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.

Geometry & Topology msp Volume 27 (2023) A higher-rank rigidity theorem for convex real projective manifolds A real projective structure on a d -manifold M is an open cover M D S ˛U˛a long with coordinate charts ' ˛W U ˛! P .R d C1 / such that each transition function ' ˛ı ' 1 čoincides with the restriction of an element in PGL d C1 .R/.A real projective manifold is a manifold equipped with a real projective structure.
An important class of real projective manifolds is the convex real projective manifolds, which are defined as follows.First, a subset P .R d C1 / 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 is Aut./ WD fg 2 PGL d C1 .R/ W g D g: If Ä Aut./ is a discrete subgroup that acts freely and properly discontinuously on , then the quotient manifold n is called a convex real projective manifold.Notice that local inverses to the covering map !n provide a real projective structure on the quotient.In the case when there exists a compact quotient, the domain is called divisible.For more background see the expository papers by Benoist [7], Marquis [22] and Quint [25].
When d Ä 3, the structure of closed convex real projective d -manifolds is very well understood thanks to deep work of Benzécri [9], Goldman [16] and Benoist [6].But, when d 4, their general structure is mysterious.

Andrew Zimmer
We establish a dichotomy for convex real projective manifolds inspired by the theory of nonpositively curved Riemannian manifolds.In particular, a compact Riemannian manifold .M; g/ with nonpositive 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 [2] and Burns and Spatzier [11;12] states that every compact irreducible Riemannian manifold with nonpositive 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 Ballmann [1] and Knieper [20]).
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 a definition of higher rank: Definition 1.1 (i) A properly convex domain P .R d / has higher rank if for every p; q 2 there exists a properly embedded simplex S with dim.S / 2 and OEp; q S. (ii) If a properly convex domain P .R d / does not have higher rank, then we say that has rank one.
There are two basic families of properly convex domains with higher rank: reducible domains (see Section 2.4) and symmetric domains with real rank at least two.
A properly convex domain P .R d / is called symmetric if there exists a semisimple Lie group G Ä PGL d .R/ which preserves and acts transitively.In this case, the real rank of 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 Faraut and Korányi [15], Koecher [21] and Vinberg [28; 29].Borel [10] proved that every semisimple Lie group contains a cocompact lattice, which implies that every symmetric properly convex domain is divisible.
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: 3 Given a properly convex domain P .R d / the (possibly infinite valued) simplicial distance on @ is defined by s @ .x;y/ D inffk W 9a 0 ; : : : ; a k with x D a 0 ; y D a k and OEa j ; a j C1 @ for 0 Ä j Ä k 1g: We will prove a characterization of higher rank in the context of convex real projective manifolds: Theorem 1.4 (see Section 9) Suppose that P .R d / is an irreducible properly convex domain and Ä Aut./ is a discrete group acting cocompactly on .Then the following are equivalent: is symmetric with real rank at least two.
(ii) has higher rank.
(iii) The extreme points of form a closed proper subset of @ .
(vii) has higher rank in the sense of Prasad and Raghunathan (see Section 8).
(viii) For every g 2 with infinite order, the cyclic group g Z has infinite index in the centralizer of g in .
(ix) Every g 2 with infinite order has at least three fixed points in @ .
(x) OE`C g ; ` g @ for every biproximal element g 2 .
M Islam [18] 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 [1]) and are defined as follows: Definition 1.5 (Islam [18]) Suppose that P .R d / is a properly convex domain.An element g 2 Aut./ is a rank-one isometry if g is biproximal and s @ .`Cg ; ` g / > 2.
Remark 1.6 (1) When g 2 Aut./ is a rank-one isometry, the properly embedded line segment .`Cg ; ` g / is preserved by g.Further, g acts by translations on .`Cg ; ` g / in the following sense: if H is the Hilbert metric on , then there exists T > 0 such that H .g n .x/;x/ D nT for all n 0 and x 2 .`Cg ; ` g /.
(2) Islam [18,Proposition 6.3] also proved a weaker characterization of rank-one isometries: g 2 Aut./ is a rank-one isometry if and only if g acts by translations on a properly embedded line segment .a;b/ and s @ .a;b/ > 2.
As an immediate consequence of Theorem 1.4: Corollary 1.7 Suppose that P .R d / is an irreducible properly convex domain and Ä Aut./ is a discrete group acting cocompactly on .Then the following are equivalent: has rank one.
(ii) contains a rank-one isometry.
Islam has also established a number of remarkable results when the automorphism group contains a rank-one isometry; see [18] for details.For instance: Corollary 1.8 (consequence of Theorem 1.4 and [18, Theorem 1.5]) Suppose that P .R d / is an irreducible properly convex domain and Ä Aut./ is a discrete group acting cocompactly on .If d 3 and is not symmetric with real rank at least two, then is an acylindrically hyperbolic group.
1.1 Outline of the proof of Theorem 1.4 The difficult part is showing that any one of conditions (ii)-(xi) 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 [5,  We then prove: Theorem 3.1 Suppose that P .R d / is an irreducible properly convex domain and Ä Aut./ is a discrete group acting cocompactly on .If is nonsymmetric, F @ is a boundary face, V WD Span F R d , and dim.V / 2, then: F ;? g is a nondiscrete Zariski-dense semigroup in P .End.V //.Using Theorem 3.1 we will show that any one of Theorem 1.4(ii)-(xi) implies that the domain is symmetric with real rank at least two.Here is a sketch of the argument: First suppose that P .R d / is an irreducible properly convex domain, Ä Aut./ is a discrete group acting cocompactly on , and any one of Theorem 1.4(ii)-(xi) is true.
Then let E @ denote the extreme points of .We will show that there exists a boundary face F @ such that (1) F \ E D ∅: By choosing F minimally, we can also assume that E intersects every boundary face of strictly smaller dimension.As before, let V WD Span F .Then using (1) we show that T j V 2 Aut.F / for every T 2 End F ;? .Therefore Theorem 3.1 implies that either is symmetric or Aut.F / is a nondiscrete 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 must be symmetric.

