Positive crossratios, barycenters, trees and applications to maximal representations

We study metric properties of maximal framed representations of fundamental groups of surfaces in symplectic groups over real closed fields, interpreted as actions on Bruhat-Tits buildings endowed with adapted Finsler norms. We prove that the translation length can be computed as intersection with a geodesic current, give sufficient conditions guaranteeing that such a current is a multicurve, and, if the current is a measured lamination, construct an isometric embedding of the associated tree in the building. These results are obtained as application of more general results of independent interest on positive crossratios and actions with compatible barycenters.


Introduction
Maximal framed representations in real closed fields. Let Σ := Γ \H 2 be the quotient of the Poincaré upper half plane H 2 by a torsion-free lattice Γ < PSL(2, R) and let G be a simple real algebraic group. The aim of Higher Teichmüller Theory is to single out and study special components or specific semialgebraic subsets of the representation variety Hom(Γ , G) that consist of injective homomorphisms with discrete image; such components thus generalize In the study of appropriate compactifications of character varieties, representations of Γ into algebraic groups over non-Archimedean real closed fields play an important role [Bru88,Ale08,Par12,BP17,BIPP21c]: for example, ultralimits of representations in PSp(2n, R) can be understood as representations ρ ω,σ : Γ → PSp(2n, R ω,σ ), where R ω,σ is a Robinson field (see §8.1 and [Par12]).
However, the viewpoint of the real spectrum compactification of character varieties leads typically to the study of representations into real closed fields F that are small when compared to Robinson fields, for example F is often of finite transcendence degree over the field of real algebraic numbers.
Given a representation ρ : Γ → PSp(2n, F) where F is a general real closed field, having a maximal Toledo invariant as defined in [BIPP21c,Definition 18], admitting a maximal framing defined on a Γ -invariant non-empty subset of ∂H 2 , or admitting a maximal framing defined on the set of fixed points of hyperbolic elements are all equivalent conditions [BIPP21b] (see [BIPP21c,Theorem 20] for a precise statement). This paper solely relies upon the third definition, which we now recall. If F 2n is endowed with the standard symplectic form, let L(F 2n ) be the space of Lagrangians in F 2n . The Maslov cocycle (see §6.4 and [LV80]) classifies the orbits of PSp(2n, F) on triples of pairwise transverse Lagrangians.
Such a triple is maximal if the cocycle takes its maximal value n. Let ∂H 2 be the boundary of the hyperbolic plane, which we endow with the cyclic ordering on We assume that the real closed field F admits an order compatible R-valued valuation v with value group Λ = v(F × ) < R. Then the group PSp(2n, F) acts by isometries on a Λ-metric space 1 B F n obtained as quotient of the Siegel upper half space S n F associated to PSp(2n, F) (see § 6.2 and [BIPP21c,§ 3.4]). If F = R, B R n coincides with S n R , while if F is non-Archimedean this metric space B F n sits naturally inside the Bruhat-Tits building associated to PSp(2n, F) as a dense subset. 2 The latter relationship will play no role in this paper, but will be discussed in detail in [BIPP21a] (see also [BIPP21c,§2.1] and [KT04]). The translation length of an element g ∈ PSp(2n, F) acting on B F n induces then the length function where λ 1 , . . . , λ n , λ −1 n , . . . , λ −1 1 ∈ F( √ −1) are the eigenvalues of a representative g ∈ Sp(2n, F) of g ∈ PSp(2n, F) counted with multiplicity and ordered in such a way that |λ 1 | · · · |λ n | 1. Here we denote by | · | : F( √ −1) → F + the absolute value, that is the square root of the norm function on the quadratic extension In our first result we construct a geodesic current on Σ encoding the length Theorem 1.2. Let F be a real closed field with an order compatible valuation v and let ρ : Γ → PSp(2n, F) be maximal framed. Then there is a geodesic current µ ρ such that, 1 In [BIPP21c] denoted with B PSp(2n,F) . 2 Note, however, that the metric that we consider here is only biLipschitz to the restriction of the CAT(0) metric on the Bruhat-Tits building. See §6.2 for the Finsler metric relevant to our purposes.
If n = 1, F = R and ρ : Γ → PSL(2, R) is the lattice embedding, then µ ρ is the Liouville current [Bon88], that is the unique PSL(2, R)-invariant geodesic current. If Σ is compact and F = R, Theorem 1.2 was proven by Martone is a quasi-isometric embedding.
We now give a robust criterion guaranteeing that the current µ ρ is atomic.
We say that a geodesic current is a multicurve if it is a finite sum of Γ -orbits of Dirac masses on (lifts of) closed geodesics and geodesics with endpoints in cusps.
We will see that if Q(ρ) has discrete valuation, then V(µ ρ ) is the vertex set of a simplicial tree (see § 7.3 for a general statement).
