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A higher rank rigidity theorem for convex real projective manifolds

For convex real projective manifolds we prove an analogue of the higher rank rigidity theorem of Ballmann and Burns-Spatzier.


Introduction
A real projective structure on a d-manifold M is an open cover M = ∪ α U α along with coordinate charts ϕ α : U α → P(R d+1 ) such that each transition function ϕ α • ϕ −1 β coincides with the restriction of an element in PGL d+1 (R). A real projective manifold is a manifold equipped with a real projective structure.
An important class of real projective manifolds are the convex real projective manifolds, which are defined as follows. First, a subset Ω ⊂ P(R d+1 ) is called a properly convex domain if there exists an affine chart which contains it as a bounded convex open set. In this case, the automorphism group of Ω is Aut(Ω) := {g ∈ PGL d+1 (R) : gΩ = Ω}.
If Γ ≤ Aut(Ω) is a discrete subgroup that acts freely and properly discontinuously on Ω, then the quotient manifold Γ\Ω is called a convex real projective manifold. Notice that local inverses to the covering map Ω → Γ\Ω provide a real projective structure on the quotient. In the case when there exists a compact quotient the domain Ω is called divisible. For more background see the expository papers [Ben08,Qui10,Mar14].
When d ≤ 3, the structure of closed convex real projective d-manifolds are very well understood thanks to deep work of Benzécri [Ben60], Goldman [Gol90], and Benoist [Ben06]. But when d ≥ 4 their general structure is mysterious.
In this paper we establish a dichotomy for convex real projective manifolds inspired by the theory of non-positively curved Riemannian manifolds. In particular, a compact Riemannian manifold (M, g) with non-positive curvature is said to have higher rank if every geodesic in the universal cover is contained in a totally geodesic subspace isometric to R 2 . Otherwise (M, g) is said to have rank one. An important theorem of Ballmann [Bal85] and Burns-Spatzier [BS87b,BS87a] states that every compact irreducible Riemannian manifold with non-positive curvature and higher rank is a locally symmetric space. This foundational result reduces many problems about nonpositively curved manifolds to the rank one case. Further, rank one manifolds possess very useful "weakly hyperbolic behavior" (see for instance [Bal82,Kni98]).
In the context of convex real projective manifolds, the natural analogue of isometrically embedded copies of R 2 are properly embedded simplices, see Section 2.6 below, which leads to the following definition of higher rank. (1) A properly convex domain Ω ⊂ P(R d ) has higher rank if for every p, q ∈ Ω there exists a properly embedded simplex S ⊂ Ω with dim(S) ≥ 2 and [p, q] ⊂ S.
(2) If a properly convex domain Ω ⊂ P(R d ) does not have higher rank, then we say that Ω has rank one.
There are two basic families of properly convex domains with higher rank: reducible domains (see Section 2.4 below) and symmetric domains with real rank at least two.
A properly convex domain Ω ⊂ P(R d ) is called symmetric if there exists a semisimple Lie group G ≤ PGL d (R) which preserves Ω and acts transitively. In this case, the real rank of Ω is defined to be the real rank of G. Koecher and Vinberg characterized the irreducible symmetric properly convex domains and proved that G must be locally isomorphic to either For details see [Koe99,Vin63,Vin65,FK94]. Borel [Bor63] proved that every semisimple Lie group contains a co-compact lattice, which implies that every symmetric properly convex domain is divisible.
In this paper we prove that these two families of examples are the only divisible domains with higher rank. In fact, we show that being symmetric with real rank at least two is equivalent to a number of other "higher rank" conditions. Before stating the main result we need a few more definitions.
Next we define a distance on the boundary using projective line segments. We will prove the following characterization of higher rank in the context of convex real projective manifolds.