Outline of the paper
In Section 2 we recall some preliminary material.In Section 3 we prove Theorem 3.1.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 and Raghunathan.Finally, in Section 9 we prove Theorem 1.4.

Acknowledgments
I would like to thank Ralf Spatzier and Mitul Islam for helpful conversations.I would also like to thank the referee for their very useful corrections and suggestions.This material is based upon work supported by the National Science Foundation under grants DMS-1904099, DMS-2105580, and DMS-2104381.

Notation
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 OEo be the projective equivalence class of o.For instance: We also identify P .R d / D Gr 1 .R d /, so for instance if x 2 P .R d / and V R d is a linear subspace, then x 2 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 some topological definitions: Definition 2.1 Let C P .R d / be 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 WD C nrel-int.C /.
A line segment in P .R d / is a connected subset of a projective line.Given two points x; y 2 P .R d / there is no canonical line segment with endpoints x and y, but we will use the convention that if C P .R d / is a properly convex set and x; y 2 C , then (when the context is clear) we will let OEx; y denote the closed line segment joining x to y which is contained in C .In this case, we will also let .x;y/ D OEx; y n fx; yg, OEx; y/ D OEx; y n fyg, and .x;y D OEx; y n fxg.

Irreducibility
A subgroup Ä PGL d .R/ is irreducible if f0g and R d are the only -invariant linear subspaces of R d , and strongly irreducible if every finite-index subgroup is irreducible.
We will use the following observation several times: are linear subspaces, then there exists g 2 such that gx j … P .V j / for all 1 Ä j Ä k.
Proof Let G D Zar denote the Zariski closure of in PGL d .R/ and let G 0 Ä G denote the connected component of the identity of G (in the Zariski topology).Then G 0 \ is a finite-index subgroup of and hence G 0 is irreducible.So each set Otherwise, C is said to be irreducible.The preimage in R d of a properly convex domain P .R d / is the union of a cone and its negative; when this cone is reducible (respectively irreducible) we say that is reducible (respectively irreducible).