Currents associated to positive crossratios. The proof of Theorem 1.2 relies on an abstract framework that is applicable to more general situations and that we shortly describe here. Let X ⊂ ∂H 2 be a Γ -invariant non-empty subset, such as, for example the set H Γ of fixed points of hyperbolic elements in Γ and let X [4] denote the set of positively ordered quadruples in X. A positive crossratio is a and satisfies the property If γ ∈ Γ is hyperbolic and {γ − , γ + } ⊂ X, the period 3 per(γ) of γ with respect to [ · , · , · , · ] is defined by where x ∈ X is any point such that (γ − , x, γx, γ + ) ∈ X [4] . We show: Theorem 1.6. Let X ⊂ ∂H 2 be a Γ -invariant non-empty subset and [ · , · , · , · ] a positive crossratio on X. Then there is a geodesic current µ on Σ such that for all hyperbolic γ ∈ Γ per(γ) = i(µ, δ c ).
The theorem has been previously shown by Martone-Zhang under the hypothesis that X = ∂H 2 and the crossratio is continuous [MZ19]. For the last statement we consider the space CR + (X) of positive crossratios as a closed 3 See § 3.3 for a more general definition of the period without the restriction that {γ − , γ + } ⊂ X.
convex cone in the topological vector space of crossratios on X with the topology of pointwise convergence. This last property will be used in the proof of the continuity of the map which to a point in the real spectrum compactification of maximal representations associates a geodesic current [BIPP21a]; see also [BIPP21c,Theorem 36].
The proof of Theorem 1.6 bypasses the possible discontinuities of the crossratio [ · , · , · , · ] by forcing inner and outer regularity of the current µ and using its σ-additivity. As an application of the explicit construction we obtain: Corollary 1.7. If the crossratio [ · , · , · , · ] is integral valued, then the current µ is a multicurve.
Maximal representations are not the only class of representations whose length function is given by the periods of a positive crossratio: this is the case for all positively ratioed representations [MZ19] -a class that also includes Hitchin representations [Lab07] -, representations satisfying property H k [BP20] and Θpositive representations [BP21]. Corollary 1.7 can be used to study asymptotic properties of these representations as well.
Our approach using σ-additivity of geodesic currents has interesting applications even for representations in PSp(2n, R), for which we cannot always assume that the crossratio is continuous. The simplest instance is for PSL(2, R) if ρ sends an element representing a cusp of Γ to a hyperbolic element. Corollary 1.8. Let ρ : Γ → PSp(2n, R) be a maximal representation and let K ⊂ Σ = Γ \H 2 be a compact subset. Then there are constants 0 < c 1 c 2 such that for every In particular this holds uniformly for all γ representing simple closed geodesics.
This corollary is well known for Anosov representations. However if Σ is not compact, a maximal representation is not necessarily Anosov since the images of parabolic elements can be unipotent (see for instance §8.3).
Actions with compatible barycenters. The proof of Theorem 1.5 is carried out in the framework of actions with compatible barycenters that we now define.
Given an isometric Γ -action on a metric space (X, d), we say that a map We show then: Theorem 1.9. Let X ⊂ ∂H 2 be a Γ -invariant non-empty subset and [ · , · , · , · ] a positive crossratio on X. Assume that the geodesic current µ associated by Theorem 1.6 to the positive crossratio [ · , · , · , · ] corresponds to a measured lamination. Then for every isometric Γ -action on a metric space X admitting a barycenter compatible with the We will see that Theorem 1.9 always applies to a framed action of Γ on an R-tree T if the crossratio [ · , · , · , · ] induced by the action is positive (Proposition 5.8). This crossratio is always positive in the case of the action on B F 1 induced by a maximal framed representation in SL(2, F) (Theorem 8.1).
Structure of the paper. In § 2 we discuss preliminaries on geodesic currents and measured laminations. The new result is Proposition 2.1 that gives an useful 4point characterization of measured laminations among geodesic currents that only involves a dense subset of ∂H 2 . In § 3 we introduce positive crossratios and the associated periods. In § 4 we construct the geodesic current associated to such a crossratio. Theorem 1.6 follows directly combining Propositions 4.3, 4.8 and 4.10, which are proven in this section. Corollary 1.7 follows from Proposition 4.12. In § 5 we discuss barycenter maps, and prove Theorem 1.9. In § 6 we review the geometry of the Siegel space over real closed fields from [BP17], using this we associate to a maximal framed action on the Λ-metric space B F n a positive crossratio (Proposition 6.5), as well as a compatible barycenter map ( § 6.7). In § 7 we prove the results on maximal framed representations: Theorems 1.2, 1.4, Given a subset A ⊂ ∂H 2 , we will denote: A [4] := {(x, y, z, t) ∈ A 4 : (x, y, z, t) is positively oriented} .