Theorem 1.4 (see Section 9) Suppose that Ω ⊂ P(R d ) is an irreducible properly convex domain and Γ ≤ Aut(Ω) is a discrete group acting co-compactly on Ω. Then the following are equivalent: (1) Ω is symmetric with real rank at least two, (2) Ω has higher rank, (3) the extreme points of Ω form a closed proper subset of ∂Ω, s ∂Ω (x, y) < +∞ for all x, y ∈ ∂Ω, (7) Γ has higher rank in the sense of Prasad-Raghunathan (see Section 8), (8) for every g ∈ Γ with infinite order the cyclic group g Z has infinite index in the centralizer of g in Γ, (9) every g ∈ Γ with infinite order has at least three fixed points in ∂Ω, for every bi-proximal element g ∈ Γ, and (11) s ∂Ω (ℓ + g , ℓ − g ) < +∞ for every bi-proximal element g ∈ Γ.
M. Islam [Isl19] has recently defined and studied rank one isometries of a properly convex domain. These are analogous to the classical definition of rank one isometries of CAT(0) spaces (see [Bal82]) and are defined as follows.
Remark 1.6 (1) When g ∈ Aut(Ω) is a rank one isometry, then the properly embedded line segment (ℓ + g , ℓ − g ) ⊂ Ω is preserved by g. Further, g acts by translations on (ℓ + g , ℓ − g ) in the following sense: if H Ω is the Hilbert metric on Ω, then there exists T > 0 such that for all n ≥ 0 and x ∈ (ℓ + g , ℓ − g ). (2) M. Islam [Isl19, Proposition 6.3] also proved the following weaker characterization of rank one isometries: g ∈ Aut(Ω) is a rank one isometry if and only if g acts by translations on a properly embedded line segment (a, b) ⊂ Ω and s ∂Ω (a, b) > 2.
As an immediate consequence of Theorem 1.4 we obtain the following.
Corollary 1.7 Suppose that Ω ⊂ P(R d ) is an irreducible properly convex domain and Γ ≤ Aut(Ω) is a discrete group acting co-compactly on Ω. Then the following are equivalent: (1) Ω has rank one, and Ω is not symmetric with real rank at least two, then Γ is an acylindrically hyperbolic group.
1.1 Outline of the proof of Theorem 1.4 The difficult part is showing that any one of the conditions (2) through (11) implies that the domain is symmetric with real rank at least two.
One key idea is to construct and study special semigroups in P(End(R d )) associated to each boundary face. This is accomplished as follows. First, motivated by a lemma of Benoist [Ben03, Lemma 2.2], we consider the following compactification of a subgroup of PGL d (R).
denote the closure of G in P(End(R d )).
Next for a dividing group we introduce the following subsets of this compactification.
Definition 1.10 Suppose that Ω ⊂ P(R d ) is a properly convex domain and Γ ≤ Aut(Ω) is a discrete group acting co-compactly on Ω. If F ⊂ ∂Ω is a boundary face and V := Span F ⊂ R d , then define We then prove the following result about these subsets.
Theorem 1.11 (see Theorem 3.1 below) Suppose that Ω ⊂ P(R d ) is an irreducible properly convex domain and Γ ≤ Aut(Ω) is a discrete group acting co-compactly on Ω.
If Ω is non-symmetric, F ⊂ ∂Ω is a boundary face, V := Span F ⊂ R d , and dim(V) ≥ 2, then: is a non-discrete Zariski dense semigroup in P(End(V)).
Using Theorem 1.11 we will show that any one of the conditions (2) through (11) in Theorem 1.4 implies that the domain is symmetric with real rank at least two. Here is a sketch of the argument, first suppose that Ω ⊂ P(R d ) is an irreducible properly convex domain, Γ ≤ Aut(Ω) is a discrete group acting co-compactly on Ω, and any one of the conditions (2) through (11) in Theorem 1.4 is true. Then let E Ω ⊂ ∂Ω denote the extreme points of Ω. We will show that there exists a boundary face F ⊂ ∂Ω such that By choosing F minimally, we can also assume that E Ω intersects every boundary face of strictly smaller dimension. As before let V := Span F . Then using Equation (1) we show that T| V ∈ Aut(F) for every T ∈ Γ End F,⋆ . Then Theorem 1.11 implies that either Ω is symmetric or Aut(F) is a non-discrete Zariski dense subgroup of PGL(V). In the latter case, it is fairly easy to deduce that PSL(V) ⊂ Aut(F), see Lemma 4.5 below, which is impossible. So Ω must be symmetric.