Benoist determined the Zariski closures of discrete groups acting cocompactly on irreducible properly convex domains:
Theorem 2.3 (Benoist [5]) Suppose that P .R d / is an irreducible properly convex domain and Ä Aut./ is a discrete group acting cocompactly on .Then either

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 2 L we define the cross ratio by OEa; x; y; b D jx bjjy aj jx ajjy bj ; where j j is some (any) norm in some (any) affine chart of P .R d / containing a, x, y and b.
Next, for x; y 2 P .R d / distinct, let L x;y P .R d / denote the projective line containing x and y.
Definition 2.4 Suppose that P .R d / is a properly convex domain.The Hilbert distance on , denoted by H , is defined as follows: if x; y 2 are distinct, then H .x; y/ D 1 2 logOEa; x; y; b; where @ \ L x;y D fa; bg with the ordering a; x; y; b along L x;y .
The following result is classical; see for instance [13,Section 28].
Geometry & Topology, Volume 27 (2023) Proposition 2.5 Suppose that P .R d / is a properly convex domain.Then H is a complete Aut./-invariant metric on which generates the standard topology on .Moreover, if p; q 2 , then there exists a geodesic joining p and q whose image is the line segment OEp; q.

Properly embedded simplices
In this subsection we recall the definition of properly embedded simplices.
In this case, we write dim.S / D k (notice that S is homeomorphic to R k ).Definition 2.7 Suppose that A B P .R d /.Then A is properly embedded in B if the inclusion map A ,! B is a proper map (relative to the subspace topology).
By [23,Proposition 1.7], [17], or [26] the Hilbert metric on a simplex is isometric to a normed space, and so: Observation 2.8 Suppose that P .R d / is a properly convex domain and S is a properly embedded simplex.Then .S; H / is quasi-isometric to R dim S .

Limits of linear maps
Every T 2 P .End.R d // induces a map P .R d / n P .kerT / !P .R d / defined by x !T .x/.We will frequently use: for all x 2 P .R d / n P .kerT /.Moreover, the convergence is uniform on compact subsets of P .R d / n P .kerT /. (iv) If y 2 @F .x/,then F .y/ @F .x/and F .y/ D F F .x/ .y/.
Proof These are all simple consequences of convexity.
We will also use results about the action of the automorphism group: Proposition 2.12 [19, Proposition 5.6] Suppose that P .R d / is a properly convex domain, p 0 2 , and .gn / n 1 is a sequence in Aut./ such that (i) g n .p0 / !x 2 @ , (ii) g 1 n .p0 / !y 2 @ , and (iii) g n converges in P .End.R d // to T 2 P .End.R d //.
In the case of "nontangential" convergence we can say more: Proposition 5.7 in [19] is stated differently, so we provide the proof: Proof Proposition 2.12 implies T ./ F .x/, so we have to prove T ./ F .x/.
Fix y 2 F .x/.Then we can pick a sequence .yn / n 1 in OEp 0 ; y/ such that sup n 1 H .y n ; p n / < 1: H .g 1 n .yn /; p 0 / < 1: So there exists n j ! 1 such that the limit exists in .Notice that q … P .kerT / by Proposition 2.12 and so the "moreover" part of Observation 2.9 implies that y n j D y: Since y was arbitrary, F .x/ T ./.

