Gromov flow space
$\partial^{(2)}_{\infty}\hat{M}\times{\mathbb{R}}$ as follows. Consider the
lifted geodesic flow $\hat{\phi}_{t}$ on
$\partial^{(2)}_{\infty}\hat{M}\times{\mathbb{R}}$, and define a linear flow
on
$\hat{E}:=(\partial^{(2)}_{\infty}\hat{M}\times{\mathbb{R}})\times{\mathbb{R}}^{n}$
by: $\hat{\psi}_{t}(x,\vec{v})=(\phi_{t}(x),\vec{v})$ for
$x\in\partial^{(2)}_{\infty}\hat{M}\times{\mathbb{R}}$. Now consider the
action of $\Gamma_{M}$ on $\hat{E}$ given by:

$\gamma\cdot\left(x,\vec{v}\right):=\left(\gamma(x),\rho(\gamma)(\vec{v})\right).$

It follows that $\hat{\psi}_{t}$ induces in a flow on
$E:={\mathbb{R}}^{n}_{\rho}=\hat{E}/\Gamma_{M}$.

When the representation $\rho$ is $k$-convex, by the equivariance, there
exists a $\phi_{t}$-invariant splitting of the form $E_{\rho}=Z\oplus\Theta$;
it is obtained by taking the quotient of the bundles

$\hat{Z}(x)=\zeta(x_{+})\hbox{ and }\hat{\Theta}(x)=\theta(x_{-})\hbox{ for
}x\in\partial^{(2)}_{\infty}\hat{M}\times{\mathbb{R}}$

where $x_{+}$ is the forward endpoint of the complete isometric geodesic
through $x$ and $x_{-}$ is the backward endpoint of the complete isometric
geodesic through $x$.

We say that a $k$-convex representation is $k$-Anosov in the bundle sense if
the splitting $E_{\rho}=Z\oplus\Theta$ is a dominated splitting for the linear
bundle automorphism $\psi_{t}$, with $Z$ dominating $\Theta$ in the
terminology of [3]. This is equivalent to the fact that the bundle ${\rm
Hom}(Z,\Theta)$ is uniformly contracted by the flow induced by $\psi_{t}$.

Suppose that we further require for a $k$-dominated representation $\rho$ for
$k\leq n/2$:

  * •

$a_{p}(\rho(g))\geq C^{-1}\exp(Aw(g))$ for $p\leq k$, and

  * •

$a_{r}(\rho(g))\leq C\exp(-Aw(g))$ for $r\geq n-k+1$ for some constants
$C>1,A>0$.

Then we say that $\rho$ is partially hyperbolic in the singular value sense
with index $k$.

We obtain by Lemma 3.1:

######  Lemma 4.1.

If $\rho$ is partially hyperbolic for an index $k$, $k\leq n/2$, in the
singular value sense, then $\rho$ is $P_{\theta}$-Anosov for the index set
$\theta$ containing $k$ and $n-k$. ∎

Recall that we can identify $\partial_{\infty}\hat{\Gamma}_{M}$ for the Cayley
graph $\hat{\Gamma}_{M}$ of $\Gamma_{M}$ with $\partial_{\infty}\hat{M}$ by
Theorem 11.108 of [10]. Hence, we can identify
$\partial_{\infty}^{(2)}\hat{\Gamma}_{M}$ with
$\partial^{(2)}_{\infty}\hat{M}$. The Gromov flow space
$\partial_{\infty}^{(2)}\hat{\Gamma}_{M}\times{\mathbb{R}}$ is identified with
$\partial^{(2)}_{\infty}\hat{M}\times{\mathbb{R}}$. By the direct following of
Proposition 4.9 of Bochi-Potrie-Sambarino [3], we obtain:

######  Theorem 4.2.

Let $\rho$ be partially hyperbolic with an index $k$, $k<n/2$, in the singular
value sense. Then $\rho$ is partially hyperbolic in the bundle sense with
index $k$.

###### Proof.

By Lemma 4.1 and Proposition 3.1, $\rho$ is $k$-dominated.

Bochi, Potrie, and Sambarino [3] construct subbundles $Z$ and $\Theta$ of
$E_{\rho}$ where $Z$ dominates $\Theta$ and $\dim Z=k,\dim\Theta=n-k$.