Geodesic currents. A geodesic current is a flip-invariant Γ -invariant positive
Radon measure on the set of (oriented) geodesics in H 2 , which we identify with Given a (non-oriented) geodesic c ⊂ Σ that is either closed or joining two cusps, δ c will be the geodesic current given by where we sum on the set of oriented geodesics (a, b) ∈ (∂H 2 ) (2) lifting c. (1) supp(µ) is a lamination ; 4 We recall that the carrier of a geodesic current µ is the closed subset pr( g∈supp(µ) g) ⊂ Σ.
The hypothesis that ν has compact carrier ensures that i(µ, ν) < ∞ and is needed in the proofs of the continuity of the Bonahon intersection.
Since A × B ⊂ G, in particular g and g ′ intersect in a point, hence supp(µ) is not a lamination.
Suppose now (4). Let g, g ′ be two geodesics in supp(µ). If they are transverse, then g = (x, y), g ′ = (x ′ , y ′ ) with (x, x ′ , y, y ′ ) positively oriented. It follows from Proposition 2.1 that if the current µ is of lamination type, its support consists of µ-short geodesics. 5 While in the introduction we identified with a slight abuse of notation measured laminations with currents with zero self intersection, we prefer to keep the objects distinct for the rest of the paper.
2.3. The tree associated to a current of lamination type. We now recall the construction of the tree T(µ) associated to a current µ of lamination type. We chose here a description adapted to our purposes, but this agrees with the standard construction described, for example, in [MS91] and [Kap09,§11.12].
Given a geodesic current µ, we consider the straight pseudodistance on H 2 where for a possibly empty geodesic segment I ⊂ H 2 we define as the set of geodesics g that intersect transversely the geodesic segment I.
If µ is of lamination type, then the quotient metric space X µ = H 2 /∼, obtained by identifying points at d µ -distance zero, is 0-hyperbolic in the sense of Gromov and can therefore be canonically embedded in a minimal R-tree T(µ). We will denote by V(µ) the image in T(µ) of the complementary regions R of supp(µ).

It corresponds to the set of branching points of T(µ).
Since µ is Γ -invariant, the group Γ acts on T(µ) and therefore on V(µ) by isometries. A direct consequence of the definition of Bonahon intersection is that, for this action, for hyperbolic γ representing a closed geodesic c.

Positive crossratios
In this section we introduce the notion of positive crossratio [ · , · , · , · ], prove that its periods are well defined, and discuss examples.
satisfying the following properties: is a crossratio according to Definition 3.1. However we do not require continuity, and our crossratio is defined only on a smaller set. As a direct consequence of (CR3), positive crossratios have the following monotonicity property and x 3 x 2 x 4 x 1 To gain some intuition on the properties (CR2) and (CR3), we recall that if is the logarithm of the usual crossratio, the Liouville measure L has the property that Thus (CR2) corresponds to the flip- and (CR3) to additivity Examples. There are two natural crossratios associated to a geodesic current: Example 3.4. If µ is a current, it is easily checked that defines a positive crossratio on ∂H 2 . Similarly, defines a positive crossratio on ∂H 2 . Note that these two crossratios may be different (for example this is the case if µ = δ c for some closed geodesic c).