Outline of the paper
In Section 2 we recall some preliminary material. In Section 3 we prove Theorem 1.11. In Section 4 we prove the rigidity result mentioned in the previous subsection.
The rest of the paper is devoted to the proof of the various equivalences in Theorem 1.4. In Sections 5, 6, and 7 we prove some new results about the action of the automorphism group. In Section 8 we consider the rank of a group in the sense of Prasad-Raghunathan. Finally, in Section 9 we prove Theorem 1.4.

Notations
Given a linear subspace V ⊂ R d , we let P(V) ⊂ P(R d ) denote its projectivization. In all other cases, given some object o we will let [o] be the projective equivalence class of o, for instance: We also identify P(R d ) = Gr 1 (R d ), so for instance: if x ∈ P(R d ) and V ⊂ R d is a linear subspace, then x ∈ P(V) if and only if x ⊂ V .
Finally, given a subset X of R d (respectively P(R d )) we will let Span X ⊂ R d denote the smallest linear subspace containing X (respectively, the preimage of X ).

Convexity and line segments
A subset C ⊂ P(R d ) is called convex if there exists an affine chart which contains it as a convex subset. A subset C ⊂ P(R d ) is called properly convex if there exists an affine chart which contains it as a bounded convex subset. For convex subsets, we make the following topological definitions.
Definition 2.1 Suppose that C ⊂ P(R d ) is a convex set. The relative interior of C , denoted by rel-int(C), is the interior of C in its span and the boundary of C is ∂C := C \ rel-int(C).
A line segment in P(R d ) is a connected subset of a projective line. Given two points x, y ∈ P(R d ) there is no canonical line segment with endpoints x and y, but we will use the following convention: if C ⊂ P(R d ) is a properly convex set and x, y ∈ C, then (when the context is clear) we will let [x, y] denote the closed line segment joining x to y which is contained in C. In this case, we will also let (x, y) = [x, y] \ {x, y},

Irreduciblity
and R d are the only Γ-invariant linear subspaces of R d and strongly irreducible if every finite index subgroup is irreducible.
We will use the following observation several times.
is strongly irreducible, x 1 , . . . , x k ∈ P(R d ), and V 1 , . . . , V k R d are linear subspaces, then there exists g ∈ Γ such that gx j / ∈ P(V j ) for all 1 ≤ j ≤ k.
Proof Let G = Γ Zar denote the Zariski closure of Γ in PGL d (R) and let G 0 ≤ G denote the connected component of the identity of G (in the Zariski topology). Then, G 0 ∩ Γ is a finite index subgroup of Γ and hence G 0 is irreducible. So each set

Zariski closures
An open convex cone C ⊂ R d is reducible if there exists a non-trivial vector space decomposition R d = V 1 ⊕ V 2 and convex cones C 1 ⊂ V 1 and C 2 ⊂ V 2 such that C = C 1 + C 2 . Otherwise, C is said to be irreducible. The preimage in R d of a properly convex domain Ω ⊂ P(R d ) is the union of a cone and its negative, when this cone is reducible (respectively irreducible) we say that Ω is reducible (respectively irreducible).
Benoist determined the Zariski closures of discrete groups acting co-compactly on irreducible properly convex domains. Theorem 2.3 (Benoist [Ben03]) Suppose that Ω ⊂ P(R d ) is an irreducible properly convex domain and Γ ≤ Aut(Ω) is a discrete group acting co-compactly on Ω. Then either (1) Ω is symmetric or (2) Γ is Zariski dense in PGL d (R).

The Hilbert distance
In this section we recall the definition of the Hilbert metric, but first some notation.
Given a projective line L ⊂ P(R d ) and four distinct points a, x, y, b ∈ L we define the cross ratio by where |·| is some (any) norm in some (any) affine chart of P(R d ) containing a, x, y, b.
Next, for x, y ∈ P(R d ) distinct let L x,y ⊂ P(R d ) denote the projective line containing x and y.
where ∂Ω ∩ L x,y = {a, b} with the ordering a, x, y, b along L x,y .
The following result is classical (see for instance [BK53, Section 28]).
Then H Ω is a complete Aut(Ω)-invariant metric on Ω which generates the standard topology on Ω. Moreover, if p, q ∈ Ω, then there exists a geodesic joining p and q whose image is the line segment [p, q].

Properly embedded simplices
In this subsection we recall the definition of properly embedded simplices.
In this case, we write dim(S) = k (notice that S is homeomorphic to R k ).
Then A is properly embedded in B if the inclusion map A ֒→ B is a proper map (relative to the subspace topology).
The Hilbert metric on a simplex is isometric to a normed space (see [Nus88, Proposition 1.7], [dlH93], or [Ver14]) and so we have the following observation.

Limits of linear maps
Every T ∈ P(End(R d )) induces a map defined by x → T(x). We will frequently use the following observation.
for all x ∈ P(R d )\P(ker T). Moreover, the convergence is uniform on compact subsets of P(R d ) \ P(ker T).

The faces and extreme points of a properly convex domain
∃ an open line segment in Ω containing x and y .
denote the set of all extreme points.
These subsets have the following basic properties.
Observation 2.11 Suppose that Ω ⊂ P(R d ) is a properly convex domain.
for all p ∈ F Ω (x) and q ∈ F Ω (y).
Proof These are all simple consequences of convexity.
We will also use the following results about the action of the automorphism group.
In the case of "non-tangential" convergence we can say more.
Proposition 5.7 in [IZ21] is stated differently, so we provide the proof.
Fix y ∈ F Ω (x). Then we can pick a sequence (y n ) n≥1 in [p 0 , y) such that sup n≥1 H Ω (y n , p n ) < ∞.
So there exists n j → ∞ so that the limit exists in Ω. Notice that q / ∈ P(ker T) by Proposition 2.12 and so the "moreover" part of Observation 2.9 implies that T(q) = lim n→∞ g n (q) = lim j→∞ g n j (q) = lim j→∞ g n j g −1 n j (y n j ) = lim j→∞ y n j = y.