Now, $\rho$ is also $n-k$-dominated. By Lemma 4.1, we obtain a new splitting
of the bundle $E_{\rho}=Z^{\prime}\oplus\Theta^{\prime}$ over
$\partial_{\infty}^{(2)}\hat{M}\times{\mathbb{R}}$ where $\dim Z^{\prime}=n-k$
and $\dim\Theta^{\prime}$ is $k$. Furthermore, $Z^{\prime}$ dominates
$\Theta^{\prime}$. Then $Z$ is a subbundle of $Z^{\prime}$, and
$\Theta^{\prime}$ is a subbundle of $\Theta$ by Proposition 2.1 of [3]. We now
form bundles $Z,Z^{\prime}\cap\Theta,\Theta^{\prime}$ over
$\partial_{\infty}^{(2)}\hat{M}\times{\mathbb{R}}$. The dominance property (1)
of Definition 1.1 follows from those of $Z,\Theta$ and
$Z^{\prime},\Theta^{\prime}$.

There are exponential expansions of $a_{i}(g)$ for $1\leq i\leq k$ by the
partially hyperbolic condition of the premise. Now (iii)(a) of Definition 1.1
follows: Define $U_{p}(A)$ as the sum of eigenspaces of $\sqrt{AA^{*}}$
corresponding to $p$ largest eigenvalues. Define
$S_{n-p}(A):=U_{n-p}(A^{-1})$. Since $\hat{\Gamma}_{M}/\Gamma_{M}$ is compact,
Theorem 2.2 of [3] shows

$U_{k}(\psi^{t}_{\phi^{-1}_{t}(x)})\rightarrow\hat{Z}(x)\hbox{ and
}S_{n-k}(\psi^{t}_{x})\rightarrow\hat{\Phi}(x)\hbox{ uniformly for
$x\in\hat{\Gamma}_{M}$ as }t\rightarrow\infty.$

Since by the paragraph above Theorem 2.2 of [3], we obtain

$\psi^{t}(S_{n-k}(\psi^{t}_{\phi^{-1}_{t}(x)})^{\perp})=U_{k}(\psi^{t}_{\phi^{-1}_{t}(x)}),$

where the unit vectors of $S_{n-k}(\psi^{t}_{\phi^{-1}_{t}(x)})^{\perp}$ goes
to vectors of norm at least the $k$-th singular value for $\psi_{t}$. The
angle between $\hat{Z}(x)$ and $\hat{\Phi}(x)$ is uniformly bounded below by a
positive constant by the compactness of $\hat{\Gamma}_{M}/\Gamma_{M}$. Hence,
unit vectors in $\hat{Z}({\phi^{-1}_{t}(x)})$ have components in
$S_{n-k}(\psi^{t}_{\phi^{-1}_{t}(x)})^{\perp}$ whose lengths are uniformly
bounded below by a positive constant for sufficiently large $t$.

As $t\rightarrow\infty$, $\phi_{t}$ passes the images of the fundamental
domain under $\Gamma_{M}$. Let $g_{i}$ denote the sequence of elements of
$\Gamma_{M}$ arising in this way. Since the $a_{k}(g_{i})$ grows
exponentially, the expansion properties of $Z$ follows. (iii)(b) of the
definition follows by reversing the flow.

By the quasi-isometry of $\mathbf{UC}\hat{M}$ with
$\partial^{(2)}\hat{M}\times{\mathbb{R}}$ in Theorem 2.6, we construct our
pulled-back partially hyperbolic bundles over $\mathbf{UC}\hat{M}$. ∎

###  4.2. Promoting $P$-Anosov representations to partially hyperbolic ones

######  Theorem 4.3 (Hirsch-Kostant-Sullivan [21]).

Let $N$ be a complete affine manifold. Let
$\rho:\pi_{1}(N)\rightarrow{\mathsf{GL}}(n,{\mathbb{R}})$ be the linear part
of the affine holonomy representation $\rho^{\prime}$. Then $g$ has an
eigenvalue equal to $1$ for each $g$ in the Zariski closure
$Z(\rho(\pi_{1}(N)))$ of $\rho(\pi_{1}(N))$.