Framed actions on trees give other fundamental examples of crossratios:
Example 3.5. If T is a real tree, we denote by [ · , · , · , · ] T the usual crossratio on the boundary ∂ ∞ T of the tree T: where d denotes the distance in T.
A framed action of Γ on T is an action by isometries ρ : Γ → Isom(T) admitting Example 3.6. An example of such situation is given by the Γ -action on the Rtree T(µ) associated to a current of lamination type. Let X ⊂ ∂H 2 be the set of fixed point of hyperbolic elements whose axis are transverse to the geodesic lamination supp(µ). Then for every such 3) and has thus an attractive fixed point ϕ(γ + ) and a repulsive one ϕ(γ − ) in is a framing and it follows from the definition of the distance on T(µ) that It follows from the discussion recalled in §2.2 that the crossratio is positive.
Example 3.5 inspires the following definition: Definition 3.7. We say that a crossratio is ultrametric if it satisfies : The following is clear.

Proposition 3.8. The crossratio induced by a framed action on a R-tree is ultrametric.
The following is a corollary of Proposition 2.1.
Proposition 3.9. The crossratio associated to a lamination type current µ is ultrametric.
3.3. The periods of the crossratio. Let now [ · , · , · , · ] be a positive crossratio defined on X ⊂ ∂H 2 , and let γ ∈ Γ be hyperbolic such that This justifies the following: The purpose of this section is to extend the definition of the period of a crossratio defined on a Γ -invariant set X ⊂ ∂H 2 to hyperbolic elements γ ∈ Γ whose endpoints do not necessarily belong to the set X.
This is achieved by the following: Proposition 3.11. Let [ · , · , · , · ] be a positive crossratio on X, and γ ∈ Γ be a hyperbolic Proof. Up to passing to a subsequence we can and will assume that (x 0 , x, γx, y 0 ) is positive (see Figure 1). Since by (CR2) and (CR3) we have and the analogous statement for [y ′ n , x, γx, y n ]. Since, by (CR4), the crossratio is positive, and γ −n x 0 → γ − , it is in turn Here in the last equality we used that the crossratio is Γ -invariant. The claim for [y ′ n , x, γx, y n ] follows analogously.

The geodesic current associated to a positive crossratio
In this section we prove Theorem 1.6 and Corollary 1.7. The proof of Theorem 1.6 is carried out in three steps: in §4.1 we use a crossratio to construct a geodesic current µ [ · , · , · , · ] ; in §4.2 we relate the periods of the crossratio [ · , · , · , · ] and the intersection of curves with µ [ · , · , · , · ] ; in §4.3 we conclude the proof of Theorem 1.6 by showing that µ [ · , · , · , · ] depends continuously on the crossratio [ · , · , · , · ]. The fact that an integer valued crossratio leads to a multicurve (Corollary 1.7) is shown in §4.4. We conclude the section discussing in §4.5 how crossratios and geodesic currents can be restricted to subsurfaces; this is for future reference and will be used in the study of the real spectrum compactification of maximal representations. Fix a Γ -invariant non-empty subset X ⊂ ∂H 2 , and a positive crossratio and I and J have non-empty interior. The vertices of R are then the unique ( Remark 4.2. The function r : R(X) → R may not be σ-additive, even restricting to the family of left half open rectangles for a closed curve e corresponding to some hyperbolic γ ∈ Γ , we get a positive crossratio [ · , · , · , · ] on X [4] = ∂H 2 whose associated function r is not σ-additive contradicting σ-additivity. The problem is due to the fact that this crossratio is not continuous at (a, γ + , c, γ − ).
We now construct the mesure µ. Recall that for a rectangle R ∈ R(X) with R) the open (resp. closed) rectangle with the same vertices as R. (1) µ(R) r(R) µ(R) for any (proper) rectangle R ∈ R(X).