Proximal elements
In this section we recall some basic properties of proximal elements. For more background we refer the reader to [BQ16].
A straightforward calculation provides the following characterization of proximality.
Observation 2.15 Suppose that g ∈ PGL d (R) and x is a fixed point of the g action on P(R d ). Then the following are equivalent: (1) x is an attractive fixed point of g, (2) g is proximal and x = ℓ + g .
Next we explain the global dynamics of a proximal element.
When g ∈ PGL d (R) is proximal, H − g is usually called the repelling hyperplane of g. This is motivated by the following observation.
We observe the following.
Proof Proposition 2.12 implies that ℓ + g ∈ ∂Ω and P( Since Aut(F) acts properly on F and ℓ + g ∈ F , the cyclic group must be relatively compact. This implies that every eigenvalue of h has the same absolute value. Then, since h is proximal, V must be one-dimensional and so F = {ℓ + g }. Thus ℓ + g is an extreme point.
The following result can be viewed as a converse to Observation 2.17 and will be used to construct proximal elements.

Proposition 2.19
Suppose that (g n ) n≥1 is a sequence in PGL d (R) and then for n sufficiently large g n is proximal and for all x ∈ P(R d )\P(ker T). Moreover, the convergence is uniform on compact subsets of P(R d ) \ P(ker T).
By assumption image T / ∈ P(ker T), so we can find a compact neighborhood U of image T in P(R d ) such that U is homeomorphic to a closed ball and Then by passing to a tail, we can assume that g n (U) ⊂ U for all n. So by the Brouwer fixed-point theorem, each g n has a fixed point x n ∈ U . Since U can be chosen arbitrarily small we also have We claim that for n large, x n is an attractive fixed point of g n . By Lemma 2.15 this will finish the proof. Let f n : P(R d ) → P(R d ) be the diffeomorphism induced by g n , that is f n (x) = g n (x) for all x. Then, since each g n acts by projective linear transformations, we see that f n converges locally uniformly in the C ∞ topology on Hence for n large x n is an attractive fixed point of g n .

Rank one isometries
In this section we state a characterization of rank one isometries established in [Isl19]. and γ fixes two points x, y ∈ ∂Ω with s ∂Ω (x, y) > 2, then In particular, γ is a rank one isometry. (2) The only points fixed by γ in ∂Ω are ℓ + γ and ℓ − γ .

A semigroup associated to a boundary face
In this section we prove Theorem 1.11, which we restate here.
is an irreducible properly convex domain and Γ ≤ Aut(Ω) is a discrete group acting co-compactly on Ω.
If Ω is non-symmetric, F ⊂ ∂Ω is a boundary face, V := Span F , and dim(V) ≥ 2, then: The proof of Theorem 3.1 will follow from a series of lemmas, many of which hold in greater generality.
For the rest of the section fix a properly convex domain Ω ⊂ P(R d ) and a subgroup Γ ≤ Aut(Ω). Notice that we are not (currently) assuming that Ω is irreducible, that Γ is discrete, or that Γ acts co-compactly on Ω.
We begin with an observation.
Proof Part (a) follows immediately from Proposition 2.12.
We first show that ker( Next, by definition, So by Equation (2) and dimension counting we have Then, since T 1 , T 2 ∈ Γ End F,⋆ were arbitrary, we see that is a semigroup in P(End(V)).
The next lemma requires a definition. Definition 3.6 A point x ∈ ∂Ω is a conical limit point of Γ if there exist p 0 ∈ Ω, a sequence (p n ) n≥1 in [p 0 , x) with p n → x, and a sequence (γ n ) n≥1 in Γ with sup n≥1 H Ω (γ n (p 0 ), p n ) < +∞.
Notice that if Γ acts co-compactly on Ω, then every boundary point is a conical limit point. Lemma 3.7 Suppose that x ∈ ∂Ω is a conical limit point of Γ, F = F Ω (x), V = Span F , and dim(V) = k. If k ≥ 2 and the image of Γ ֒→ PGL(∧ k R d ) is strongly irreducible (e.g. Γ is Zariski dense in PGL d (R)), then there exists a sequence (g n ) n≥1 in Γ with the following properties: (1) g n → T in P(End(R d )) where T ∈ Γ End F,⋆ .
Then define g n := γ n φ. Then Next we claim that g n V = V for n sufficiently large. Notice that g n V = V if and only if g −1 n V = V if and only if γ −1 n V = φV . But γ −1 n V → V ∞ and φV = V ∞ . So g n V = V for n sufficiently large.
Finally, since g n V → V and g n V = V for n sufficiently large, we can pass to a subsequence so that V, g 1 V, g 2 V, . . . are pairwise distinct subspaces. Thus g 1 | V , g 2 | V , . . . must be pairwise distinct. Lemma 3.8 Suppose that x ∈ ∂Ω is a conical limit point of Γ, F = F Ω (x), V = Span F , and dim(V) = k. If k ≥ 2 and the image of Γ ֒→ PGL(∧ k R d ) is strongly irreducible (e.g. Γ is Zariski dense in PGL d (R)), then the set Proof Let T ∈ Γ End F,⋆ and (g n ) n≥1 be as in the previous lemma. Since g 1 | V , g 2 | V , . . . are pairwise distinct and each g n | V is determined by its values on any set of dim V + 1 points in general position, after passing to a subsequence we can find a point x 0 ∈ F such that g 1 (x 0 ), g 2 (x 0 ), . . . are pairwise distinct.
Since g 1 (x 0 ), g 2 (x 0 ), . . . are pairwise distinct, by passing to another sequence we can assume that g n (x 0 ) = T(x 0 ) for all n. Then, for each n there exists a unique projective line L n containing T(x 0 ) and g n (x 0 ). By passing to a subsequence we can suppose that L n converges to a projective line L. Then let W ⊂ R d be the two dimensional linear subspace with L = P(W).
Notice that TϕT = lim n→∞ g n ϕg n is in Γ End F,⋆ . Then replacing (g n ) n≥1 with a tail, we can assume that S n := Tϕg n ∈ Γ End F,⋆ for all n.
We claim that the set {S n (x 0 ) : n ≥ 0} ⊂ F is infinite. For this calculation we fix an affine chart A of P(R d ) which contains Ω. We then identify A with R d−1 such that T(x 0 ) = 0 and for some sequence (t n ) n≥1 converging to 0. Then in these coordinates Since d(Tϕ) 0 (1, 0, . . . , 0) = 0 and t n → 0, we see that the set {S n (x 0 ) : n ≥ 0} is infinite.
Finally, since S n | V → TϕT| V , this implies that is non-discrete in P(End(V)). Lemma 3.9 Suppose that x ∈ ∂Ω is a conical limit point of Γ, F = F Ω (x), V = Span F , and dim(V) = k. If k ≥ 2 and Γ is Zariski dense in PGL d (R), then is Zariski dense in P(End(V)).
Proof Let Z 0 be the Zariski closure of