###### Proof.

We define function $f:Z(\rho(\pi_{1}(N)))\rightarrow{\mathbb{R}}$ by
$f(x)=\det(x-{\rm I})$ for each element $x$. Suppose that $f(\rho(g))\neq 0$,
$g\in\Gamma_{M}$. Then $\rho(g)(x)+b=x$ has a solution for each $b$ by
Kramer’s rule. Hence $\rho^{\prime}(g)$ has a fixed point, which is a
contradiction. We obtain that $f$ is zero on the Zariski closure, implying
that each element has at least one eigenvalue equal to $1$. ∎

For a reductive Lie group $Z$, let $A_{Z}$ denote the maximal
${\mathbb{R}}$-split maximal torus of $Z$ with corresponding Lie algebra
$\mathfrak{a}_{Z}$, and let $\mathfrak{a}_{Z}^{+}$ denote the Weyl chamber of
$\mathfrak{a}_{Z}$. Also, we denote the Jordan (or Lyapunov) projection by
$\log\vec{\lambda}_{Z}:Z\rightarrow{\rm Cl}(\mathfrak{a}^{+}_{Z})$. Let
$\Gamma$ be Zariski dense in $Z$. (See Section 2.4 of [14].) The Benoist cone
$\mathcal{BC}_{Z}(\Gamma)$ of a linear group $\Gamma\subset Z$ is the closure
of the set of all positive linear combinations of
$\log\vec{\lambda}_{Z}(g),g\in\Gamma$ in ${\rm Cl}(\mathfrak{a}^{+}_{Z})$.

######  Proposition 4.4 (Choi-Stecker).

Assume the following:

  * •

$G$ is a finitely generated discrete group.

  * •

$\rho:G\rightarrow{\mathsf{GL}}(n,{\mathbb{R}})$ is a $P_{\theta}$-Anosov
representation for some index set $\theta$ containing $k$ and $n-k$ for $1\leq
k\leq n/2$.

  * •

$\rho(g)$ for each $g\in G$ has an eigenvalue equal to $1$.

  * •

he Zariski closure $Z$ of the image of $\rho$ is a connected reductive Lie
group.

Then $\rho$ is partially hyperbolic with index $k$ in the singular value
sense, and $k<n/2$.

###### Proof.

(I) We begin with some preliminary: The embedding
$Z\hookrightarrow{\mathsf{GL}}(n,{\mathbb{R}})$ induces a map $\iota_{Z}:{\rm
Cl}(\mathfrak{a}_{Z}^{+})\rightarrow{\rm Cl}(\mathfrak{a}^{+}_{n})$ for the
Weyl chamber $\mathfrak{a}^{+}_{n}$ of ${\mathsf{GL}}(n,{\mathbb{R}})$. Let
$\log\lambda_{1},\dots,\log\lambda_{n}$ denote the coordinates of the Weyl
chamber $\mathfrak{a}^{+}_{n}$. Of course,
$\iota_{Z}\circ\log\vec{\lambda}_{Z}|\rho(G)=\log\vec{\lambda}|\rho(G)$.

Let $B:=\mathcal{BC}_{Z}(\rho(G))$ denote the Benoist cone of $\rho(G)$ in
$Z$. The image $\iota_{Z}(B)$ is not convex in general. Now, $B$ is a convex
cone with nonempty interior in ${\rm Cl}(\mathfrak{a}_{Z}^{+})$. By Theorem
4.3, for each $b\in B$, there is an index $i$ where the log of the $i$-th norm
of the eigenvalue equals $0$. Let $S([1,...,n])$ denote the collection of
subsets of consecutive elements. We define a function

$I:B-\\{0\\}\rightarrow S([1,\dots,n])$

given by sending $b\in B$ to the set of consecutive indices where the
coordinates are $0$. We have

(5) $\lim_{i\rightarrow\infty}I(x_{i})\subset I(x)\hbox{ whenever
}x_{i}\rightarrow x,x_{i},x\in B.$

Let $S_{n}$ denote the unit sphere in the Lie algebra $\mathfrak{a}_{n}$ of
${\mathsf{GL}}(n,{\mathbb{R}})$. Let $S$ denote a finite set of generators of
$\rho(G)$ and their inverses. By Theorems 1.3 and 1.4 of [5], the sequence
$\frac{1}{n}\log\vec{\lambda}_{Z}(S^{n})\subset J(S)$ geometrically converges
to a compact convex set $J(S)\subset{\rm Cl}(\mathfrak{a}_{Z}^{+})$ (see
Section 3.1 of [5]). Moreover, the cone spanned by $0$ and $J(S)$ is the
Benoist cone $B$. This implies that the set of directions of
$\log\vec{\lambda}_{Z}(\rho(G))$ is dense in $B$. (See also [2]). Since

$\log\lambda_{k}(g)-\log\lambda_{k+1}(g)\geq
C\left|\mkern-2.0mu\left|\log\vec{\lambda}(g)\right|\mkern-2.0mu\right|,g\in\rho(G),\hbox{
for a constant }C>0$

by Theorem 4.2(3) of [14], we obtain

(6) $\log\lambda_{k}(b)-\log\lambda_{k+1}(b)\geq C$

for every $b\in B$ going to an element of $\iota_{Z}(B)\cap S_{n}$ for a
constant $C>0$.