Proximal elements
In this section we recall some basic properties of proximal elements.For more background we refer the reader to [8].
A straightforward calculation provides a characterization of proximality: Observation 2.15 Suppose that g 2 PGL d .R/ and x is a fixed point of the g action on P .R d /.Then the following are equivalent: x is an attractive fixed point of g.
(ii) g is proximal and x D `C g .
Next we explain the global dynamics of a proximal element./ is proximal , then `C g is an extreme point of @ and P .H g / \ @ D ∅.
Proof Proposition 2.12 implies that `C g 2 @ and P .H g /\@ D ∅.Let F D F .`C g / and V D Span F .Then g.V / D V .Let N g 2 GL d .R/ be a lift of g 2 PGL d .R/ and let h 2 GL.V / denote the element obtained by restricting N g to V .Notice that h is proximal since `C g V .Further OEh 2 Aut.F / and h.`C g / D `C g .Since Aut.F / acts properly on F and `C g 2 F , the cyclic group OEh Z Ä Aut.F / Ä PGL.V / 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 D f`C g g.Thus `C 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.x n : We claim that, for n large, x n is an attractive fixed point of g n .By Observation 2.15 this will finish the proof.Let f n W P .R d / !P .R d / be the diffeomorphism induced by g n , that is f n .x/D g n .x/for all x.Then, since each g n acts by projective linear transformations, we see that the f n converge locally uniformly in the C 1 topology on P .R d / n P .kerT / to the constant map f Á image T .So, fixing a Riemannian metric on P .R d /, we have lim 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 [18]: and fixes two points x; y 2 @ with s @ .x;y/ > 2, then: is biproximal and f`C ; ` g D fx; yg.In particular, is a rank-one isometry.
(ii) The only points fixed by in @ are `C and ` .
3 A semigroup associated to a boundary face Theorem 3.1 Suppose that P .R d / is an irreducible properly convex domain and Ä Aut./ is a discrete group acting cocompactly on .If is nonsymmetric, F @ is a boundary face, V WD 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 P .R d / and a subgroup Ä Aut./.Notice that we are not (currently) assuming that is irreducible, that is discrete, or that acts cocompactly on .
and so T 1 ı T 2 2 End by Observation 3.2.
We first show ker.T 1 ıT 2 /\V D f0g.Suppose v 2 ker.T 1 ıT 2 /\V .Then T 2 .v/ 2 ker T 1 .But image T 2 D V and ker T 1 \ V D f0g, so T 2 .v/D 0 and so v 2 ker T 2 \ V D f0g.So v D 0, and thus (2) f0g D ker.T 1 ı T 2 / \ V: So by ( 2) and dimension counting image.T 1 ı T 2 / D V: Then, since T 1 ; T 2 2 End F ;? were arbitrary, we see that The next lemma requires a definition.Definition 3.6 A point x 2 @ is a conical limit point of if there exist p 0 2 , a sequence .pn / n 1 in OEp 0 ; x/ with p n !x, and a sequence .n / n 1 in with sup n 1 H . n .p0 /; p n / < C1: Notice that if acts cocompactly on then every boundary point is a conical limit point.Lemma 3.7 Suppose x 2 @ is a conical limit point of , F D F .x/, V D Span F , and dim.V / D k.If k 2 and the image of ,! PGL V k R d is strongly irreducible (eg is Zariski dense in PGL d .R/), then there exists a sequence .gn / n 1 in with : (i) g n !T in P .End.R d //, where T 2 End F ;? .(ii) g 1 j V ; g 2 j V ; : : : are pairwise distinct elements of P .Lin.V; R d //.
Proof By hypothesis there exist p 0 2 , a sequence .pn / n 1 in OEp 0 ; x/ with p n !x, and a sequence .n / n 1 in with sup n 1 H . n .p0 /; p n / < C1: After passing to a subsequence we can suppose that the limit and so S 2 End F .By passing to another subsequence we can suppose that

S D lim
Let V D Spanfv 1 ; : : : ; v k g, V 1 D Spanfu 1 ; : : : ; u k g, and ker S D Spanfs 1 ; : : : ; s d k g, and let W 1 D OEu 1 ^ ^uk and that there exists some 2 such that OEv g n V: Next we claim that g n V ¤ V for n sufficiently large.Notice that g n V D V if and only if 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 j V ; g 2 j V ; : : : must be pairwise distinct.Lemma 3.8 Suppose x 2 @ is a conical limit point of , F D F .x/, V D Span F , and dim.V / D k.If k 2 and the image of ,! PGL.^kR d / is strongly irreducible (eg is Zariski dense in PGL d .R/), then the set fT j V W T 2 End F ;? g is nondiscrete in P .End.V //.

Proof Let T 2 End
F ;? and .gn / n 1 be as in the previous lemma.Since g 1 j V ; g 2 j V ; : : : are pairwise distinct and each g n j V is determined by its values on any set of dim V C 1 points in general position, after passing to a subsequence we can find a point x 0 2 F such that g 1 .x0 /; g 2 .x0 /; : : : are pairwise distinct.Since x 0 2 F and P .kerT / \ F D ∅, T .x0 / D lim n!1 g n .x0 /: Since g 1 .x0 /; g 2 .x0 /; : : : are pairwise distinct, by passing to another sequence we can assume that g n .x0 / ¤ T .x0 / for all n.Then, for each n there exists a unique projective line L n containing T .x0 / and g n .x0 /.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 D P .W /.
Fix some W 0 2 Gr k .R d / with W W 0 and suppose that V D Spanfv 1 ; : : : ; v k g, W 0 D Spanfw 1 ; : : : ; w k g, and ker T D Spanft 1 ; : : : ; t d k g.Let for all n.
We claim that the set is infinite.For this calculation we fix an affine chart A of P .R d / which contains .We then identify A with R d 1 so that T .x0 / D 0 and A \ L D f.t; 0; : : : ; 0/ W t 2 Rg: Since ker T \ 'V D f0g, in these coordinates the map T ' is smooth in a neighborhood of 0 D T .x0 /.Further, since ker T \ 'W D f0g, in these coordinates d.T '/ 0 .1;0; : : : ; 0/ ¤ 0: Now, since L n !L and g n .x0 / !T .x0 / in these coordinates, Since d.T '/ 0 .1;0; : : : ; 0/ ¤ 0 and t n !0, we see that the set fS n .x0 / W n 0g is infinite.
Finally, since S n j V !T 'T j V , this implies that is nondiscrete in P .End.V //.Lemma 3.9 Suppose x 2 @ is a conical limit point of , F D F .x/, V D Span F , and dim.V / D k.If k 2 and is Zariski dense in PGL d .R/, then fT j V W T 2 End F ;? g is Zariski dense in P .End.V //.
Proof Let Z 0 be the Zariski closure of fT j V W T 2 End F ;? g in P .End.V //.