Lemma 3.7 implies that Γ
End F,⋆ is non-empty. So fix T ∈ Γ End F,⋆ . Then define Notice that Z 1 is a proper Zariski closed set in PGL d (R) since rank(T) = dim(V). Also define Notice that Z 2 is a Zariski closed subset of PGL d (R).
Then, since Z 1 is a proper Zariski closed subset of PGL d (R) and Γ is Zariski dense in PGL d (R), we see that Z 2 = PGL d (R). Then

Proof of Theorem 3.1 Parts (1) and (2) follow from Lemmas 3.3 and 3.4 respectively.
Since Γ acts co-compactly on Ω every point in ∂Ω is a conical limit point and Theorem 2.3 implies that Γ is Zariski dense in PGL d (R). So Part (3) follows from Lemmas 3.3, 3.8, and 3.9.

The main rigidity theorem
Recall that E Ω ⊂ ∂Ω denotes the set of extreme points of a properly convex domain Ω. In this section we prove the following rigidity result.
Theorem 4.1 Suppose that Ω ⊂ P(R d ) is an irreducible properly convex divisible domain and there exists a boundary face F ⊂ ∂Ω such that Then Ω is symmetric with real rank at least two.
The rest of the section is devoted to the proof of the theorem. So suppose Ω ⊂ P(R d ) satisfies the hypothesis of the Theorem. Then let Γ ≤ Aut(Ω) be a discrete group acting co-compactly on Ω.
We assume for a contradiction that Ω is not symmetric with real rank at least two.

Lemma 4.2 Ω is not symmetric.
Proof If Ω were symmetric, then by assumption it would have real rank one. Then, by the characterization of symmetric convex divisible domains, Ω coincides with the unit ball in some affine chart. Then E Ω = ∂Ω which is impossible since there exists a boundary face F ⊂ ∂Ω such that Then define V := Span F .
is in Aut(F).
Proof Notice that T| V ∈ PGL(V) since T(V) ⊂ V and ker T ∩ V = {0}. So we just have to show that T(F) = F . Theorem 3.1 part (2) says that T(F) ⊂ F and so we just have to show that F ⊂ T(F).
Fix y ∈ F . Since the set T(F) ∩ F is closed in F , there exists x 0 ∈ T(F) ∩ F such that Since Thus we can find z ∈ F ′ and a sequence (z n ) n≥1 in E Ω such that z n → z. Since z ∈ F ′ , there exists an open line segment L in F which contains z and x ′ 0 . Then T(L) is an open line segment in F since T| V ∈ PGL(V). Then, since T(x ′ 0 ) ∈ F , we also have T(z) ∈ F . Since T ∈ Γ End F,⋆ ⊂ Γ End there exists a sequence (g n ) n≥1 in Γ such that g n → T in P(End(R d )). Now z / ∈ P(ker T) since ker T ∩ V = {0}. So by the "moreover" part of Observation 2.9 However, g n (z n ) ∈ E Ω and so Thus we have a contradiction. Hence y = x 0 ∈ T(F). Since y ∈ F was arbitrary we have F ⊂ T(F).