(II) Our first major step is to prove $I(B-\\{0\\})\subset[k+1,n-k]$: Let
$\iota_{n}$ denote the involution of $\mathfrak{a}_{n}$. First, suppose that
the rank of $Z$ is $\geq 2$, and hence $\dim\mathfrak{a}_{Z}^{+}\geq 2$.

A diagonal subspace

$\Delta_{i,j}\subset\mathfrak{a}_{n}\hbox{ for }i\neq j,i,j=1,\dots,n,$

is a subspace of the maximal abelian algebra $\mathfrak{a}_{n}$ given by
$\log\lambda_{i}-\log\lambda_{j}=0$ for some indices $i$ and $j$. The inverse
images under $\iota_{Z}$ in ${\rm Cl}(\mathfrak{a}_{Z}^{+})$ of the diagonal
subspaces of $\mathfrak{a}$ may meet $B$. We let $\mathcal{I}_{B}$ the maximal
collection of indices $(i,j),i<j$, where $\iota_{Z}^{-1}(\Delta_{i,j})$
contains $B$, which may be empty. Let $\mathcal{I}^{\prime}_{B}$ the
collection of indices $(k,l)$, $k<l$, of $\Delta_{k,l}$ not in
$\mathcal{I}_{B}$. We call the closures of the components of

$B-\bigcup_{(k,l)\in\mathcal{I}^{\prime}_{B}}\iota_{Z}^{-1}(\Delta_{k,l})$

the generic flat domains of $B$. There are finitely many of these to be
denoted $B_{1},\dots,B_{m}$ which are convex domains in $B$.

In the interior of each $B_{i}$, $I$ is constant since the $I$-value is never
empty in the interior of $B_{i}$ and the change in $I$ will happen only in the
inverse images of some of $\Delta_{k,l}$.

For adjacent regions, their images under $I$ differ by adding or removing some
top or bottom consecutive sets.

We now aim to prove that the following never occurs –(*):

> There exists a pair of elements $b_{1},b_{2}\in B$ with $b_{1}\in B_{p}^{o}$
> and $b_{2}\in B_{q}^{o}$ for some $p,q$ where $I(b_{1})$ has an element
> $\leq k$ and $I(b_{2})$ has an element $>k$.

We use the discrete version of continuity: Suppose that this is true. Since
$B$ is convex with $\dim B\geq 2$, there is a chain of adjacent polyhedral
images $B_{l_{1}},\dots,B_{l_{m}}$ where $b_{1}\in B_{l_{1}}^{o}$ and
$b_{2}\in B_{l_{m}}^{o}$ where $B_{l_{j}}\cap B_{l_{j+1}}$ contains a nonzero
point. For adjacent $B_{l_{p}}$ and $B_{l_{p+1}}$, we have by (5)

(7) $I(B_{l_{p}}^{o})\cup I(B_{l_{p+1}}^{o})\subset I(x)\hbox{ for }x\in{\rm
Cl}(B_{l_{p}})\cap{\rm Cl}(B_{l_{p+1}})-\\{O\\}.$

Considering $I(B_{l_{j}}^{o})$, $j=1,\dots,m$, and the $I$-values of the
intersections of adjacent generic flat domains, we have $I(b^{\prime})\ni
k,k+1$ for some nonzero element $b^{\prime}$ of $B$. This follows easily by
considering the changes on $I$-values as we change generic flat domains to
adjacent ones according to (7). This means

(8) $\log\lambda_{k}(b^{\prime})=0\hbox{ and
}\log\lambda_{k+1}(b^{\prime})=0.$

Since we can assume $b^{\prime}\in S_{n}$ for $b^{\prime}$ in (8) by
normalizing, we obtain a contradiction to (6).

Hence, we proved (*), and one of the following holds:

  * •

for all $b\in B$ in the interiors of generic flat domains, $I(b)$ contains an
element $\leq k$ or

  * •

for all $b\in B$ in the interiors of generic flat domains, $I(b)$ has all the
elements $>k$ only.

