Cusped Borel Anosov representations with positivity

We show that if a cusped Borel Anosov representation from a lattice $\Gamma \subset \mathsf{PGL}_2(\mathbb{R})$ to $\mathsf{PGL}_d(\mathbb{R})$ contains a unipotent element with a single Jordan block in its image, then it is necessarily a (cusped) Hitchin representation. We also show that the amalgamation of a Hitchin representation with a cusped Borel Anosov representation that is not Hitchin is never cusped Borel Anosov.


Introduction
Let Γ be a (word) hyperbolic group and let θ ⊂ ∆ := {1, . . ., d − 1} be any subset.In his seminal work, Labourie [Lab06] defined what it means for a representation ρ : Γ → PGL d (R) to be P θ -Anosov.This notion of Anosov representations has proven to be very useful: It is strong enough for general theorems to be proven for the entire class of Anosov representations, but at the same time is also flexible enough to admit many interesting examples.For this reason, the theory of Anosov representations has been heavily studied and developed in the last twenty years [GW12, KLP17, GGKW17, BPS19].
Recently, there has been a successful push to generalize the notion of Anosov representations to the setting where Γ is a relatively hyperbolic group.These include the relatively asymptotically embedded and relatively Morse representations defined by Kapovich and Leeb [KL18], the relatively dominated representations defined by Zhu [Zhu21], and the extended geometrically finite representations defined by Weisman [Wei22].Most recently, Zhu and Zimmer [ZZ22] defined the notion of relatively Anosov representations, and clarified the relationship between their notion and the other notions mentioned above.
In the case when Γ ⊂ PGL 2 (R) is a geometrically finite subgroup i.e. a finitely generated, non-elementary, discrete subgroup, Canary, Zhang and Zimmer [CZZ22a] defined the notion of a cusped Anosov representation ρ : Γ → PGL d (R).If we view Γ as a hyperbolic group relative to the cusp subgroups in Γ, then cusped Anosov representations are a special case of all the above notions.Canary, Zhang and Zimmer [CZZ22b] also defined the notion of transverse representations, which extends the notion of cusped Anosov representations to allow for Γ to be any non-elementary, discrete subgroup of PGL 2 (R) (and more generally, any projectively visible group), see Remark 1.2.In this article, we will focus exclusively on transverse representations of non-elementary, discrete subgroups of PGL 2 (R), which we now define.
For any non-elementary, discrete subgroup Γ ⊂ PGL 2 (R), let Λ(Γ) denote its limit set, i.e.Λ(Γ) is the set of accumulation points in ∂ H 2 of some/any Γ-orbit in H 2 .Note that Λ(Γ) is an infinite, Γ-invariant, compact subset of ∂ H 2 .For any subset θ ⊂ ∆, let F θ (R d ) denote the corresponding partial flag manifold, i.e. if θ = {k 1 , . . ., k s } with k 1 < • • • < k s , then In the case when θ = ∆, we will simply denote ) that satisfies all of the following properties: In the above definition, the strongly dynamics preserving property of ξ ensures that it is unique to ρ.We thus refer to ξ as the limit map of ρ.
In the case when θ = ∆, P ∆ -transverse representations and cusped P ∆ -Anosov representations are also called Borel transverse representations and cusped Borel Anosov representations respectively.When Γ ⊂ PGL 2 (R) is a convex cocompact free subgroup, (cusped) Borel Anosov representations from Γ to PGL d (R) can be constructed via a ping pong type argument.However, when Γ ⊂ PGL 2 (R) is a lattice, there are currently only two known families of cusped Borel Anosov representations: the Hitchin representations and the Barbot examples, see Section 2.2 and Appendix B respectively.The search for more examples of cusped Borel Anosov representations can be formulated as the following question: The two main results of this paper are rigidity results about Borel transverse representations of non-elementary discrete subgroups of PGL 2 (R) whose limit set is all of ∂ H 2 .When specialized to lattices in PGL 2 (R), they can be interpreted as providing supporting evidence to a negative answer to the above question.