Lemma 4.4 Aut(F) is non-discrete and Zariski dense in PGL(V).
Proof This follows immediately from Lemma 4.3 and Theorem 3.1 part (3).

Lemma 4.5 PSL(V) ⊂ Aut(F).
Proof Let Aut 0 (F) denote the connected component of the identity in Aut(F) and let g ⊂ sl(V) denote the Lie algebra of Aut 0 (F). Then g = {0} since Aut(F) is closed and non-discrete. Also Aut 0 (F) is normalized by Aut(F) and so Ad(g) g = g for all g ∈ Aut(F). Then, since Aut(F) is Zariski dense in PGL(V), we see that Ad(g) g = g for all g ∈ PGL(V). Since the representation Ad : PGL(V) → GL(sl(V)) is irreducible, we must have g = sl(V). Thus Aut 0 (F) = PSL(V).
Proof of Theorem 4.1 The previous lemma immediately implies a contradiction: fix x ∈ F , then So F = P(V) which contradicts the fact that Ω is properly convex.

Density of bi-proximal elements
In this section we prove a density result for the attracting and repelling fixed points of bi-proximal elements. To state the result we need one definition. If Ω ⊂ P(R d ) is a properly convex domain and Γ ≤ Aut(Ω), then the limit set of Γ is Equivalently, a point x ∈ ∂Ω is in L Ω (Γ) if and only if there exist p ∈ Ω and a sequence (γ n ) n≥1 in Γ such that γ n (p) → x.
Theorem 5.1 Suppose that Ω ⊂ P(R d ) is a properly convex domain and Γ ≤ Aut(Ω) is a strongly irreducible group. If x, y ∈ L Ω (Γ) are extreme points of Ω and (x, y) ⊂ Ω, then there exists a sequence of bi-proximal elements (g n ) n≥1 in Γ such that lim n→∞ ℓ + gn = x and lim n→∞ ℓ − gn = y.
Before proving the theorem we state and prove one corollary. Fix some x ′ ∈ image T − and y ′ ∈ image S − . Since Γ is strongly irreducible, by Observation 2.2 there exists h ∈ Γ such that (4) hT − (y) / ∈ P(ker S + ).
and hence, by Proposition 2.19, g n is proximal for n sufficiently large and ℓ + gn → x. Similar reasoning shows that g −1 n is proximal for n sufficiently large and ℓ − gn = ℓ + g −1 n → y.

North-South dynamics
In this section we prove a stronger version of Theorem 5.1 for pairs of extreme points in the limit set whose simplicial distance is greater than two. Theorem 6.1 Suppose that Ω ⊂ P(R d ) is a properly convex domain and Γ ≤ Aut(Ω) is strongly irreducible. Assume x, y ∈ L Ω (Γ) are extreme points of Ω and s ∂Ω (x, y) > 2. If A, B ⊂ Ω are neighborhoods of x, y, then there exists g ∈ Γ with g Ω \ B ⊂ A and g −1 Ω \ A ⊂ B. Before proving the theorem we state and prove one corollary. Corollary 6.3 Suppose that Ω ⊂ P(R d ) is an irreducible properly convex domain and Γ ≤ Aut(Ω) is a discrete group acting co-compactly on Ω. Assume x, y ∈ ∂Ω are extreme points and s ∂Ω (x, y) > 2. If A, B ⊂ Ω are neighborhoods of x, y, then there exists g ∈ Γ with g Ω \ B ⊂ A and g −1 Ω \ A ⊂ B.
Proof Observation 2.17 implies that for all x ∈ P(R d ) − P(H − g ) and the convergence is locally uniform. We claim that ∩ Ω and hence [y, ℓ − g ] ⊂ ∂Ω. Then by Theorem 2.20 part (2) we have y = ℓ + g . So P(H − g ) ∩ Ω ⊂ {ℓ − g } and the claim is established.
Then by the locally uniform convergence in Equation (3), there exists N 1 > 0 such that Repeating the same argument with γ −1 shows that there exists N 2 > 0 such that Then N = max{N 1 , N 2 } satisfies the conclusion of the lemma.
Proof of Theorem 6.1 By Theorem 5.1 there exists a sequence of bi-proximal elements (g n ) n≥1 in Γ such that lim n→∞ ℓ + gn = x and lim n→∞ ℓ − gn = y.
Next fix n sufficiently large such that ℓ + gn ∈ A and ℓ − gn ∈ B. Then by Lemma 6.4 there exists m ≥ 0 such that g m n Ω \ B ⊂ A and g −m n Ω \ A ⊂ B.
So g = g m n satisfies the theorem.

