Balanced ideals and domains of discontinuity of Anosov representations

We consider the action of Anosov subgroups of a semi-simple Lie group on the associated flag manifolds. A systematic approach to construct cocompact domains of discontinuity for this action was given by Kapovich, Leeb and Porti in arXiv:1306.3837. For $\Delta$-Anosov representations, we prove that every cocompact domain of discontinuity arises from this construction, up to a few exceptions in low rank. Then we compute which flag manifolds admit these domains and, in some cases, the number of domains. We also find a new compactification for locally symmetric spaces arising from maximal representations into $\mathrm{Sp}(4n+2, \mathbb{R})$.


Introduction
Let Γ be a word hyperbolic group and let G be a semi-simple Lie group.A particularly well-behaved subset of the representations Hom(Γ, G) are the Anosov representations.For instance, they have a discrete image and a finite kernel and they form an open subset of Hom(Γ, G).A definition can be found in Section 2. Examples of Anosov representations include all discrete injective representations into SL(2, R), quasi-Fuchsian representations into SL(2, C), representations in the Hitchin component of Hom(π 1 S, SL(n, R)) for a closed surface S, and maximal representations from such a group π 1 S into a Hermitian Lie group.
We want to study the action of an Anosov representation ρ on a flag manifold associated to G, that is a homogeneous space F = G/P where P ⊂ G is a parabolic subgroup.A special case is the full flag manifold F ∆ = G/B, with B being the minimal parabolic subgroup.If G = SL(n, R) the elements of flag manifolds are identified with sequences of nested subspaces of fixed dimensions in R n .The ρ-action on G/P is generally not proper, but in [GW12] Guichard and Wienhard described a way of removing a "bad set" from a suitable flag manifold such that ρ acts properly discontinuously and cocompactly on the complement.In other words, they constructed cocompact domains of discontinuity: Definition 1.1.A domain of discontinuity Ω ⊂ F for ρ is a ρ(Γ)-invariant open subset such that the action Γ ρ Ω is proper.It is called cocompact if the quotient Γ\Ω is compact.
Note that we require domains of discontinuity to be open subsets.In contrast, Danciger, Gueritaud and Kassel [DGK17; DGK18] and Zimmer [Zim17] recently proved that the Anosov property is equivalent to the existence of certain cocompact domains in RP n or H p,q .These domains are closed subsets.
A systematic construction of (open) domains of discontinuity for Anosov representations was given by Kapovich, Leeb and Porti in [KLP18].Say we have an Anosov representation ρ.It comes with a ρ-equivariant limit map ξ : ∂ ∞ Γ → F from the boundary of Γ into some flag manifold F. We want to find a cocompact domain of discontinuity in a flag manifold F , which may be different from F. To construct such domains, Kapovich, Leeb and Porti use a combinatorial object called a balanced ideal.That is a subset I of the finite set G \ (F × F ) satisfying certain conditions (see Section 2).They prove that, for every balanced ideal I, the set In this paper, we prove a type of converse to this statement in the case of ∆-Anosov representations.∆-Anosov, or Anosov with respect to the minimal parabolic, is the strongest form of the Anosov property, and far more restrictive than general Anosov representations.
Theorem 1.2.Let ρ : Γ → G be a ∆-Anosov representation and Ω ⊂ F a cocompact domain of discontinuity for ρ in some flag manifold F. Then there is a balanced ideal I ⊂ G\(F ∆ × F ∆ ) such that the lift of Ω to the full flag manifold F ∆ is a union of connected components of Ω ρ,I .
We can say more if the dimension of the bad set is not too big.To calculate it, we associate to a semi-simple Lie group G a number mbic(G) which gives a lower bound on the codimension of the set we have to remove for every limit point x ∈ ∂ ∞ Γ.It increases with the rank of G, for example mbic(SL(n, R)) = (n + 1)/2 .The general definition is given in Section 3.5.With this we get Theorem 1.3.Let ρ : Γ → G be a ∆-Anosov representation and assume that dim ∂ ∞ Γ ≤ mbic(G) − 2. Then there is a 1:1 correspondence of balanced ideals in G\(F ∆ × F) and non-empty cocompact domains of discontinuity in F.
Remark 1.4.A key point for these theorems is that cocompact domains are maximal among all domains of discontinuity, at least if they are connected.We then establish a correspondence between minimal fat ideals and maximal domains of discontinuity, even if they are not cocompact.This approach only works for ∆-Anosov representations: Section 3.6 shows an example of an Anosov, but not ∆-Anosov representation which admits infinitely many maximal domains of discontinuity.They are not cocompact.
Remark 1.5.For some choices of flag manifold F there are no balanced ideals in G\(F ∆ × F) and therefore no cocompact domain of discontinuity in F. But even in these cases one can sometimes find a cocompact domain of discontinuity in a finite cover F of F, an oriented flag manifold.In [ST18], a theory analogous to [KLP18] is developed for oriented flag manifolds.Many arguments of this paper can potentially be extended to this setting, to show that essentially all cocompact domains in oriented flag manifolds are of the type described in [ST18].
• representations in a Hitchin component of Hom(π 1 S, SL(n, R)) for a closed surface S.These are the connected components containing the composition of discrete injective representations π 1 S → SL(2, R) with the irreducible embedding SL(2, R) → SL(n, R).
• the composition of a Fuchsian representation with the reducible embedding of SL(2, R) into SL(3, R) and small deformations thereof.These were studied in [Bar10].A similar construction works for all SL(n, R) with odd n.
Next we combine Theorem 1.3 with a criterion for the existence of balanced ideals in the case G = SL(n, R) or G = SL(n, C).This gives a full description which flag manifolds admit cocompact domains of discontinuity.
Theorem 1.6.Let Γ be a hyperbolic group, K ∈ {R, C}, and ρ : Γ → SL(n, K) a ∆-Anosov representation.Choose integers i 0 , . . ., i k+1 with Denote by F the corresponding flag manifold In particular, for surface group representations into SL(n, R) acting on Grassmannians we get Corollary 1.7.Let n ≥ 5 and let ρ : π 1 S → SL(n, R) be a ∆-Anosov representation from the fundamental group of a surface S with or without boundary.Then the induced action π 1 S ρ Gr(k, n) on the Grassmannian of k-planes in R n admits a non-empty cocompact domain of discontinuity if and only if n is odd and k is even.For small n, the number of different such domains is Remark 1.8.While we don't know a general formula for these numbers, they are just the numbers of different balanced ideals.This is a combinatorial problem and needs no information about the representation except that it is ∆-Anosov.So we can use a computer program to enumerate all balanced ideals.Appendix A shows more results from this enumeration.
In low ranks, e.g.SL(n, R) with n ≤ 4 if Γ is a surface group, the existence of cocompact domains of discontinuity depends on more information about the geometry of ρ.So we can't make general lists like above in these cases.But at least for Hitchin representations the cocompact domains of discontinuity are also known in low ranks.We will briefly discuss this in Section 4.2.
Another observation from studying the list of balanced ideals is the existence of a balanced ideal in Sp(2n, C)\(Lag(C 2n )×Lag(C 2n )) whenever n is odd.This allows us to construct a compactification for locally symmetric spaces associated to {α n }-Anosov representations, which is modeled on the bounded symmetric domain compactification of the symmetric space.In particular, this includes maximal representations.
Recall that the symmetric space X = Sp(2n, R)/ U(n), like any Hermitian symmetric space, can be realized as a bounded symmetric domain D ⊂ C n(n+1)/2 [Hel79, Theorem VIII.7.1].Concretely, we can take as D the set of symmetric complex matrices Z such that 1 − ZZ is positive definite.Its closure D in C n(n+1)/2 is the bounded symmetric domain compactification of X.
Then there exists a subset D ⊂ C n(n+1