Note that the opposition involution $\iota_{n}$ sends $\iota_{Z}(B)$ to itself
since the inverse map $g\rightarrow g^{-1}$ preserves $Z$ and the set of
eigenvalues of $g$ is sent to the set of the eigenvalues of $g^{-1}$. In the
first case, by the opposition involution, $I(\iota_{n}(b))$ for $b\in B$
contains an element $\geq n-k+1$. This again contradicts the above. Hence, we
conclude $I(b)\subset[k+1,\dots,n]$ for all $b\in B$ in the interiors of
generic flat domains. Acting by the opposition involution and the similar
arguments, we obtain $I(b)\subset[k+1,\dots,n-k]$ for all $b\in B$ in the
interiors of generic flat domains.

Furthermore, we can show that $I(b)\subset[k+1,n-k]$ for all $b\in B-\\{0\\}$
since otherwise we have $I(b^{\prime})\ni k,k+1$ for some $b^{\prime}\neq 0$
by a similar reason to the above applied to lower-dimensional strata. This is
a contradiction as before. Also, the argument shows $[k+1,n-k]\neq\emptyset$
and $k<n/2$.

Suppose that $Z$ has rank $1$. Then the maximal abelian group $A_{Z}$ is
$1$-dimensional. For each $g\in Z$, there is an index $i(g)$ for which
$\log\lambda_{i(g)}(g)=0$. Thus, $\mathfrak{a}_{Z}$ is in the null space of
$\log\lambda_{i}$ for some index $i$. Since $g\mapsto g^{-1}$ preserves
$\mathfrak{a}_{Z}$, we have $\log\lambda_{n-i+1}=0$ on $\mathfrak{a}_{Z}$.
Since ${\rm Cl}(\mathfrak{a}_{Z}^{+})=\mathfrak{a}_{Z}\cap{\rm
Cl}(\mathfrak{a}^{+})$, we also have $\log\lambda_{n-i+1}=0$ on
$\mathfrak{a}_{Z}^{+}$. Hence,

$I(B-\\{0\\})=[i,\dots,n-i+1]\hbox{ for some index }i,0<i\leq n/2.$

Now (6) and $k<i$ imply $I(B-\\{0\\})\subset[k+1,n-k]$.

(III) To complete the proof, we show the partial hyperbolicity property: We
let $\log\vec{a}_{Z}|Z:Z\rightarrow\mathfrak{a}_{Z}^{+}$ denote the Cartan
projection of $Z$. Recall

$\iota_{Z}\circ\log\vec{a}_{Z}|\rho(G)=\log\vec{a}|\rho(G).$

Since $\rho(G)$ has a connected reductive Zariski closure, the sequence

$\left\\{\frac{\log\vec{a}(\rho(g_{l})))}{w(g_{l})}\right\\}$

have limit points only in $\iota_{Z}(B)$ by Theorem 1.3 of [5]. By the
conclusion of step (II), for any sequence $\\{g_{l}\\}$,

$\left\\{\frac{\min\\{|\log a_{i}(\rho(g_{l}))|\\}_{i=k+1,\dots
n-k}}{w(g_{l})}\right\\}\rightarrow 0,$

uniformly in terms of $w(g_{l})$. This implies that

$\min\\{a_{i}(\rho(g))\\}_{i=k+1,\dots,n-k}\geq A\exp(-Cw(g))$

for any small constant $C>0$ and a constant $A>0$. By the $P_{\theta}$-Anosov
property of $\rho$ and Lemma 3.1, we have

$\frac{a_{k}(\rho(g))}{a_{k+1}(\rho(g))}\geq A^{\prime}\exp(C^{\prime}w(g))$

for constants $A^{\prime},C^{\prime}>0$. Since

$a_{k+1}(\rho(g))\geq a_{i}(\rho(g)),i=k+1,\cdots,n-k,\hbox{ for every }g\in
G,$

it follows that

$a_{k}(\rho(g))\geq A^{\prime\prime}\exp(C^{\prime\prime}w(g))$

for some constants $C^{\prime\prime},A^{\prime\prime}>0$. Taking inverses
also, we showed that $\rho^{\prime}$ is partially hyperbolic in the singular
value sense with index $k$.

Finally, since $[k+1,n-k]$ is not empty, $k<n/2$. ∎

For a general representation $\phi:\Gamma\rightarrow G$, we define the
semisimplification $\phi^{ss}:\Gamma\rightarrow G$ by post-composing $\phi$
with the projection to the Levi factor of the Zariski closure of
$\phi(\Gamma)$. (See Section 2.5.4 of [14] for details.) Also, the Zariski
closure of a finite index group is reductive if so was the Zariski closure of
the original discrete group. (See Remark 2.38 of [14].)