Fixed points and centralizers
In this section we prove the following result connecting the number of boundary fixed points of an element with the size of its centralizer.
Theorem 7.1 Suppose that Ω ⊂ P(R d ) is an irreducible properly convex domain and Γ ≤ Aut(Ω) is a discrete group that acts co-compactly on Ω. If g ∈ Γ has infinite order, then the following are equivalent: (1) there exist two distinct points x, y ∈ ∂Ω fixed by g with s ∂Ω (x, y) < +∞, (2) g fixes at least three points in ∂Ω, (3) the cyclic group g Z has infinite index in its centralizer.
As a corollary we will obtain the following.
Corollary 7.2 Suppose that Ω ⊂ P(R d ) is an irreducible properly convex domain and Γ ≤ Aut(Ω) is a discrete group that acts co-compactly on Ω. If g ∈ Γ is bi-proximal, then the following are equivalent: g has at least three fixed points in ∂Ω, (4) the cyclic group g Z has infinite index in its centralizer.
We will first recall some results established in [IZ21], then prove the theorem and corollary.

Maximal Abelian subgroups and minimal translation sets
We have the following description of maximal Abelian subgroups. Moreover, A has a finite index subgroup isomorphic to Z dim(S) .

Remark 7.4
The above result is a special case of Theorem 1.6 in [IZ21] which holds in the more general case when Γ ≤ Aut(Ω) is a naive convex co-compact subgroup.
Cooper-Long-Tillmann [CLT15] showed that the minimal translation length of an element can be determined from its eigenvalues. Proposition 7.6 [CLT15, Proposition 2.1] If Ω ⊂ P(R d ) is a properly convex domain and g ∈ Aut(Ω), then Remark 7.7 Recall, that denote the absolute values of the eigenvalues of some (hence any) lift of g to SL ± d (R) := {h ∈ GL d (R) : det h = ±1}.
As a consequence of Proposition 7.6 we observe the following.
Next, given a group G and an element g ∈ G, let C G (g) denote the centralizer of g in G. Then given a subset X ⊂ G, define In [IZ21], the following result about centralizers and minimal translation sets of Abelian groups was established.

Remark 7.10
The above result is a special case of Theorem 1.9 in [IZ21] which holds in the more general case when Γ ≤ Aut(Ω) is a naive convex co-compact subgroup.
Finally, we will use the following observations. Proposition 7.11 Suppose that S ⊂ P(R d ) is a simplex. If g ∈ Aut(S) fixes every vertex of S, then Min S (g) = S.
Observation 7.12 Suppose that Ω ⊂ P(R d ) is a properly convex domain and Γ ≤ Aut(Ω) is a discrete group. If g ∈ Γ is bi-proximal and (ℓ + g , ℓ − g ) ⊂ Ω, then g Z has finite index in C Γ (g).

Proof of Theorem 7.1
Fix a maximal Abelian subgroup A ≤ Γ which contains g. Then by Theorem 7.3 there exists S ⊂ Ω such that • S is a properly embedded simplex, • A acts co-compactly on S, • A fixes every vertex of S, and • A has a finite index subgroup isomorphic to Z dim(S) .
We consider a number of cases and prove that in each case (1), (2), and ( Case 2 (a): Assume s ∂Ω (v + , v − ) > 2. Then Theorem 2.20 implies that g is a rank one isometry and v ± = ℓ ± g . Theorem 2.20 also implies that v + , v − are the only fixed points of g in ∂Ω and s ∂Ω (v + , v − ) = ∞. Hence (1) and (2) are false. Observation 7.12 implies that g Z has finite index in C Γ (g) and hence (3) is false.
Passing to a further sequence we can suppose that g ±n j (y 0 ) → y ± . Then for all x ∈ P(R d ) \ P(ker T ± ) we must have y ∈ P(ker T + ∩ ker T − ). Thus the set is non-empty. Then g has a fixed point y ∈ C since C is g-invariant, closed, and convex. So g has at least three fixed points in ∂Ω and (2) is true.
Recall that v ∓ ∈ P(ker T ± ) and P(ker T ± ) ∩ Ω = ∅, hence Then by Equation (4) and Proposition 7.11 Now we claim that g Z has infinite index in C Γ (g). Theorem 7.9 implies that there is a compact set K ⊂ Ω such that Thus g Z has infinite index in C Γ (g) and so (3) is true.

Rank in the sense of Prasad-Raghunathan
In this section we consider the rank of a group in the sense of Prasad and Raghunathan [PR72]. Prasad and Raghunathan computed the rank of lattices in semisimple Lie groups which implies the following.
is an irreducible properly convex domain. If Ω is symmetric with real rank r and Γ ≤ Aut(Ω) is a discrete group acting co-compactly on Ω, then rank PR (Γ) = r.
As a corollary to Selberg's lemma we have the following lower bound on the Prasad-Raghunathan rank. Proof By Selberg's lemma, there exists a finite index torsion-free subgroup Γ * ≤ Γ.
In this section we will show that the existence of a rank one isometry implies that the Prasad-Raghunathan rank is one.