Flag manifolds and Anosov representations
In this section we fix some notation and conventions.In particular, we define Anosov representations and the relative position of flags.
Let G be a connected semi-simple Lie group with finite center and g its Lie algebra.Choose a maximal compact subgroup K ⊂ G with Lie algebra k ⊂ g and let p = k ⊥ be the Killing orthogonal complement in g.Further choose a maximal subspace a ⊂ p on which the Lie bracket vanishes.Denote by the set of restricted roots.Also choose a simple system ∆ ⊂ Σ and let Σ ± ⊂ Σ be the corresponding positive and negative roots.Note that any choice of the triple (K, a, ∆) is equivalent by conjugation in G (see [Hel79,Theorem 2.1] and [Kna02, Theorems 2.63, 6.51, 6.57]).
The Weyl group of G is the group W = N K (a)/Z K (a).It can be viewed as a group of linear isometries of a equipped with the Killing form.A natural generating set of W is given by ∆, identifying every α ∈ ∆ with the orthogonal reflection along ker α.
As (W, ∆) is a finite Coxeter system there is a unique longest element w 0 ∈ W , which squares to the identity.The opposition involution is the map ι(w) = w 0 ww 0 on W .It restricts to an involution ι : ∆ → ∆ of the simple roots.
Define for ∅ = θ ⊂ ∆ the Lie subalgebras The flag manifolds of G are the spaces F θ = G/P θ for non-empty subsets θ ⊂ ∆.For two such subsets θ, η ⊂ ∆ the set of relative positions is W θ,η = ∆\θ \W/ ∆\η , where A denotes the subgroup generated by A ⊂ W .The relative position map is the unique map pos θ,η : which is invariant by the diagonal action of G and satisfies pos θ,η ([1], [w]) = w for all w ∈ N K (a).We will often omit the subscripts θ, η.Two flags f ∈ F θ and f ∈ F η are said to be transverse if pos θ,η (f, f ) = w 0 .
For any x ∈ F θ and w ∈ W θ,η the space The Bruhat order ≤ on W θ,η is the inclusion order on the closures of Bruhat cells, i.e. for all w, w ∈ W θ,η and any (and therefore all) Now assume that ι(θ) = θ.Then w 0 acts on W θ,η by left-multiplication.
be the Cartan projection defined by the KAK-decomposition.That is, for every element g ∈ G there are k, ∈ K and a unique µ(g) ∈ a + = {X ∈ a | α(X) ≥ 0 ∀α ∈ ∆} such that g = k e µ(g) .k and are uniquely defined up to an element in the centralizer of µ(g).
Definition 2.2.Let θ ⊂ ∆ be non-empty and ι(θ where | • | is the word length in Γ with respect to any finite generating set.This is only one out of many equivalent definitions of Anosov representations [Lab06; GW12; KLP14b; KLP14a; GGKW17; BPS16].See [KLP17, Theorem 1.1] for an overview and [KLP17; BPS16] for proofs of the equivalences. The main fact we need about Anosov representations (which is part of many definitions) is that a θ-Anosov representation ρ : Γ → G admits a unique limit map which is continuous, ρ-equivariant, transverse (meaning that ξ(x) and ξ(y) are transverse for all x = y) and which maps the attracting fixed point of every infinite order element γ ∈ Γ to the attracting fixed point of ρ(γ).