3) For all (proper) open rectangles
Definition 4.4. We call the measure µ in Proposition 4.3 the geodesic current Proposition 4.3 implies the following "outer and inner" continuity properties of the current.
Proposition 4.5. Let (a, b, c, d) be a positively oriented quadruple in ∂H 2 . Let (a n , b n , c n , d n ) n 1 be a sequence in X [4] converging to (a, b, c, d). Then (1) If d n , a n ∈ I (d,a) and b n , c n ∈ I (b,c) for all n 1, then µ(I (d,a) × I (b,c) ) = lim n [a n , b n , c n , d n ]; (2) If a n , b n ∈ I (a,b) and c n , d n ∈ I (c,d) for all n 1, Proof. We prove the first assertion (the second is similar). We have by (3) and hence which by (5) implies (1).
Proof of Proposition 4.3. We begin by proving that the conditions are equivalent.
First observe that for every R ′ in R(X) such that R ′ ⊂ R, we have by (2) that By density of X in ∂H 2 we can now take an increasing sequence of rectangles R n in R(X) with union R such that R n ⊂R n+1 . We have by (2) that in particular by σ-additivity of µ we have µ(R) = lim n µ(R n ) = lim n r(R n ).
We now prove that (3)  Let now R n be a decreasing sequence of open rectangles in R(X) with intersection R, such that R n+1 ⊂ R n . Then µ(R) = lim n µ(R n ), and by (3) we have r(R n ) µ(R n−1 ) and hence µ(R) = lim n r(R n ).
We finally check that (4) implies (1). Consider any rectangle R in R(X). As r(R) r(R ′ ) for all R ′ containing R, taking infimum on R ′ containing R in their interior we get by (4) that r(R) µ(R). Now write the open rectangleR as a increasing unionR = ∪ ↑ R n of closed rectangles R n in R(X) with R n ⊂R n+1 .
We now prove the existence of µ satisfying (2). The strategy of the construction of µ is to use the finitely additive function r to define the integral of compactly supported continuous functions. This leads by the Riesz representation theorem to a Radon measure µ.
A simple function is a linear combination g = n i=1 α i χ R i of characteristic functions of rectangles R i in R(X). Define E(g) by The additivity property of r on R(X) (Proposition 4.1) shows that E(g) is independent of the representation of g as linear combination of characteristic functions of proper rectangles in R(X). It implies that if g 1 and g 2 are simple functions, then This property also shows that if g 1 , g 2 are simple and g 1 g 2 , then E(g 1 ) . We now prove that µ satisfies (2). Let R, R ′ be rectangles with R ′ ⊂R. As there is a continuous function f with compact support such that

Uniqueness comes from (3), as the class of proper open rectangles in (∂H
is stable under finite intersection and generates the Borel σ-algebra. In the following proposition we will use the well known fact that if γ is a hyperbolic element representing a closed geodesic c ⊂ Σ, and µ is a geodesic current, since I (γ + ,γ − ) × I (x,γx] is a Borel fundamental domain for the γ -action on I (γ + ,γ − ) × I (γ − ,γ + ) , the intersection i(µ, δ c ) can be computed as Proof. In the notation of Proposition 3.11 (see also Figure 1) we have that and also where x ∈ I (γ − ,γ + ) ∩ X is arbitrary.
For any x ∈ I (γ − ,γ + ) ∩ X and n ∈ N we have where the inequality follows from Proposition 4.3 (1). By (6) this implies that Next we have: Using Lemma 4.9 below this equals which, again by Proposition 4.3 (1), implies Then it follows from (7) that and thus for any x ∈ I (γ − ,γ + ) ∩ X.
4.3. The current depends continuously on the crossratio. The vector space CR(X) of crossratios on X is a topological vector space for the topology of pointwise convergence and the space CR + (X) of positive crossratios is a closed convex cone in it. We observe moreover that the map from positive crossratios to the space C(Σ) of geodesic currents is surjective.
In fact, if µ is a geodesic current, one verifies using the regularity of µ that µ [ · , · , · , · ] + µ = µ for the crossratio [ · , · , · , · ] + µ of Example 3.4. Let H Γ denote the subset of ∂H 2 consisting of the fixed points of hyperbolic elements in Γ . Recall that if γ ∈ Γ is hyperbolic we let γ + and γ − denote respectively the attractive and the repulsive fixed point of γ. For every a ∈ H Γ , we choose γ such that a = γ − and we denote by a the point γ + .
The following simple lemma is crucial: We call a Γ -invariant subset S Γ ⊂ H Γ symmetric if ξ ∈ S Γ whenever ξ ∈ S Γ .
is continuous.
Observe that Proposition 4.10 applies to S Γ = H Γ in particular. In the proof of Proposition 4.10 we will focus on a special subset of S Fix some distance inducing the topology on (∂H 2 ) (2) , fix ǫ > 0 and let δ > 0 be such that if R ⊂ (∂H 2 ) (2) is any closed rectangle of diameter diam(R) < δ,