###### Proof of Theorem 1.1.

Suppose that $\rho$ is $P$-Anosov in the sense of [14]. Hence,
$\Gamma:=\rho(\pi_{1}(N))$ is word-hyperbolic by [14].

First, suppose that the Zariski closure of the image of the linear holonomy
group is reductive. By Lemma 3.1, $\rho$ is a $k$-dominated representation of
index $k$. We take a finite index normal subgroup $\Gamma^{\prime}$ of
$\Gamma$. Assume that the Zariski closure $Z$ of $\rho(\Gamma^{\prime})$ is a
connected reductive Lie group. By Theorem 4.3 and Proposition 4.4,
$\rho|\Gamma^{\prime}$ is partially hyperbolic in the singular value sense
with index $k$, $k<n/2$. So is $\rho$ since $\Gamma$ is a finite-index
extension of $\Gamma^{\prime}$. By Theorem 4.2, $\rho$ is a partially
hyperbolic representation in the bundle sense with index $k$.

Now, we drop the reductive condition of the Zariski closure. Mainly, we have
to prove the expanding and contracting properties in the subbundles. The
semisimplification $\rho^{ss}$ is $P$-Anosov by Proposition 1.8 of [14]. Since
the projection to the Levi factor does not change eigenvalues by the proof of
Lemma 2.40 of [14], $\rho^{ss}(g)$ for each $g\in\Gamma$ has an eigenvalue
equal to $1$ by Theorem 4.3. By Proposition 4.4 and Theorem 4.2, $\rho^{ss}$
is a partially hyperbolic representation in the bundle sense with index $k$,
$k<n/2$. By Proposition 1.3 of [14], $\rho$ is $P$-Anosov with respect to a
continuous Anosov map $\zeta:\partial_{\infty}\Gamma\rightarrow{\mathcal{F}}$
for the flag variety $\mathcal{F}:={\mathsf{GL}}(n,{\mathbb{R}})/P$.

We can conjugate $\rho$ as close to $\rho^{ss}$ as one wishes in ${\rm
Hom}(\Gamma,{\mathsf{GL}}(n,{\mathbb{R}}))$ since the orbit of $\rho^{ss}$ is
the closed orbit in the closure of the orbit of $\rho$ under the conjugation
action. (See Section 2.5.4 of [14]. ) Let
$\zeta_{i}:\partial_{\infty}\Gamma\rightarrow{\mathcal{F}}$ denote the Anosov
map for $\rho_{i}$ conjugate to $\rho$ converging to $\rho^{ss}$. Theorem 5.13
of [15] generalizes to ${\mathsf{GL}}(n,{\mathbb{R}})$ since we can multiply a
representation by a homomorphism $\Gamma\rightarrow{\mathbb{R}}^{+}$ not
changing the P-Anosov property to
${\mathsf{SL}}_{\pm}(n,{\mathbb{R}})$-representations. (See Section 2.5.3 of
[14]). This implies that $\zeta_{i}$ converges to
$\zeta^{ss}:\partial_{\infty}\Gamma\rightarrow{\mathcal{F}}$ the map for
$\rho^{ss}$ in the $C^{0}$-sense.

Let ${\mathbb{V}}^{ss+},{\mathbb{V}}^{ss0}$, and ${\mathbb{V}}^{ss-}$ denote
the bundles of the partially hyperbolic decomposition of
${\mathbb{R}}^{n}_{\rho^{ss}}$ as in Definition 1.1. Let
${\mathbb{V}}^{+}_{i},{\mathbb{V}}^{0}_{i}$, and ${\mathbb{V}}^{-}_{i}$ denote
the bundles of the decomposition of ${\mathbb{R}}^{n}_{\rho_{i}}$ of
respective dimensions $k$, $n-2k$, and $k$ obtainable from $\zeta_{i}$ since
$\rho_{i}$ is $k$-dominated. The single product bundle
$\mathbf{UC}\hat{M}\times{\mathbb{R}}^{n}$ covers the direct sums of these
bundles.

  * •

Let $\hat{\mathbb{V}}^{ss+},\hat{\mathbb{V}}^{ss0}$, and
$\hat{\mathbb{V}}^{ss-}$ denote the cover of
${\mathbb{V}}^{ss+},{\mathbb{V}}^{ss0}$, and ${\mathbb{V}}^{ss-}$.