Domains of discontinuity
In this section, we prove the main results, Theorem 1.2 (which is Theorem 3.18) and Theorem 1.3 (as Theorem 3.24), after some lemmas.

Divergent sequences
We first consider the behaviour of divergent sequences (g n ) ∈ G N in the semi-simple Lie group G.As before, let θ, η ⊂ ∆ be non-empty subsets of the simple restricted roots and assume ι(θ) = θ.
∈ F θ are the repelling and attracting limits of this sequence.
Remark 3.3.By compactness of K, every θ-divergent sequence in G has a simply θdivergent subsequence.The limits (g − , g + ) do not depend on the choice of decomposition.
In [KLP18], the following characterization of θ-divergent sequences is used.
locally uniformly as functions from F θ to F θ (where g + is the constant function).
Proof.By assumption, we can write All of the roots α appearing in this sum are linear combinations of simple roots with only non-positive coefficients, and with at least one coefficient of a root in θ being negative.So α(A n ) → −∞ for all such roots α, and therefore For the action of a discrete group Γ on a manifold X, there is a useful reformulation of properness.By [Fra05, Proposition 1], the action is proper if and only if it has no dynamical relations, in the following sense: Definition 3.5.Let Γ be a discrete group acting smoothly on a manifold X.Let (γ n ) ∈ Γ N be a divergent sequence (i.e.no element occurs infinitely many times) and x, y ∈ X.
Then x is dynamically related to y via ∼ y, if there is a sequence (x n ) ∈ X N such that x n → x and γ n x n → y.
We say x and y are dynamically related, x ∼ y, if they are dynamically related via any divergent sequence in Γ.
Lemma 3.6 is the key step of the proof of proper discontinuity in [KLP18] (Proposition 6.5).It states that flags in F η can only be dynamically related by the action of ρ if their relative positions satisfy the inequality (1).Lemma 3.8 is a converse to this statement in the case θ = ∆: It says that whenever two flags f, f satisfy a relation like (1), then they are indeed dynamically related.
Lemma 3.6.Let (g n ) ∈ G N be a simply θ-divergent sequence and let (g − , g + ) ∈ F 2 θ be its limits.Let f, f ∈ F η be dynamically related via (g n ).Then Proof.As f, f are dynamically related, there exists a sequence We get the following relative positions: Now h n F → F and since F and almost all of the h n F are in C w 0 (g − ), we get by Lemma 3.4 that g n h n F → g + .So Proof.We can write n − = e X − for X − ∈ n − and n + = e X + with X + ∈ n.Let H n = X − + e − ad An X + and h n = e Hn .For all α ∈ Σ + and X α ∈ g α we know that e − ad An X α = e −α(An) X α converges to 0. As X + is a linear combination of these, H n → X − and thus h n → n − .On the other hand e An e Hn e −An = exp(Ad e An H n ) = exp(e ad An H n ) = exp(e ad An X − + X + ) which converges to n + = e X + by a similar argument.
Lemma 3.8.Let (g n ) ∈ G N be a simply ∆-divergent sequence with limits (g − , g Then f is dynamically related to f via (g n ).
Proof.Fix some representative in N K (a) for w and w 0 .Let g n = k n e An n be a KAKdecomposition, such that (k n ), ( n ) ∈ K N converge to k, ∈ K and α(A n ) → ∞ for all α ∈ ∆.Then the limits can be written as Because of (2) there exist h, h ∈ G with since elements of A and Z K (a) commute with Weyl group elements.By Lemma 3.7 there is a sequence