Lemma 3.7 implies that End
F ;? is nonempty, so fix T 2 End F ;? .Then define So .T ı g/.V / D V , which implies that T ı g 2 End F ;? , and hence that g 2 Proof of Theorem 3.1 Parts (a) and (b) follow from Lemmas 3.3 and 3.4, respectively.Since acts cocompactly on , every point in @ is a conical limit point, and Theorem 2.3 implies that is Zariski dense in PGL d .R/.So part (c) follows from Lemmas 3.3, 3.8, and 3.9.

The main rigidity theorem
Recall that E @ denotes the set of extreme points of a properly convex domain .In this section we prove the following rigidity result: Theorem 4.1 Suppose that P .R d / is an irreducible properly convex divisible domain and there exists a boundary face F @ such that Then is symmetric with real rank at least two.
The rest of the section is devoted to the proof of the theorem, so suppose P .R d / satisfies the hypothesis of the theorem.Then let Ä Aut./ be a discrete group acting cocompactly on .
We assume, for a contradiction, that is not symmetric with real rank at least two.Lemma 4.2 It holds that is not symmetric.
Proof If were symmetric, then by assumption it would have real rank one.Then, by the characterization of symmetric convex divisible domains, coincides with the unit ball in some affine chart.Therefore E D @ , which is impossible since there exists a boundary face F @ such that Now we fix a boundary face F @ , where F ;? , then the map F ! P .V /; p 7 !T .p/; is in Aut.F /.
Proof Notice that T j V 2 PGL.V / since T .V / V and ker T \ V D f0g.So we just have to show that T .F / D F .Theorem 3.1(b) says that T .F / F , and so we just have to show that F T .F /.
Fix y 2 F .Since the set T .F /\F is closed in F , there exists x 0 2 T .F /\F such that Since T j V 2 PGL.V /, the set T .F / is open in F .So we either have y D x 0 2 T .F / or x 0 2 T .@F/.Suppose for a contradiction that x 0 2 T .@F/.Then let x 0 0 2 @F be the point where T .x 00 / D x 0 .Next, let F 0 @F be the face of Thus we can find z 2 F 0 and a sequence .
there exists an open line segment L in F which contains z and x 0 0 .Then T .L/ is an open line segment in F since T j V 2 PGL.V /.So, since T .x 00 / 2 F , we also have T .z/ 2 F , and since T 2 End F ;?
End ; there exists a sequence .gn / n 1 in such that g n !T in P .End.R d //.Now note that z … P .kerT / since ker T \ V D f0g.So by the "moreover" part of Observation 2.9, However, g n .zn / 2 E , and so Thus we have a contradiction.Hence y D x 0 2 T .F /, and since y 2 F was arbitrary we have F T .F /.
Proof This follows immediately from Lemma 4.3 and Theorem 3.1(c).
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 ¤ f0g since Aut.F / is closed and nondiscrete.Also Aut 0 .F / is normalized by Aut.F /, and so Ad.g/g D g for all g 2 Aut.F /.Then, since Aut.F / is Zariski dense in PGL.V /, we see that Ad.g/g D g for all g 2 PGL.V /.Since the representation AdW PGL.V / !GL.sl.V // is irreducible, we must have g D sl.V /.Thus Aut 0 .F / D PSL.V /.
Proof of Theorem 4.1 The previous lemma immediately implies a contradiction: fix x 2 F , then P .V / F Aut.F / x PSL.V / x D P .V /: So F D P .V /, which contradicts the fact that is properly convex.