If Γ ⊂ PGL 2 (R) is a non-elementary, discrete subgroup and ρ : Γ → PGL d (R) is a Hitchin representation, then it follows from the work of Canary, Zhang and Zimmer [CZZ22a] that ρ sends every (non-identity) parabolic element in Γ to a unipotent element with a single Jordan block, see Theorem 2.4 and Remark 2.5.Our first theorem resolves Question 1.3 under the additional assumption that the image of ρ contains a unipotent element with a single Jordan block.
) is a Borel transverse representation whose image contains a unipotent element with a single Jordan block, then ρ is a Hitchin representation.
Remark 1.5.If ρ : Γ → PGL d (R) is a Barbot example, then d is necessarily odd, and ρ sends every parabolic element in Γ to a unipotent element in PGL d (R) with two Jordan blocks, one of size j and the other of size d − j for some j ∈ {1, . . ., d−1 2 }, see Appendix B. As such, the hypothesis of Theorem 1.4 rules out the need to consider the Barbot examples.
One might attempt to construct new examples of cusped Borel Anosov representations on a lattice Γ ⊂ PGL 2 (R) via the following "amalgamation" procedure.
Step 1: Realize Γ as a free product of two non-elementary, geometrically finite subgroups Γ 1 and Γ 2 , amalgamated over a cyclic subgroup γ .
Step 2: Specify a Barbot example ρ 1 : Γ 1 → PGL d (R) and a Hitchin representation Step 3: Find a cusped Borel Anosov representation ρ : Γ → PGL d (R) so that ρ| Γ 1 = ρ 1 and ρ| Γ 2 = ρ 2 .There are situations (see for example [GW10,CLS17]) where this amalgamation procedure allows one to construct new classes of P θ -Anosov representations from existing ones.However, our next theorem implies that the amalgamation process described above will never yield a Borel transverse representation.
By Remark 1.2 above, Theorems 1.4 and 1.6 hold for cusped Borel Anosov representations as well; one simply imposes the additional condition that Γ is geometrically finite.
A key tool used in the proofs of Theorem 1.4 and Theorem 1.6 (and also in the definition of Hitchin representations) is Fock and Goncharov's notion of positivity for n-tuples in F(R d ) for any integer n 3, see Section 2.1.With this, one can then define the notion of a positive map from a subset Λ ⊂ S 1 (with #Λ 3) to F(R d ): we say that a map ξ : Λ → F(R d ) is positive if for any integer n 3, the tuple (ξ(a 1 ), . . ., ξ(a n )) is positive for all a 1 < • • • < a n < a 1 in Λ (according to the clockwise cyclic order on S 1 ).The proofs of both Theorem 1.4 and Theorem 1.6 rely on the following result about continuous, positive maps, which is a special case of more general results of Guichard, Labourie and Wienhard [GLW21, Lemma 3.5 and Proposition 3.15] in the setting of Θ-positive maps.
In Section 2, we will recall Fock and Goncharov's notion of positivity of k-tuples of complete flags and the definition of Hitchin representations.Then, in Section 3, we provide an elementary and self-contained proof of Proposition 1.7.Finally, we use Proposition 1.7 to prove Theorems 1.4 and 1.6 in Section 4. In the appendices, we give an elementary proof of a well-known fact about positive triples of flags that was used to prove Proposition 1.7, and also describe the Barbot examples mentioned above.
Acknowledgements.We are thankful for helpful conversations with Fanny Kassel and Jaejeong Lee.

Positive tuples and positive maps
2.1.Fock-Goncharov positivity.We say that an upper triangular, unipotent matrix is totally positive if its non-trivial minors (i.e.those that are not forced to be 0 by virtue of the matrix being upper triangular) are positive.Then given an (ordered) basis B = (e 1 , . . ., e d ) of R d , we say that a unipotent element in PGL d (R) is totally positive with respect to B if it is represented in the basis B by an upper triangular, unipotent, totally positive matrix.Let denote the set of unipotent elements that are totally positive with respect to B, and let denote the closure of U >0 (B).Note that the elements in U 0 (B) are exactly the ones where all the non-trivial minors are non-negative.Using well-known formulas for how minors behave under products, it is straightforward to verify that both U >0 (B) and n for all i ∈ {1, . . ., d}, and Recall from the introduction that given a subset Λ of S 1 , a map ξ : Λ → F(R d ) is positive provided that if n 3 and (x 1 , . . ., x n ) is a cyclically ordered subset of pairwise distinct points in Λ, then (ξ(x 1 ), . . ., ξ(x n )) is a positive n-tuple of flags.