Limit sets
Now let Γ be a non-elementary hyperbolic group and ρ : Γ → G a representation.
Definition 3.9.Let θ ⊂ ∆ be non-empty and ι-invariant.The limit set of ρ is the set and the set of limit pairs is These limit sets are particularly well-behaved for Anosov representations.Namely, we have the following well-known facts: The first part can be found e.g. in [GGKW17, Theorem 5.3(3)].We give a detailed proof of the second part.We first need two short lemmas.
Lemma 3.11.Let ρ : Γ → G be a θ-Anosov representation with limit map ξ : locally uniformly.Then on F θ we also have the locally uniform convergence Proof.By restricting to a subsequence we can assume that ρ(γ n ) is simply θ-divergent with limits (g − , g Finally, since any subsequence of (γ n ) has a subsequence satisfying (3), it actually holds for the whole sequence (γ n ).
Proof.Fix a metric on ∂ ∞ Γ.Let P ⊂ ∂ ∞ Γ 2 be the set of fixed point pairs of infinite order elements of Substituting each γ n by a sufficiently high power, we can assume that γ n maps the complement of a divergent sequence satisfying (4), which exists by Lemma 3.12.By Lemma 3.11 this implies Let ρ(γ n k ) be any simple subsequence of the θ-divergent sequence (ρ(γ n )) and (g − , g + ) ∈ F 2 θ its limits.Then by Lemma 3.4 , and repeating the argument for this sequence shows that ρ,θ .

Maximal domains of discontinuity
Recall that we call Ω ⊂ F η a domain of discontinuity if it is an open Γ-invariant subset on which Γ acts properly.In this section, we deal with maximal domains of discontinuity, i.e. those which are not contained in any strictly larger domain of discontinuity.Definition 3.13.Let Ω ⊂ F η , Λ ⊂ F θ , and I ⊂ W θ,η .We define Proof.We first prove that I is an ideal.If not, there are w ≤ w with w ∈ I and w ∈ I.So there exist ∈ Λ and x ∈ Ω such that pos( , x) = w , i.e. x ∈ C w ( ).But C w ( ) ⊂ F η \ Ω which is closed, so x ∈ C w ( ) ⊂ C w ( ) ⊂ F η \ Ω, a contradiction.So I is an ideal.
Corollary 3.16.Every maximal domain of discontinuity of a ∆-Anosov representation with limit set Λ is of the form Ω(Λ, I) for a minimal fat ideal I ⊂ W ∆,η .
Proof.Let Ω be a maximal domain of discontinuity.Then by Proposition 3. 15 I(Λ, Ω) is a fat ideal and Ω = Ω(Λ, I(Λ, Ω)).In general, there could be other ideals generating the same domain.Let I be minimal among all fat ideals I with Ω(Λ, I) = Ω.Then I is in fact minimal among all fat ideals, as otherwise there would be another fat ideal I with Ω = Ω(Λ, I) Ω(Λ, I ), contradicting maximality of Ω.

Cocompactness
The most important fact we need about cocompact domains of discontinuity is that they are essentially maximal.More precisely: Lemma 3.17.Let ρ : Γ → G be a representation and Ω ⊂ F η a cocompact domain of discontinuity.Then Ω is a union of connected components of a maximal domain of discontinuity.
Proof.By Zorn's lemma Ω is contained in some maximal domain of discontinuity Ω ∈ F η and it is an open subset.Then also Γ\Ω ⊂ Γ\ Ω, where Γ\Ω is compact and Γ\ Ω is Hausdorff.So Γ\Ω is closed in Γ\ Ω and therefore Ω is also closed in Ω.
This immediately leads to our first main theorem.
Theorem 3.18.Let ρ : Γ → G be a ∆-Anosov representation with limit map ξ : ∂ ∞ Γ → F ∆ and let Ω ⊂ F η be a cocompact domain of discontinuity for ρ.Then there is a balanced ideal Proof.The natural projection π η : F ∆ → F η is smooth, G-equivariant and proper.This implies that Ω = π −1 η (Ω) is also a cocompact domain of discontinuity.So by Lemma 3.17 there is a maximal domain of discontinuity Ω ⊂ F ∆ and Ω is a union of connected components of Ω.By Corollary 3.16 Ω = Ω(Λ, I) for a minimal fat ideal I ⊂ W .But since the action of w 0 on W has no fixed points, every minimal fat ideal in W is balanced.This is proved e.g. in [ST18, Lemma 3.34].
We know from [KLP18] that domains constructed from a balanced ideal are cocompact.The combination of the next two lemmas shows that if the domain is dense, the converse also holds.That is, if a domain constructed from a fat ideal is cocompact, then this ideal must be balanced.
Then Ω 0 can be cocompact only if Proof.Assume that Ω 0 is cocompact and x ∈ ∂Ω 0 .Take a sequence (x n ) ∈ Ω N 0 with x n → x.Let (h n ) ∈ G N be a sequence converging to the identity such that x n = h n x.By cocompactness, a subsequence of (x n ) converges in the quotient.Passing to this subsequence, there is (g n ) ∈ ρ(Γ) N such that g n x n → x ∈ Ω 0 .Clearly g n → ∞ as otherwise a subsequence of (g n x n ) would converge to something in ∂Ω 0 .Passing to a subsequence another time we can also assume that (g n ) is simply θ-divergent with limits (g − , g + ) ∈ Λ 2 .Now let ∈ Λ \ {g − }.Then h n → and thus g n h n → g + by Lemma 3.4 since ∈ C w 0 (g − ) and this is an open set.So pos(g + , x ) ≤ pos(g n h n , g n h n x) = pos( , x).
Lemma 3.20.In the setting of Lemma 3.19 an ideal I ⊂ W θ,η is slim if and only if D(Λ, I) = ∅.
Proof.First assume that I is not slim, i.e. there is w ∈ I with w 0 w ∈ I. Let = ∈ Λ.Since , are transverse there is g ∈ G such that g = [1] and g and since [a −1 k na k w] → [w] we get pos( , x) = pos([w 0 ], [nw]) ≥ w 0 w and thus w 0 w ∈ I.But also w = pos([1], [nw]) = pos( , x) ∈ I, so I is not slim.