  * •

Let $\hat{\mathbb{V}}^{+}_{i},\hat{\mathbb{V}}^{0}_{i}$, and
$\hat{\mathbb{V}}^{-}_{i}$ denote the cover of
${\mathbb{V}}^{+}_{i},{\mathbb{V}}^{0}_{i}$, and ${\mathbb{V}}^{-}_{i}$.

Then the above convergence $\zeta_{i}\rightarrow\zeta^{ss}$ means that

(9)
$\hat{\mathbb{V}}^{+}_{i}(x)\rightarrow\hat{\mathbb{V}}^{ss+}(x),\hat{\mathbb{V}}^{0}_{i}(x)\rightarrow\hat{\mathbb{V}}^{ss0}(x),\hbox{
and }\hat{\mathbb{V}}^{-}_{i}(x)\rightarrow\hat{\mathbb{V}}^{ss-}(x)\hbox{ as
}i\rightarrow\infty$

in the respective Grassmann spaces pointwise for each
$x\in\mathbf{UC}\hat{M}$.

We may construct a sequence of the fiberwise metrics
$\left|\mkern-2.0mu\left|\cdot\right|\mkern-2.0mu\right|_{{\mathbb{R}}^{n}_{\rho_{i}}}$
so that the sequence of the lifted norms of
$\left|\mkern-2.0mu\left|\cdot\right|\mkern-2.0mu\right|_{{\mathbb{R}}^{n}_{\rho_{i}}}$
uniformly converging to that of
$\left|\mkern-2.0mu\left|\cdot\right|\mkern-2.0mu\right|_{{\mathbb{R}}^{n}_{\rho^{ss}}}$
on $\mathbf{UC}\hat{M}\times{\mathbb{R}}^{n}$ over each compact subset of
$\mathbf{UC}\hat{M}$ using a fixed set of partition of unity subordinate to a
covering with trivializations.

The flow $\phi^{ss}$ on ${\mathbb{R}}_{\rho^{ss}}$ satisfies the expansion and
contraction properties in Definition 1.1. Note that the lift $\Phi^{ss}$ of
$\phi^{ss}$ and that $\Phi_{i}$ of the flow $\phi_{i}$ for
${\mathbb{R}}^{n}_{\rho_{i}}$ in the product bundle
$\mathbf{UC}\hat{M}\times{\mathbb{R}}^{n}$ are the same by construction
induced from trivial action on the second factor. Since $M$ is compact, there
exists $t_{0}$ such that for $t>t_{0}$,

  * •

$\Phi^{ss}_{t}|{\mathbb{V}}^{ss+}$ is expanding by a factor $c^{+}>1$

  * •

$\Phi^{ss}_{t}|{\mathbb{V}}^{ss-}$ is contracting by a factor $c^{-}<1$ with
respect to the metric
$\left|\mkern-2.0mu\left|\cdot\right|\mkern-2.0mu\right|_{{\mathbb{R}}^{n}_{\rho^{ss}}}$.

Hence for sufficiently large $i$, the flow $\Phi_{i}$ on
${\mathbb{R}}^{n}_{\rho_{i}}$ satisfies the corresponding properties as well
as the domination properties for
$\left|\mkern-2.0mu\left|\cdot\right|\mkern-2.0mu\right|_{{\mathbb{R}}^{n}_{\rho_{i}}}$
by (9). Hence, $\Phi_{i}$ is partially hyperbolic with index $k$.

For converse, suppose that $\rho$ is partially hyperbolic with index $k$.
Proposition 4.5 of [3] shows that $\rho$ is $k$-dominated. Lemma 3.1 shows
that $\rho$ is $P_{\theta}$-Ansov for $k\in\theta$. Also, $k<n/2$ since the
neutral bundle exists. ∎