The following proposition summarizes the basic properties of positive tuples of flags.It follows easily from a well-known parameterization result of Fock and Goncharov [FG06, Theorem 9.1(a)] (see Kim-Tan-Zhang [KTZ22, Observation 3.20]).
(1) If n 3, then the following are equivalent: and so F i and F j are transverse for all distinct pairs i, j ∈ {1, . . ., n}.
Let P denote the set of positive triples of flags in F(R d ), and let T denote the set of pairwise transverse triples of flags in F(R d ).The following theorem is also a well-known property of positive triples of flags, which has been generalized to the setting of triples of Θ-positive flags by Guichard, Labourie and Wienhard [GLW21, Proposition 2.5(1)].We provide an elementary proof in Appendix A.
Theorem 2.2.Let F , G and H be complete flags in F (R d ) such that both G and H are transverse to F .Let u ∈ PGL d (R) be the unipotent element that fixes F and sends H to G, and let B = (e 1 , . . ., e d ) be any basis of R d such that e k ∈ F k ∩ H d−k+1 for all k ∈ {1, . . ., d}.If u ∈ U 0 (B)− U >0 (B), then G and H are not transverse.In particular, P is a union of connected components of T .
2.2.Hitchin representations.Suppose for now that Γ ⊂ PGL 2 (R) is surface group (i.e.Γ is cocompact and torsion-free).Then the discrete and faithful representations from Γ to PGL 2 (R) form a single connected component of Hom(Γ, PGL 2 (R))/PGL 2 (R), known as the Teichmüller component.Hitchin [Hit92] noticed that for all d 2, there is a distinguished connected component of Hom(Γ, PGL d (R))/PGL d (R) that is analogous to the Teichmüller component.Today, this connected component is commonly known as the Hitchin component, and the Hitchin representations are the ones whose conjugacy class lies in the Hitchin component.Fock and Goncharov [FG06] characterized the Hitchin representations as the representations for which there exists a ρ-equivariant positive map ξ : Λ(Γ) → F(R d ), and Labourie [Lab06] showed that every Hitchin representation is (cusped) Borel Anosov.
Motivated by Fock and Goncharov's characterization of Hitchin representations, Canary, Zhang and Zimmer [CZZ22b] extended the notion of Hitchin representations to the case when Γ is a discrete subgroup of PGL 2 (R).
Labourie's result can also be generalized to this case using the proof of [CZZ22a, Theorem 1.4].
Theorem 2.4.Every Hitchin representation ρ : Γ → PGL d (R) is Borel transverse, and the continuous, ρ-equivariant, positive map is the limit map of ρ (and hence is unique).Furthermore, ρ sends parabolic elements in Γ to unipotent elements in PGL d (R) with a single Jordan block.
Remark 2.5.Even though [CZZ22a, Theorem 1.4] is stated only in the case when Γ ⊂ PGL 2 (R) is geometrically finite, the proof does not use the geometric finiteness of Γ.

Proof of Proposition 1.7
To prove Proposition 1.7, we will use the following lemma, which is already well-known to experts (see for example [GLW21, Proposition 3.15]).We give an elementary proof of the lemma for the reader's convenience.We remark that the lemma is false without the continuity assumption on ξ.
Proof.By Proposition 2.1(2), it suffices to show that (ξ(x), ξ(y), ξ(z), ξ(w)) is positive for all quadruples x, y, z, w ∈ S 1 such that x < y < z < w < x along S 1 .Pick any such quadruple x, y, z, w ∈ S 1 , and let I ⊂ S 1 denote the closed subinterval that contains z with endpoints y and w.By Proposition 2.1(1), the map ξ is transverse.Thus, for all t ∈ I, we may define the map u : I → PGL d (R) by setting u(t) ∈ PGL d (R) to be the unipotent element that fixes ξ(x) and sends ξ(w) to ξ(t).The continuity of ξ then implies that the map u is continuous.
Since we have proven that both u(z) and u(z) −1 u(y) lie in U >0 (B), the quadruple of flags is positive, so the lemma follows.