Dimensions
If the domain Ω comes from a balanced ideal, the "bad set" F η \ Ω fibers over ∂ ∞ Γ.The dimension of the fiber is bounded by the following quantity, depending only on G: Definition 3.21.For a subset A ⊂ Σ of the simple roots let The we can define the minimal balanced ideal codimension of G mbic(G) = min Dumas and Sanders showed in [DS17, Theorem 4.1] that if the Weyl group W of G has no factors of type A 1 , then w 0 w ≤ w for all w ∈ W with (w) ≤ 1, and that the same is true for (w) ≤ 2 if W also has no factors of type A 2 , A 3 or B 2 .This implies mbic(G) ≥ 2 resp.mbic(G) ≥ 3 in these cases (and even higher lower bounds if the root spaces are more than one-dimensional, e.g. in the case of complex groups).
Example 3.22.For the special linear group we have To see this, recall that the Weyl group of SL(n, R) can be identified with the symmetric group S n with its standard generating set of adjacent transpositions.There is also a simple description of the Bruhat order on S n : Define, for any permutation w ∈ S n and integers i, j w[i, j] := |{a ≤ i | w(a) ≤ j}|.
Then w ≤ w if and only if for all i, j.Since every root space g α is 1-dimensional and |Ψ w | = (w), mbic(G) is therefore the minimal word length an element w ∈ S n has to have such that there are i, j not satisfying this inequality.We can express this problem in a nice graphical way: Suppose we have n balls in a row which we can permute.What is the minimal length of a permutation such that for some choice of i and j, if the first i balls were painted red before, then after the permutation there are more red ones among the last j than among the first j?
The solution of this elementary combinatorial problem can be seen in the right picture.At least (n + 1)/2 adjacent transpositions are needed, and the minimum is obtained e.g. by choosing i = 1 and j = n/2 .The argument for SL(n, C) is the same except that the root spaces g α are 2-dimensional.
Similarly to the nonemptiness proof in [GW12, Theorem 9.1], a bound on the dimension of the limit set can ensure that the domain is dense or connected.
Lemma 3.23.Let ρ : Γ → G be a ∆-Anosov representation with limit map ξ : Proof.We can assume that η = ∆ in parts (ii) and (iii) as otherwise we could just lift to F ∆ .So let I ⊂ W be a balanced ideal.We will calculate the covering dimension of Since I is balanced, K is a continuous fiber bundle over ∂ ∞ Γ with fiber w∈I C w ([1]) (see [ST18,Lemma B.8] for details).Since the dimension can be calculated in local trivializations and the fiber is a CW-complex, To bound the latter dimension, we use that dim C w ([1]) = dim Ψ w and that all w ∈ I satisfy w 0 w ≤ w.Furthermore, it is easy to check that dim Ψ w = dim F ∆ − dim Ψ w 0 w for every w ∈ W .So we get the estimate (5) So Ω must be dense, as otherwise K would contain an open subset and therefore dim K = dim F ∆ .This proves (ii).
For part (iii), we can use Alexander duality [Hat10, Theorem 3.44]: For a compact set K of a closed manifold M , there is an isomorphism and every Čech cohomology group above the covering dimension vanishes, we have So by the long exact sequence of the pair (F ∆ , Ω) there is an isomorphsim H 0 (Ω; Z) ∼ = H 0 (F ∆ ; Z), i.e.Ω is connected.
Finally, for part (i), we just need a balanced ideal which gives equality in (5).This always exists: Take w ∈ W such that w 0 w ≤ w and which realizes the maximum.Then the ideal generated by w is slim and can therefore be extended to a balanced ideal I by [ST18, Lemma 3.34] with max w∈I dim Ψ w = dim Ψ w = max w 0 w ≤w dim Ψ w .The corresponding K then satisfies dim Theorem 3.24.Let ρ : Γ → G be a ∆-Anosov representation with limit map ξ : Then every non-empty cocompact domain of discontinuity in F η is dense and connected and there is a bijection given by I → Ω(Λ, I) and Ω → I(Λ, Ω).
We have to prove that both maps are well-defined and inverses of each other.If Ω is a non-empty cocompact domain of discontinuity, then it is a union of connected components of some maximal domain Ω ⊂ F η by Lemma 3.17.Since Ω is dense it equals Ω and is maximal itself.So by Proposition 3.15 I(Λ, Ω) is a fat ideal and Ω = Ω(Λ, I(Λ, Ω)).
Conversely, if I ⊂ W ∆,η is a balanced ideal, then Ω(Λ, I) is a cocompact domain of discontinuity by the main theorem of [KLP18].It is dense and thus non-empty by Lemma 3.23(ii).By the above, I(Λ, Ω(Λ, I)) is then a balanced ideal, and since I ⊂ I(Λ, Ω(Λ, I)), they must be equal.