Density of biproximal elements
In this section we prove a density result for the attracting and repelling fixed points of biproximal elements.To state the result we need one definition: if P .R d / is a properly convex domain and Ä Aut./, then the limit set of is

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.
Corollary 7.2 Suppose that P .R d / is an irreducible properly convex domain and Ä Aut./ is a discrete group that acts cocompactly on .If g 2 is biproximal , then the following are equivalent: (i) OE`C g ; ` g @ .(ii) s @ .`Cg ; ` g / < C1.
(iii) g has at least three fixed points in @ .
(iv) The cyclic group g Z has infinite index in its centralizer.
We will first recall some results established in [19], then prove the theorem and corollary.Cooper, Long and Tillmann [14] showed that the minimal translation length of an element can be determined from its eigenvalues: For the other inequality, fix > 0 and q 2 with H .g.q/; q/ < .g/C .Then H .g n .q/;q/ C 2H .p0 ; q/ n Ä lim sup n!1 H .g n .q/;g n 1 .q//C C H .g.q/; q/ C 2H .p0 ; q/ n D lim sup n!1 H .g.q/; q/ C 2H .p0 ; q/ n < .g/C : Since > 0 was arbitrary, the proof is complete.
Next, given a group G and an element g 2 G, let C G .g/ denote the centralizer of g in G. Then given a subset X G, define

Rank in the sense of Prasad and Raghunathan
In this section we consider the rank of a group in the sense of [24].Definition 8.1 (Prasad and Raghunathan) Suppose that is an abstract group.For i 0 let A i ./be the subset of elements whose centralizer contains a free abelian group of rank at most i as a subgroup of finite index.Next define r./ to be the minimal i 2 f0; 1; 2; : : : g [ f1g such that there exist  (iv) OEx 1 ; x 2 @ for every two extreme points x 1 ; x 2 2 @ .(v) s @ .x;y/ Ä 2 for all x; y 2 @ .(vi) s @ .x;y/ < C1 for all x; y 2 @ .
(vii) has higher rank in the sense of Prasad and Raghunathan.
(viii) For every g 2 with infinite order, the cyclic group g Z has infinite index in the centralizer C .g/ of g in .
(ix) Every g 2 with infinite order has at least three fixed points in @ .
(x) OE`C g ; ` g @ for every biproximal element g 2 .
(xii) There exists a boundary face F @ such that We verify all the implications shown in Figure 1 Proof These implications follow from direct inspection of the short list of irreducible symmetric properly convex domains.
Proof Fix a boundary face F @ of maximal dimension.We claim that E \ F D ∅: Otherwise, there exists x 2 F and a sequence x n 2 E such that x n !x 2 F .Now fix an extreme point y 2 @ n F .Then, by hypothesis, OEx n ; y @ for all n, so OEx; y @ .Proof By Theorem 7.3 every infinite-order element g 2 preserves a properly embedded simplex S with dim.S / 1. Hence g fixes the vertices v 1 ; : : : ; v k of S. By hypothesis s @ .v 1 ; v 2 / < C1 and hence, by Theorem 7.1, g Z has infinite index in the centralizer C .g/.Lemma 9.5 .x/D ) .iv/.
Proof We prove the contrapositive: if there exist extreme points x; y 2 @ with .x;y/ , then there exists a biproximal element g 2 with .`Cg ; ` g / .If such x and y exist, then by Theorem 5.1 there exist biproximal elements g n 2 with `C g n !x and ` g n !y.Then, for n large, we must have .`Cg n ; ` g n / .