Proof of Proposition 1.7.By Lemma 3.1, it suffices to show that (ξ(a), ξ(b), ξ(c)) is positive for any pairwise distinct triple a, b, c ∈ S 1 .By Proposition 2.1(1), we may assume that a < b < c and x < y < z by switching the roles of a and c and the roles of x and z if necessary.Then there are continuous maps are pairwise distinct triples for all t.
Recall that P denotes the set of positive triples of flags in F(R d ), and T denotes the set of pairwise transverse triples of flags in F(R d ).Since ξ is continuous and transverse, this implies that the map F : [0, 1] → T given by F (t) = ξ(f 1 (t)), ξ(f 2 (t)), ξ(f 3 (t)) is well-defined and continuous.Since F (0) ∈ P by hypothesis, Theorem 2.2 implies that F (1) ∈ P.

Proof of Theorems 1.4 and 1.6
Using Proposition 1.7, we will now prove Theorems 1.4 and 1.6.
Proof of Theorem 1.4.The d-th upper triangular Pascal matrix Q d is the d × d upper triangular matrix whose (i, j)-th entry (with i j) is the integer j−1 i−1 .To prove the theorem, we will first recall some basic properties of Q d .
Lemma 4.1.Q d is totally positive, unipotent, and has a single Jordan block Proof.The claim that Q d is unipotent is obvious, and the claim that Q d has a single Jordan block is a straightforward calculation: one simply verifies that Q d has a unique eigenvector.
To prove that Q d is totally positive, observe that the natural GL 2 (R) action on the symmetric tensor Sym d−1 (R 2 ) given by Here, the identification GL(Sym given by identifying the standard basis (e 1 , . . ., e d ) of R d with the basis (e d−1 1 , e d−2 1 e 2 , . . ., e 1 e d−2 2 , e d−1 2 ) of Sym d−1 (R 2 ) induced by the standard basis (e 1 , e 2 ) of R 2 .Note that the representation ι d descends to a representation, also denoted If we take B to be the standard basis of R d , then by [FG06, Proposition 5.7], It is also straightforward to verify that [Q d ] = ι d 1 1 0 1 and that 1 1 0 1 clearly lies in The proof of this theorem relies on the following lemma, which demonstrates the inherent positive nature of a unipotent element in PGL d (R) with a single Jordan block.
Lemma 4.2.Let u ∈ PGL d (R) be a unipotent element with a single Jordan block, and let F be the fixed flag of u.Then for any flag G that is transverse to F and for any sufficiently large t, the triple (F, u t • G, G) is positive.
Proof.By Lemma 4.1, Q d is a unipotent upper triangular matrix with a single Jordan block, so we may choose a basis , which is upper triangular, and whose (i, j)-th entry (with i j) is j−1 i−1 t j−i .Furthermore, for all k ∈ {1, . . ., d − 1}, the subspace for sufficiently large t.Indeed, if this were the case, then the observation that In fact, we will show that if v ′ and v are two unipotent elements that fix F , then v ′ u t v ∈ U >0 (B) for sufficiently large t.Observe that since v ′ and v are represented in the basis B by upper triangular matrices whose diagonal entries are all 1, the product v ′ u t v is also represented in the basis B by an upper triangular matrix M t whose diagonal entries are all 1.Furthermore, for each i < j, the (i, j)-th entry of M t is a polynomial in the variable t whose leading term is j−1 i−1 t j−i , which is the (i, j)-th entry of Q t d .By Lemma 4.1, Q t d is totally positive, so the leading term of any minor of M t is the corresponding minor of Q t d .Hence, for sufficiently large t, we have Let γ ∈ Γ be the element such that ρ(γ) is unipotent with a single Jordan block.Since ρ is Borel transverse, the strongly dynamics preserving property of its limit map ξ : Λ(Γ) → F(R d ) ensures that γ is parabolic.Let x ∈ Λ(Γ) be the unique fixed point of γ and let y ∈ Λ(Γ) − {x}.Then ξ(x) is the fixed flag of ρ(γ).By Lemma 4.2 and the ρ-equivariance and transversality of ξ, the triple of flags ξ(x), ξ(γ n y), ξ(y) is positive for sufficiently large n.Proposition 1.7 then implies that ξ is a positive map, so ρ is a Hitchin representation.