A representation into Sp(4, R) with infinitely many maximal domains
In this section, we describe an example of a representation (of a free group into Sp(4, R)) which is Anosov (but not ∆-Anosov) and where the analogue of Corollary 3.16 does not hold, i.e. there are maximal domains of discontinuity which do not come from a balanced ideal.In fact, it will have infinitely many maximal domains of discontinuity, which are however not cocompact.It is unclear whether the cocompact domains for general Anosov representations can still be classified using balanced ideals.
Let Γ = F m be a free group in m generators and ρ 0 : Γ → SL(2, R) the holonomy of a compact hyperbolic surface with boundary.Such a representation is Anosov with a limit map ξ 0 : with 1 being the 2 × 2 identity matrix.Here we chose the symplectic form ω = 0 1 −1 0 .Then ρ is {α 2 }-Anosov (where α 2 is the simple root mapping a diagonal matrix to twice its lowest positive eigenvalue), but not ∆-Anosov.Therefore, it carries a limit map ξ : ∂ ∞ Γ → Lag(R 4 ) to the manifold of Lagrangian subspaces.
Conversely, let y ∈ Λ 0 and p, q ∈ RP 1 .Since |Λ 0 | ≥ 3 we find x ∈ Λ 0 \ {p, q}, and by Lemma 3.12 there is a sequence (g n ) ∈ ρ 0 (Γ) N which is simply divergent with limits (x, y).Then p (gn) ∼ y and q (gn) ∼ y, which proves the first relation in (6).For the second relation let x, y ∈ Λ 0 and p, q, r ∈ RP 1 .If x = p or y = q then it follows from the first relation.Otherwise, take a simply divergent sequence (g n ) ∈ ρ 0 (Γ) N with limits (x, y).

Then p (gn)
∼ y and q ∼ x.This shows the second relation in (6).
Proposition 3.26.Let A ⊂ RP 1 be a minimal closed subset such that A ∪ RA = RP 1 . Then is a maximal domain of discontinuity for ρ.
Proof.Ω A is open since (RP 1 ) 3 is compact and Θ therefore is a closed map.
Assume that there was a dynamical relation within Ω A .Then following (6) it would either be of the form Θ(p, q, r) ∼ Θ(x, x, r) or Θ(p, x, r) ∼ Θ(y, q, r) with x, y ∈ Λ 0 and p, q, r ∈ RP 1 .In the first case, Θ(x, x, r) is independent of r, so we can assume r ∈ A, and Θ(x, x, r) can thus not be in Ω A .In the second case, Θ(p, x, r) = Θ(x, p, Rr) can be in Ω A only if r ∈ RA and Θ(y, q, r) ∈ Ω A implies r ∈ A. But by assumption both can not hold at the same time.
Then A A and A is closed.Since A is minimal among closed sets with A∪RA = RP 1 , there has to exist some r ∈ RP 1 \(A ∪RA ).But since r, Rr ∈ A then there are x, y ∈ Λ 0 and p, q ∈ RP 1 with Θ(y, q, r), Θ(x, p, Rr) ∈ Ω.But these are dynamically related by Lemma 3.25, a contradiction.
Through the accidental isomorphism PSp(4, R) ∼ = SO 0 (2, 3) the space Lag(R 4 ) can be identified with the space of isotropic lines in R 2,3 .The form of signature (2, 3) restricts to a Lorentzian metric on this space, which is why it is also called the (2 + 1) Einstein universe.A detailed explanation of its geometry can be found in [ Bar+08,Section 5].
We can use this to visualize Θ and the construction of Ω A above: The limit set ξ(∂ ∞ Γ) ⊂ Lag(R 4 ) is a Cantor set on the line {Θ(x, x, * ) | x ∈ RP 1 }.If we take two different points on this line, described by x, y ∈ RP 1 , their light cones intersect in the circle {Θ(x, y, r) | r ∈ RP 1 }, where r acts as a global angle coordinate.If x, y ∈ Λ 0 then every point on the future pointing light ray emanating from x in a direction r is dynamically related to every point on the past pointing light ray from y in direction r (the red and blue lines in Figure 1).So by choosing the set A ⊂ RP 1 , we decide for every angle whether to take out from our domain all the future or all the past pointing light rays emanating from the points in the limit set in this direction.4. Representations into SL(n, R) or SL(n, C)