Observation 3 . 2 1 g 1 g n .p/ 2 for all p 2 .Lemma 3 . 4 2 .
(a) If T 2 End , then P .kerT / \ D ∅.(b) If S; T 2 End and image T n ker S ¤ ∅, then S ı T 2 End .Proof Part (a) follows immediately from Proposition 2.12.For part (b), fix S; T 2 End with image T n ker S ¤ ∅.By hypothesis S ı T is a welldefined element of P .End.R d //.To show that S ı T 2 End , fix sequences .gn / n 1 and .hn / n 1 in such that S D lim n!n and T D lim n!1 h n in P .End.R d //.Then, since S ı T ¤ 0, S ı T D lim n!1 g n h n in P .End.R d //.So S ı T 2 End .Geometry & Topology, Volume 27 (2023) Lemma 3.3 If F @ is a boundary face and T 2 End F , then T ./ F .Proof Suppose T 2 End F .Then there exists a sequence .gn / n 1 in such that T D lim n!1 g n in P .End.R d //.Since P .kerT / \ D ∅, T .p/D lim n!So T ./ .Since image.T / V , T ./ P .V / \ D F : If F @ is a boundary face and T 2 End F ;? , then T .F / is an open subset of F .Proof By definition and Observation 3.[ F / \ P .kerT / .[ P .V // \ P .kerT / D ∅: So T induces a continuous map on [F .Since F , the previous lemma implies that T .F / T ./ F : Since V \ ker T D f0g, T .F / is an open subset of P .V /.So T .F / rel-int.F / D F: Lemma 3.5 If F @ is a boundary face, then the set