Fix now a cover of supp(f) by closed rectangles
(2) the interiors of the rectangles are pairwise disjoint; ( Observe that for every 1 i N, the quadruple (x i , z i , w i , y i ) is good. As a result, we have (see Lemma 4.11) It follows then that From these inequalities, the assumption that lim n [ · , · , · , · ] n = [ · , · , · , · ] and (5), we deduce that and hence The second case cannot happen since g is in the support of µ. As a result µ is purely atomic with N-valued atoms.
We now show that all geodesics in the support of µ are either closed or connect two cusps. Let (a, b) ∈ (∂H 2 ) (2) be such an atom. Then Γ · (a, b) meets every compact subset of (∂H 2 ) (2) in only finitely many points. As a result, if g ⊂ H 2 is the geodesic connecting (a, b), pr(g) ⊂ Σ is a closed subset where, as always, pr : H 2 → Σ denotes the universal covering map. Thus either g corresponds to a periodic geodesic, or pr(g) is a geodesic connecting two cusps.
There is a compact subset K ⊂ Σ such that every biinfinite geodesic, as well as every closed geodesic, meets K. Thus if A denotes the set of atoms of µ there is a compact subset C ⊂ (∂H 2 ) (2) such that for all a ∈ A, Γ · a ∩ C = ∅. This implies that µ = c∈F n c δ c where F is a finite set of geodesics either periodic or connecting to cusps, δ c is the geodesic current corresponding to c, and n c ∈ N * . 4.5. Restriction to a subsurface. We conclude the section discussing how the construction of the geodesic current associated to a positive crossratio behaves with respect to restriction to subsurfaces. This will be useful in the study of maximal representations.
Let Σ ′ ⊂ Σ be a subsurface with geodesic boundary. Let G(Σ ′ ) ⊂ (∂H 2 ) (2) be the set of geodesics whose projection lies in the interiorΣ ′ of Σ ′ . If µ is a current on Σ, we define µ |Σ ′ ∈ C(Σ) by We write i(µ, ∂Σ ′ ) = 0 when i(µ, c) = 0 for every boundary component c of Σ ′ . This is the case precisely when no geodesic in the support of µ intersects ∂Σ ′ ; thus in that case we have for every closed geodesic c contained in Σ ′ .
We now choose a finite area hyperbolization Σ 0 = Γ 0 \H 2 ofΣ ′ and a corresponding identification h : Γ 0 → π 1 (Σ ′ ) < Γ . We denote by φ : ∂H 2 → ∂H 2 an injective, monotone h-equivariant map. This is a quasi-conjugacy that opens all the cusps corresponding to geodesic boundary components of Σ ′ . It follows from this discussion that: Proposition 4.13. Let µ be a current on Σ. Then defines a current µ 0 = φ * µ on Σ 0 , that we will call the current induced by µ on Σ 0 .
(5) Assume that i(µ, ∂Σ ′ ) = 0. If µ is the current associated to a positive crossratio b ∈ CR(H Γ ), then µ 0 is the current associated to the positive crossratio

Equivariant tree embeddings
In this section we discuss barycenter maps compatible with positive crossratios and prove Theorem 1.9. In §5.2 we discuss a first class of actions to which Theorem 1.9 applies: framed actions on trees.

Actions with compatible crossratio and barycenter.
Let ρ : Γ → Isom(X) be an action by isometries on a metric space (X, d X ) and X ⊂ ∂H 2 be a non-empty Γ -invariant subset. Proof. We split the proof in three easy steps.
(1) Given three connected components I 1 , I 2 , β(a, b, c) is independent of the choices a ∈ I 1 ∩ X, b ∈ I 2 ∩ X and c ∈ I 3 ∩ X.
We distinguish two cases.   β(a, b, c) , which shows the assertion.
(3) We finish now the proof of the proposition. Let {a, b, c, a ′ , b ′ , c ′ } ⊂ X with a ∈ I 1 , b ∈ I 2 , c ∈ I 3 , a ′ ∈ I ′ 1 , b ′ ∈ I ′ 2 and c ∈ I ′ 3 , where I 1 , I 2 , I 3 and I ′ 1 , I ′ 2 , I ′ 3 are distinct connected components of ∂H 2 R(∞). We can assume, up to reordering the indices that I j = I k for j = k. Then it follows from (1) and (2) that Proof. Let (x 1 , y 1 ) be the endpoints of the geodesic in ∂R 1 separating R 1 from R 2 and (x 2 , y 2 ) the endpoints of the geodesic in ∂R 2 separating R 2 from R 1 , ordered so that (x 1 , y 1 , x 2 , y 2 ) ∈ X [4] . Choose a, b ∈ I (x 1 ,y 1 ) in different connected components of ∂H 2 R 1 (∞) and c, d ∈ I (x 2 ,y 2 ) in different connected Since the geodesics bounding R 1 and R 2 are all µ-short, we have If now p i ∈ R i , the set of leaves in L that intersect the segment (p 1 , p 2 ) is exactly the set of leaves in L that separate R 1 from R 2 . This is also the same as the set of leaves in L that connect I [d,a] to I [b,c] . The assertion then follows from the above considerations, recalling that d µ (R 1 , R 2 ) is the measure of this set,