Balanced ideals
The question for which η ⊂ ∆ there exists a balanced ideal in W ∆,η is only combinatorial.For G = SL(n, K) with K ∈ {R, C} the answer is given by the following proposition.Theorem 1.6 and its corollaries then immediately follow using Theorem 3.24.
A maximal compact subgroup of SL(n, R) is K = SO(n) and for G = SL(n, C) we can choose K = SU(n).In either case, a maximal abelian subalgebra a of so(n) ⊥ resp.su(n) ⊥ are the traceless real diagonal matrices, and a simple system of restricted roots is given by {α i = λ i − λ i+1 }, where λ i : a → R maps to the i-th diagonal entry.
Proposition 4.1.Let η = {α i 1 , . . ., α i k } ⊂ ∆ be a subset of the simple roots of SL(n, K), ).This means that ww 0 w −1 ∈ ∆\η for any w ∈ W .The Weyl group of SL(n, K) can be identified with the symmetric group S n with its generators ∆ being the adjacent transpositions.Assume first that n is even.Then w 0 is the order-reversing permutation and its conjugates are precisely the fixed point free involutions in S n .So the existence of balanced ideals is equivalent to every involution in ∆\η having a fixed point.Now observe that ∆\η is a product of symmetric groups, namely ∆\η ∼ = k j=0 S i j+1 −i j , and that there are fixed point free involutions in S k if and only if k is even.So a balanced ideal exists iff at least one of the i j+1 − i j is odd, i.e. δ ≥ 1.
The same argument works if n is odd, except that the conjugates of w 0 are then involutions in S n with precisely one fixed point (every involution has at least one), and so we need δ ≥ 2 to have none of these in ∆\η .
For the action on Grassmannians, Proposition 4.1 specializes to the following simple condition: A balanced ideal exists in W ∆,{α k } if and only if n is even and k is odd.We can enumerate all balanced ideals for n ≤ 10 using a computer and obtain the following number of balanced ideals in W ∆,{α k } : In particular, a cocompact domain of discontinuity in projective space RP n−1 or CP n−1 exists if and only if n is even.Interestingly, these are also precisely the dimensions which admit complex Schottky groups by [Can08].

Hitchin representations
Let Γ be the fundamental group of a closed surface.A Hitchin representation ρ : Γ → SL(n, R) is a representation which can be continuously deformed to a representation of the form ι • ρ 0 where ρ 0 : Γ → SL(2, R) is discrete and injective and ι : SL(2, R) → SL(n, R) is the irreducible representation.Hitchin representations are ∆-Anosov [Lab06].
Theorem 3.24 together with Example 3.22 shows that if n ≥ 5 then the cocompact domains of discontinuity of a Hitchin representation in any flag manifold F η are in 1:1 correspondence with the balanced ideals in W ∆,η .These were discussed in Section 4.1.
For completeness, let us also have a look at the cases n ∈ {2, 3, 4}.
In SL(2, R) the Hitchin representations are just the discrete injective representations.
The only flag manifold is RP 1 , and since the limit maps ξ : ∂ ∞ Γ → RP 1 of Hitchin representations are homeomorphisms, there can be no non-empty domain of discontinuity in RP 1 .
In the case of SL(3, R) there is only a single balanced ideal I ⊂ W .By Theorem 3.18 the lift of any cocompact domain of discontinuity to the full flag manifold F ∆ must be a union of connected components of the corresponding domain, which is It is known (see [CG93]) that for any Hitchin representation ρ into SL(3, R) there exists a properly convex open domain D ⊂ RP 2 on which ρ acts properly discontinuously and cocompactly.The image of the limit map of ρ are then the flags consisting of a point on ∂D and the tangent line of D through this point.The domain Ω(Λ, I) ⊂ F ∆ therefore splits into three connected components: One of them (shown in red in Figure 2) consists of flags (i.e. a point and a line through it in RP 2 ) with the point inside D. The second component (blue in Figure 2) are flags whose line avoids D, and the third (green) consists of flags whose line goes through D but with the point being outside.
Only the red component descends to a domain in RP 2 , and only the blue one to Gr(2, 3), each forming the unique cocompact domain of discontinuity in these manifolds.The cocompact domains in the full flag manifold are any unions of one or more of the three components.
Finally, let's have a look at SL(4, R).There are ten balanced ideals in W in this case (see Section A.1.3),corresponding to ten maximal domains of discontinuity in F ∆ .By The topology of these domains does not change when the representation is continuously deformed, and from the Fuchsian case we can easily see that Ω 1 and Ω 2 each have two connected components, all of which are lifts of domains in RP 3 or Gr(3, 4), respectively.The quotient of one of the components in RP 3 describes a convex foliated projective structure on the unit tangent bundle of S [GW08].
Out of the other 8 cocompact domains in F ∆ , one descends to the partial flag manifolds F 1,2 , F 2,3 and F 1,3 , each.Counting all possible combinations of connected components separately, we have 14 different non-empty cocompact domains of discontinuity in F ∆ .