Fix z 2
.x; y/ @ and let C denote the convex hull of y and F .By Observation 2.11,@F .z/rel-int.C /:Then dim F .z/ > dim F;which is a contradiction.So we must have E \ F D ∅, and hence .xii/holds.Lemma 9.4 .vi/D ) .viii/.
class on the moduli space of curves PAUL NORBURY 2763 The 2-primary Hurewicz image of tmf MARK BEHRENS, MARK MAHOWALD and J D QUIGLEY 2833 Hamiltonian no-torsion MARCELO S ATALLAH and EGOR SHELUKHIN 2899 A higher-rank rigidity theorem for convex real projective manifolds ANDREW ZIMMER denote the absolute values of the eigenvalues of some (hence any) lift of g to SLḋ .R/ WD fh 2 GL d .R/ W det h D ˙1g.g 2 PGL d .R/ is proximal if 1 .g/> 2 .g/.In this case, let `C g 2 P .R d / denote the eigenline of g corresponding to 1 .g/.
g 2 PGL d .R/ is biproximal if g and g 1 are both proximal.In this case, define Lemma 2.2], we consider a compactification of a subgroup of PGL d .R/: Definition 1.9 Given a subgroup G Ä PGL d .R/ let G End P .End.R d // denote the closure of G in P .End.R d //.
let OE denote the image of in PGL d .R/. (iii) If T 2 End.R d / n f0g, let OET denote the image of T in P .End.R d //.
If x 2 @ and F .x/ D fxg, then x is called an extreme point of .Finally, let Definition 2.10 Suppose that P .R d / is a properly convex domain.For x 2 let F .x/ denote the (open) face of x; that is, F .x/ D fxg [ fy 2 W 9 an open line segment in containing x and yg: Geometry & Topology, Volume 27 (2023) (i) If x 2 , then F .x/ D .(ii) F .x/ is open in its span.(iii) y 2 F .x/ if and only if x 2 F .y/ if and only if F .x/ D F .y/.
13 [19, Proposition 5.7] Suppose that P .R d / is a properly convex domain, p 0 2 , x 2 @ , .pn / n 1 is a sequence in OEp 0 ; x/ converging to x, and .gn / n 1 is a sequence in Aut./ such that If g n converges in P .End.R d // to T 2 P .End.R d //, then T ./ D F .x/; and hence image T D Span F .x/.Geometry & Topology, Volume 27 (2023) Definition 2.16 If g 2 PGL d .R/ is proximal, then define H g 2 Gr d 1 .R PGL d .R/ is proximal, H g is usually called the repelling hyperplane of g.This is motivated by the following observation: Observation 2.17 If g 2 PGL d .R/ is proximal , then exists in P .End.R d //.Moreover, image T g D `C g , ker T g D H g , and image T g ˚ker T g D R d : d / to be the unique g-invariant linear hyperplane with`C g ˚H g D R d : If g is biproximal, then also define H C g WD H g 1 .When g 2 d / n P .H g /.Observation 2.18 Suppose P .R d / is a properly convex domain.If g 2 Aut.
Proposition 2.19 Suppose that .gn/n 1 is a sequence in PGL d .R/ and n .x/DT.x/DimageT 2 P .R d / for all x 2 P .R d / n P .kerT/.Moreover, the convergence is uniform on compact subsets of P .R d / n P .kerT/.so we can find a compact neighborhood U of image T in P .R d / such that U is homeomorphic to a closed ball andU \ P .kerT/ D ∅: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 2 U .Since U can be chosen arbitrarily small, image T D lim n exists in P .End.R d //.If dim.image T / D 1 and image T ˚ker T D R d ; then, for n sufficiently large, g n is proximal and image T D lim n!1 `C g n : Geometry & Topology, Volume 27 (2023) Proof Since g n !T in P .End.R d //, lim n!1 g n!1 [27,only if there exist p 2 and a sequence .n/n 1 in such that n .p/!x.Suppose that P .R d / is a properly convex domain and Ä Aut./ is a strongly irreducible group.If x; y 2 L ./ are extreme points of and .x;y/,thenthereexists a sequence of biproximal elements .gn/n 1 in such thatlim n!1 `C g n D xand limProof A result of Vey[27, Theorem 5]implies that is strongly irreducible and Proposition 2.13 implies that @ D L ./, so Theorem 5.1 implies the corollary.Proof of Theorem 5.1 By definition there exist p 2 and a sequence .n / n 1 in such that n .p/!x.Passing to a subsequence, we can suppose the limits exist in P .End.R d //.By Proposition 2.12 image T C Span F .x/ D Spanfxg D x; and so image T C D x. Proposition 2.12 also implies that P .kerT / \ D ∅ and x 2 P .kerT /.Notice that y … P .kerT / since .x;y/ .Similarly, we can find a sequence .n / n 1 in such that the limits exist in P .End.R d //, image S C D y, and x … P .kerS /.Fix some x 0 2 image T and y 0 2 image S .Since is strongly irreducible, by Observation 2.2 there exists h 2 such that: (i) h.y 0 / … P .kerT C /; hence, h.image S / š ker T C .(ii) hS .x/… P .kerT C /. (iii) h.x 0 / … P .kerS C /; hence, h.image T / š ker S C .(iv) hT .y/… P .kerS C /. in P .End.R d //.Notice that image.T C ı h ı S / D image T C D x and, by our choice of h, x … P .ker.T C ı h ı S //: So image.T C ı h ı S / C ker.T C ı h ı S / D x C ker.T C ı h ı S / D R d ; and hence, by Proposition 2.19, g n is proximal for n sufficiently large and `C g n !x.
7.1 Maximal abelian subgroups and minimal translation sets Theorem 7.3 (Islam and Zimmer [19, Theorem 1.6]) Suppose that P .R d / is a properly convex domain and Ä Aut./ is a discrete group that acts cocompactly on .If A Ä is a maximal abelian subgroup of then there exists a properly The above result is a special case of [19, Theorem 1.6], which holds in the more general case when Ä Aut./ is a naive convex cocompact subgroup.Suppose that P .R d / is a properly convex domain and g 2 Aut./.Define the minimal translation length of g to be denote the absolute values of the eigenvalues of some (and hence any) lift of g to SL ḋ .R/ WD fh 2 GL d .R/ W det h D ˙1g.If P .R d / is a properly convex domain, p 0 2 , and g 2 Aut./, then lim Proposition 7.6 [14, Proposition 2.1] If P .R d / is a properly convex domain and g 2 Aut./, then .g/D 1 2 log 1 .g/d .g/: Geometry & Topology, Volume 27 (2023) Fix .pn / n 1 in S 0 converging to y.Since .vC ; v / and OEv C ; y [ OEy; v @ , Observation 2.11 implies that the faces F .v C /, F .v /, and F .y/ are all distinct.Then, by the definition of the Hilbert metric, Thus g Z has infinite index in C .g/ and so (iii) is true.
a2A Min .a/ is nonempty and C .A/ acts cocompactly on the convex hull of Min .A/ in .Geometry & Topology, Volume 27 (2023) Then the Prasad-Raghunathan rank of is defined to be rank PR ./WDsupfr./W is a finite-index subgroup of g: Prasad and Raghunathan computed the rank of lattices in semisimple Lie groups, which implies: Theorem 8.2 [24, Theorem 3.9] Suppose that P .R d / is an irreducible properly convex domain.If is symmetric with real rank r and Ä Aut./ is a discrete group acting cocompactly on , then rank PR ./Dr.As a corollary to Selberg's lemma we get a lower bound on the Prasad-Raghunathan rank: Corollary 8.3 If Ä PGL d .R/ is a finitely generated infinite group, rank PR ./ 1.
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