The geometry of the Siegel spaces over real closed fields
The goal of this section is to recall facts about the geometry of Siegel spaces over real closed fields needed to show that maximal framed actions give rise to a positive crossratio and admit a compatible barycenter (Lemma 6.6 and Proposition 6.7).
6.1. Real closed fields. Recall that an ordered field is a field F endowed with a total order relation satisfying: (1) if x y then x + z y + z for all z ∈ F; (2) if 0 x and 0 y, then 0 xy.
The fields Q and R with their usual order are examples; while some fields admit no ordering, like C, others admit many, like R(X).
A basic fact is that every ordered field F admits a real closure F r , that is a maximal algebraic extension of F to which the order extends. Such a real closure is then unique up to a unique F-isomorphism. An ordered field F is then real closed if the ordering does not extend to any proper algebraic extension. Two useful characterizations are the following: (1) the field F is ordered and F(ı) is algebraically closed, with ı = √ −1; (2) the field F is ordered, every positive element is a square and any odd degree polynomial has a root.
Real closed fields have the same first order logic as the field R of the reals.
An important consequence that is implicit in most of the geometric properties of the Siegel space we use, is that any symmetric matrix with coefficient in a real closed field is orthogonally similar to a diagonal one.
Example 6.2. The following are examples of ordered fields: (1) The field R of real numbers and the field Q r of real algebraic numbers; both are real closed.
(2) Let ω be a non-principal ultrafilter on N. The quotient R ω of the ring R N by the equivalence relation (x n ) ∼ (y n ) if ω({n : x n = y n }) = 1, ordered in such a way that positive elements are the classes of the sequences (x n ) such that ω({n : x n > 0}) = 1, is a real closed field called the field of the hyperreals. It does not admit any order compatible R-valued valuation.
(3) Let σ ∈ R ω be a positive infinitesimal, that is σ can be represented by a sequence (σ n ) n 0 with lim σ n = 0 and σ n > 0. Then is a valuation ring with maximal ideal The quotient R ω,σ := O σ /I σ is a real closed field, called the Robinson field.
It admits an order compatible valuation where (x n ) n 0 represents x, that leads to a non-Archimedean norm (4) Let G be a totally ordered Abelian group and let extends to R(x, y). In this way we obtain an order on R(x, y) for which P ∈ R(x, y) is positive if and only if for some ǫ > 0, P(t, t α ) > 0 for all t ∈ (0, ǫ).
6.2. The Siegel upper half space and the space B F n . Let F be a real closed field, and ı be a square root of −1. Endow V = F 2n with the standard symplectic form where x i , y i ∈ F n . The vector space Sym n (F) of symmetric matrices admits a partial order defined by setting The Siegel upper half space is the semialgebraic set transitively. Of course this action descends to an action of PSp(2n, F). The stabilizer of ıId n ∈ S n F in Sp(2n, F) is If F = R, then S n R is the symmetric space associated to PSp(2n, R). If, instead, the real closed field F is endowed with an order compatible non-Archimedean valuation v, then PSp(2n, F) acts by isometries on a v(F)-metric space B F n : a metric quotient of S n F whose construction we now recall. See [BIPP21c,BIPP21a] for generalizations of this construction.