Symplectic Anosov representations into Sp(4n + 2, R)
In this section, we will prove Theorem 1.9, i.e. construct a compactification for locally symmetric spaces modeled on the bounded symmetric domain compactification.We first recall some facts on this compactification, which can be found in [Hel79] and [Sat80].
Every Hermitian symmetric space can be realized as a bounded symmetric domain in some C N .That is an open, connected and bounded subset D ⊂ C N such that for every point x ∈ D there is an involutive holomorphic diffeomorphism from D to itself which has x as an isolated fixed point.Concretely, to get the symmetric space Sp(2n, R)/ U(n) we can consider the bounded symmetric domain The group of holomorphic diffeomorphisms of D is isomorphic to Sp(2n, R) and acts with stabilizer U(n).We compactify the symmetric space by taking the closure D in C n(n+1)/2 .This is the bounded symmetric domain compactification of Sp(2n, R)/ U(n).
Instead of working with bounded symmetric domains, we will use an equivalent model of Sp(2n, R)/ U(n), the Siegel space.Let ω be a symplectic form on R 2n and ω C its complexification on C 2n .Together with the real structure this defines an (indefinite) Hermitian form h(v, w) The Siegel space is the subspace H n0 ⊂ Lag(C 2n ) of complex Lagrangians L such that h| L×L is positive definite.
The correspondence between these models uses the Cayley transform: Regard the symmetric complex matrices Sym(n, C) as a subset of Lag(C 2n ) by mapping X ∈ Sym(n, C) to {(Xv, v) | v ∈ C n } ∈ Lag(C 2n ).The Cayley transform on Lag(C 2n ) is just the action of the matrix e iπ/4 √ 2 It maps D to H n0 and D to H n0 , establishing an equivalence of these compactifications.
More generally, let H pq ⊂ Lag(C 2n ) be the set of Lagrangians such that h restricted to them has signature (p, q), meaning that there is an orthogonal basis with p vectors of positive norm and q vectors of negative norm (and possibly null vectors).Then makes every H pq a fiber bundle over the isotropic (n−p−q)-subspaces with fiber the semi-Riemannian symmetric space Sp(2p + 2q, R)/ U(p, q).In particular, this means H n0 = Sp(2n, R)/ U(n).A more detailed explanation can be found in [Wie16].

. The number of balanced ideals in A 4
There are 4608 balanced ideals in W , so we cannot list them all.Instead, the following table shows just how many balanced ideals exist in W θ,η for any choice of θ, η ⊂ ∆ with ι(θ) = θ.The rows correspond to different values of θ (for example 14 stands for θ = {α 1 , α 4 }) while the columns correspond to η.
g α and let A, N, N − ⊂ G be the connected Lie subgroups corresponding to a, n, and n − .Note that N − = w 0 N w −1 0 (choosing any representative for w 0 in N K (a)).Let P θ = Z G (p θ ) be the parabolic subgroup corresponding to θ.The minimal parabolic subgroup B = P ∆ decomposes as B = Z K (a)AN via the Iwasawa decomposition.

Figure 1 :
Figure1: The parametrization Θ interpreted by intersecting light cones in Lag(R 4 ).The vertical line is the set of points Θ(x, x, * ) containing the limit set.

Figure 2 :
Figure 2: The three connected components of the maximal domain of discontinuity in F ∆ for a Hitchin representation ρ : Γ → SL(3, R).One exemplary flag out of every component is shown, appearing as a point on a line in RP 2 .
and the H pq are precisely the orbits of the action of Sp(2n, R) ⊂ Sp(2n, C) on Lag(C 2n ).Furthermore, the mapH pq → Is n−p−q (R 2n ), L → L ∩ L Now let ρ : Γ → Sp(2n, R) be an {α n }-Anosov representation and ξ : Γ → F {αn} = Lag(R 2n ) its limit map.Important examples of these representations are maximal representations from a surface group Γ = π 1 S, if either S is a closed surface, or an open surface and the boundary elements map to Shilov hyperbolic elements of Sp(2n, R).The following theorem implies Theorem 1.9: Theorem 5.1.If n is odd, then there exists a balanced ideal I ⊂ W {αn},{αn} .Therefore,X := H n0 ∩ Ω(ξ(∂ ∞ Γ), I) = {L ∈ H n0 | dim C L ∩ ξ(x) C < n/2 ∀x ∈ ∂ ∞ Γ}is Γ-invariant and the quotient Γ\ X is a compactification of the locally symmetric space Γ\H n0 = Γ\ Sp(2n, R)/ U(n).A.1.1.Balanced ideals in A 1 left-invariance right-invariance dimension generators 3. Balanced ideals in A 3 left-invariance right-invariance dimension generators {α 1 , α 3 out: There is only a single balanced ideal in W {α 1 ,α 4 },∆ , and it has no right-invariances at all.In fact, we have the same situation generally in W {α 1 ,αn},∆ is even, a balanced ideal exists in W ∆,η if and only if δ ≥ 1.If n is odd, a balanced ideal exists in W ∆,η if and only if δ ≥ 2. Proof.A balanced ideal exists if and only if the action of w 0 on W ∆,η by left-multiplication fixes no element of W ∆,η (see [KLP18, Proposition 3.29] or [ST18, Lemma 3.34]