## References

  * [1] Y. Benoist, Propriétés asymptotiques des groupes linéaires, Geometric and Functional Analysis 7 (1997), no. 1, 1–47. MR 1437472
  * [2] Y. Benoist, Propriétés asymptotiques des groupes linéaires. II, Analysis on homogeneous spaces and representation theory of Lie groups, Okayama–Kyoto (1997), Adv. Stud. Pure Math., vol. 26, Math. Soc. Japan, Tokyo, 2000, pp. 33–48. MR 1770716
  * [3] J. Bochi, R. Potrie and A. Sambarino, Anosov representations and dominated splittings, Journal of the European Mathematical Society (JEMS) 21 (2019), no. 11, 3343–3414. MR 4012341
  * [4] C. Bonatti, L. J. Díaz and M. Viana, Dynamics beyond uniform hyperbolicity, Encyclopaedia of Mathematical Sciences, vol. 102, Springer-Verlag, Berlin, 2005, A global geometric and probabilistic perspective, Mathematical Physics, III. MR 2105774
  * [5] E. Breuillard and C. Sert, The joint spectrum, Journal of the London Mathematical Society. Second Series 103 (2021), no. 3, 943–990. MR 4245827
  * [6] C. Champetier, Petite simplification dans les groupes hyperboliques, Toulouse. Faculté des Sciences. Annales. Mathématiques. Série 6 3 (1994), no. 2, 161–221. MR 1283206
  * [7] S. Choi, Complete affine manifolds with Anosov holonomy groups II: partially hyperbolic holonomy and cohomological dimension, in preparation.
  * [8] M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990, Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups], With an English summary. MR 1075994
  * [9] S. Crovisier and R. Potrie, Introduction to partially hyperbolic dynamics, School on Dynamical Systems, ICTP, Trieste (July 2015), https://www.imo.universite-paris-saclay.fr/~crovisie/00-CP-Trieste-Version1.pdf, 2015\.
  * [10] C. Druţu and M. Kapovich, Geometric group theory, American Mathematical Society Colloquium Publications, vol. 63, American Mathematical Society, Providence, RI, 2018, With an appendix by Bogdan Nica. MR 3753580
  * [11] É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990, Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648
  * [12] M. Gromov, Filling Riemannian manifolds, Journal of Differential Geometry 18 (1983), no. 1, 1–147. MR 697984
  * [13] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829
  * [14] F. Guéritaud, O. Guichard, F. Kassel, and A. Wienhard, Anosov representations and proper actions, Geometry & Topology 21 (2017), no. 1, 485–584. MR 3608719
  * [15] O. Guichard and A. Wienhard, Anosov representations: domains of discontinuity and applications, Inventiones Mathematicae 190 (2012), no. 2, 357–438. MR 2981818
  * [16] Y. Guivarc’h, L. Ji and J. C. Taylor, Compactifications of symmetric spaces, Progress in Mathematics, vol. 156, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1633171
  * [17] M. Kapovich and B. Leeb, Discrete isometry groups of symmetric spaces, Handbook of group actions. Vol. IV, Adv. Lect. Math. (ALM), vol. 41, Int. Press, Somerville, MA, 2018, pp. 191–290. MR 3888689
  * [18] M. Kapovich, B. Leeb and J. Porti, Anosov subgroups: dynamical and geometric characterizations, European Journal of Mathematics 3 (2017), no. 4, 808–898. MR 3736790
  * [19] M. Kapovich, B. Leeb and J. Porti, Dynamics on flag manifolds: domains of proper discontinuity and cocompactness, Geometry & Topology 22 (2018), no. 1, 157–234. MR 3720343
  * [20] F. Kassel and R. Potrie, Eigenvalue gaps for hyperbolic groups and semigroups, arXiv:2002.07015, 2020.
  * [21] B. Kostant and D. Sullivan, The Euler characteristic of an affine space form is zero, Bulletin of the American Mathematical Society 81 (1975), no. 5, 937–938. MR 0375341
  * [22] F. Mathéus, Flot géodésique et groupes hyperboliques d’après M. Gromov, Séminaire de Théorie Spectrale et Géométrie, No. 9, Année 1990–1991, Sémin. Théor. Spectr. Géom., vol. 9, Univ. Grenoble I, Saint-Martin-d’Hères, 1991, pp. 67–87. MR 1715930
  * [23] J. Milnor, A note on curvature and fundamental group, Journal of Differential Geometry 2 (1968), 1–7. MR 0232311
  * [24] I. Mineyev, Flows and joins of metric spaces, Geometry and Topology 9 (2005), 403–482. MR 2140987
  * [25] P. Scott and T. Tucker, Some examples of exotic noncompact $3$-manifolds, The Quarterly Journal of Mathematics. Oxford. Second Series 40 (1989), no. 160, 481–499. MR 1033220
  * [26] J. R. Stallings, Embedding homotopy types into manifolds, 1965, note, Princeton University.
  * [27] A. S. Švarc, A volume invariant of coverings, Doklady Akademii Nauk SSSR 105 (1955), 32–34. MR 0075634
  * [28] W. P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997, Edited by Silvio Levy. MR 1435975
  * [29] G. A. Venema, A manifold that does not contain a compact core, Topology and its Applications 90 (1998), no. 1-3, 197–210. MR 1648326