Remark 8.5
The proof of Proposition 8.4 is a simple modification of Ballmann and Eberlein's [BE87] proof of the analogous statement for CAT(0) groups.
The rest of the section is devoted to the proof of Proposition 8.4. So suppose Ω ⊂ P(R d ), Γ ≤ Aut(Ω), and g ∈ Γ satisfy the hypothesis of the proposition. By Corollary 8.3 it is enough to fix a finite index subgroup Γ * ⊂ Γ and show that r(Γ * ) ≤ 1. Also by replacing g with a sufficiently large power we may assume that g ∈ Γ * . Lemma 8. 6 Suppose that x 1 , x 2 ∈ ∂Ω and (x 1 , x 2 ) ⊂ Ω. If A, B ⊂ ∂Ω are open sets with A ∩ B = ∅, then we can find disjoint neighborhoods V 1 , V 2 of x 1 , x 2 such that: for each ϕ ∈ Aut(Ω) at least one of the following occurs Proof The following argument is essentially the proof of Lemma 3.10 in [BE87].
Fix a distance d P on P(R d ) induced by a Riemannian metric. Then for each n and j = 1, 2, let V j,n be a neighborhood of x j whose diameter with respect to d P is less than 1/n. Suppose for a contradiction that the lemma is false. Then for each n there exists ϕ n ∈ Aut(Ω) such that ϕ n (V j,n ) ∩ A = ∅ and ϕ n (V j,n ) ∩ B = ∅ (5) for j = 1, 2. By passing to a subsequence we can suppose that T := lim n→∞ ϕ n exists in P(End(R d )). Then for all u ∈ P(R d )\P(ker T). Moreover, the convergence is uniform on compact subsets of P(R d ) \ P(ker T).
Proposition 2.12 implies that P(ker T) ∩ Ω = ∅. Then, since (x 1 , x 2 ) ⊂ Ω, it is impossible for both x 1 , x 2 to be contained in P(ker T). So after possibly relabelling, we may assume that x 1 / ∈ P(ker T).
By Equation (5), there exists sequences a n , b n ∈ ∂Ω converging to x 1 such that ϕ n (a n ) ∈ A and ϕ n (b n ) ∈ B. Then, since x 1 / ∈ P(ker T), So T(x 1 ) ∈ A ∩ B = ∅ which is a contradiction.
Proof The following argument is essentially the proof of Theorem 3.1 in [BE87].
Since Γ is strongly irreducible, Γ * is also strongly irreducible. So by Observation 2.2 there exists φ ∈ Γ * such that g are all distinct. Then h := φgφ −1 is bi-proximal, ℓ ± h = φℓ ± g , and Fix open neighborhoods A, B ⊂ ∂Ω of ℓ + h , ℓ − h such that A ∩ B = ∅. Then let V 1 , V 2 ⊂ ∂Ω be neighborhoods of ℓ + g , ℓ − g so that A, B, V 1 , V 2 satisfy Lemma 8.6. By further shrinking each V j , we can assume that each ∂Ω \ V j is homeomorphic to a closed ball.
Next let U 1 ⊂ V 1 be a closed neighborhood of ℓ + g such that: if x ∈ U 1 and y ∈ ∂Ω\V 1 , then s ∂Ω (x, y) > 2. Such a choice is possible by Theorem 2.20 part (2). In a similar fashion, let U 2 ⊂ V 2 be a closed neighborhood of ℓ − g such that: if x ∈ U 2 and y ∈ ∂Ω \ V 2 , then s ∂Ω (x, y) > 2.
By further shrinking each U j , we can assume that each U j is homeomorphic to a closed ball.
Since x, y ∈ ∂Ω were arbitrary we see that (5) holds. Proof Fix a boundary face F ⊂ ∂Ω of maximal dimension. We claim that Otherwise there exists x ∈ F and a sequence x n ∈ E Ω such that x n → x ∈ F . Now fix an extreme point y ∈ ∂Ω \ F . Then, by hypothesis, [x n , y] ⊂ ∂Ω for all n, so [x, y] ⊂ ∂Ω.
Then dim F Ω (z) > dim F which is a contradiction. So we must have E Ω ∩ F = ∅ and hence (12) holds.
Proof By Theorem 7.3 every infinite order element g ∈ Γ preserves a properly embedded simplex S ⊂ Ω with dim(S) ≥ 1. Hence g fixes the vertices v 1 , . . . , v k of S. By hypothesis s ∂Ω (v 1 , v 2 ) < +∞ and hence, by Theorem 7.1, g Z has infinite index in the centralizer C Γ (g).