On the other hand, we also have because c 1 = 0 = c k and both det(u I ′ ,J ′ ) and det(u I ′ ,J ′′ ) are positive.We thus arrive at a contradiction, so det(u I,J ) = 0. Since u ∈ U 0 (B), it follows that det(u I,J ) > 0. This completes the inductive step.
Lemma A.2. Let u ∈ U 0 (B).Suppose that there exists k ∈ {1, . . ., d} such that • all the non-trivial ℓ × ℓ-minors of u are positive for all ℓ < k; • there is a non-trivial k × k minor det(u I,J ) of u such that I and J are consecutive and det(u I,J ) = 0.
Proof.Notice that it suffices to prove that det(u I,J 0 ) = 0 and that det(u I 0 ,J ) = 0. We will only prove the former; the proof of the latter is the same.Let I = (i 1 , . . ., i k ) and J = (j 1 , . . ., j k ).By assumption, det u I ′ ,J ′ > 0, so u I ′ ,j 1 , . . ., u I ′ ,j k−1 and hence u I,j 1 , . . ., u I,j k−1 is a linearly independent collection of vectors.Since det (u I,J ) = 0, it follows that u I,j k is a linear combination of u I,j 1 , . . ., u I,j k−1 .
Since J is consecutive, we have proven that the k vectors u I,d−k+1 , u I,d−k+2 , . . ., u I,d are all linear combinations of u I,j 1 , . . ., u I,j k−1 .In particular, their span has dimension k − 1, so det(u I,J 0 ) = 0.
Proof of Theorem 2.2.Since u ∈ U 0 (B) − U >0 (B), there is some k ∈ {1, . . ., d − 1} such that some non-trivial k × k minor det(u I,J ) of u is zero, while all the non-trivial ℓ × ℓ-minors of u are positive for all ℓ < k.
By Lemma A.1, we may assume that both I and J are consecutive.Then Lemma A.2 implies det(u I 0 ,J 0 ) = 0 with I 0 = (1, . . ., k) and J 0 = (d − k + 1, . . ., d).Therefore, the span of the vectors u I 0 ,d−k+1 , . . ., u I 0 ,d has dimension at most k − 1, so This implies that G and H are not transverse.

Appendix B. The Barbot examples
Fix a lattice Γ ⊂ SL 2 (R) and some odd integer d > 2. In this appendix, we define the Barbot examples, which are representations ρ : Γ → PGL d (R) that are Borel transverse (or equivalently, cusped Borel Anosov), but not Hitchin.These are a straightforward generalization of examples (due to Barbot [Bar10]) of Borel Anosov representations of a surface group into PGL 3 (R) that are not Hitchin.
To define the Barbot examples, we need some preliminary results.First, let (e 1 , . . ., e d ) be the standard basis of R d , and equip R d with the standard inner product.For any g ∈ PGL d (R), let σ 1 (g) . . .σ d (g) > 0 denote the singular values of (any unit-determinant, linear representative of) g, and let A g := diag(log σ 1 (g), . . ., log σ d (g)).
By the singular value decomposition theorem, we may write every g ∈ PGL d (R) as the product g = m exp(A g )ℓ for some m, ℓ ∈ PO(d) (which are not necessarily unique).For every g ∈ PGL d (R), choose m g , ℓ g ∈ PO(d) such that g = m g exp(A g )ℓ g .Let F 0 ∈ F(R d ) be the flag such that F k 0 = Span(e 1 , . . ., e k ) for all k ∈ {1, . . ., d − 1}, and define One can verify that if σ k (g) > σ k+1 (g) for all k ∈ {1, . . ., d − 1}, then U (g) does not depend on the choice of m g and ℓ g , and hence is canonical to g.The following proposition is a standard linear algebra fact, see [CZZ23, Appendix A] for a proof.
The following are equivalent: ( Finally, let k 1 be an integer.Recall from the proof of Lemma 4.1 the representation One can verify that ι k restricts to a representation is a lattice, are there cusped Borel Anosov representations that are neither Hitchin representations nor the Barbot examples? and there is a basis B = (e 1 , . . ., e d ) of R d and elements u Span(e d , . . ., e d−k+1 ) + Span(e d , . . ., e k+1 ) = Span (u • e d , . . ., u • e d−k+1 , e d , . . ., e k+1 ) , . . ., e k+1 = R d .