On S n F we define an multiplicative F-valued distance function as follows. Since F is real closed, any pair (Z 1 , Z 2 ) with Z i ∈ S n F , for i = 1, 2, is PSp(2n, F)congruent to a unique pair (ı Id n , ıD), where D = diag(d 1 , . . . , d n ), d 1 · · · d n 1 in F. We then set Proposition 6.3. D is a PSp(2n, F)-invariant multiplicative distance function on S n F , namely, for all Z 1 , Z 2 , Z 3 ∈ S n F , (MD1): D(Z 1 , Z 2 ) ∈ F 1 , with equality of and only if Z 1 = Z 2 ; Proof. (MD1) and (MD2) are clear. We consider the standard action of Sp(2n, F) on W = ∧ n (F 2n ), endowed with the standard scalar product, which is Kinvariant as K ⊂ O(2n). For a = diag(a 1 , . . . , a n , a −1 1 , . . . , a −1 n ) with a 1 · · · a n 1 in F, we easily see that (the biggest eigenvalue of a in W). For g ∈ Sp(2n, F) the operator norm of g on W is given by Since g = kak ′ for some k, k ′ in K and a as before (by the Cartan decomposition), and D(ıId n , g * ıId n ) = D(ıId n , a * ıId n ) = 2|||a||| = 2|||g||| , (MD3) follows from submultiplicativity of the operator norm and transitivity of the action of Sp(2n, F).
On S n F we define an associated pseudo-distance d 1 as follows.
The triangle inequality for d 1 comes from (MD3). We denote by B F n the metric quotient of S n F with respect to the pseudo-distance d 1 . We will not need it, but note in passing that B F n can be identified with the quotient PSp(2n, F)/ PSp(2n, U), where U := {x ∈ F : x v 1}.
6.3. Embedding in K-Lagrangians. In the classical case, the Borel embedding of S n R into the complex Grassmannian provides a way to endow S n R with structures defined on the Grassmannian, such as, for example, a crossratio. We recall from [BP17] the analogous picture in the case of a general real closed field.
Let K = F(ı) be the algebraic closure of F and let us also denote by , the K-linear extension of the standard symplectic form to K 2n and by σ : K 2n → K 2n the complex conjugation. Given matrices Z 1 , Z 2 ∈ M n (K), we will denote by Z 1 Z 2 the subspace of K 2n generated by the column vectors. We denote by and sends Sym n (F) to 6.4. Maximal triples and intervals. We associate to a triple (ℓ 1 , ℓ 2 , ℓ 3 ) of pairwise transverse Lagrangians in L(F 2n ) the quadratic form Q (ℓ 1 ,ℓ 2 ,ℓ 3 ) on ℓ 1 defined by where v ′ ∈ ℓ 3 is the unique vector such that v + v ′ ∈ ℓ 2 . If ℓ = X Id n and ℓ ′ = X ′ Id n are pairwise transverse, then in the coordinates (10) For X, X ′ ∈ Sym n (F) with X ≪ X ′ , we will also denote by I (X,X ′ ) the set and set I (X,∞) := {Y ∈ Sym n (F) : X ≪ Y} .
6.5. Crossratios. In this subsection we recall the endomorphism valued crossratio from [BP17,§ 4.1] on quadruples of Lagrangians. This, together with a maximal framing, will allow us in § 7.1 to associate to any maximal framed representation ρ a positive crossratio as in § 3.
6.6. F-tubes and orthogonal projections. If ℓ, ℓ ′ ∈ L(F 2n ), the F-tube determined by ℓ, ℓ ′ is the semi-algebraic subset of S n F given by the equation where Z is the complex conjugate of Z, [BP17, § 4.2].
If F = R and n = 1, the R-tube Y ℓ,ℓ ′ is the geodesic between ℓ and ℓ ′ while, for n 1, Y ℓ,ℓ ′ is a symmetric subspace of S n R that is a Lagrangian submanifold and is isometric to the symmetric space associated to GL(n, R). In general, for all g ∈ PSp(2n, F), and if we denote We will often write Y 0,∞ for Y ℓ 0 ,ℓ ∞ .
In the case of Y 0,∞ the map pr Y 0,∞ is given by In view of (12), this implies in particular that the restrictions of pr Y ℓ 1 ,ℓ 3 to I (ℓ 1 ,ℓ 3 ) and I (ℓ 3 ,ℓ 1 ) are both bijective.
First we need to establish a formula for the (pseudo-)distance in S F 1 : Lemma 6.8. If z 1 = x 1 + ıy 1 , z 2 = x 2 + ıy 2 ∈ S F 1 , then Proof. Recall that for a general real closed field F, if z 1 , z 2 ∈ S F 1 , 2y 1 y 2 . Since F is non-Archimedean, then for a, b ∈ F, a 0, b 0, we have Hence where we took into account that n v = 1 for any n ∈ Z \ {0}.

Applications to maximal framed representations
In this section we prove Theorem 1.
It follows from Proposition 6.5 that [ · , · , · , · ] ρ is a positive crossratio on H Γ , and hence (Proposition 4.3 and 4.8) there is a geodesic current µ ρ such that for every closed geodesic c represented by a hyperbolic element γ ∈ Γ .
7.4. Lamination type currents and Theorem 1.5. Let F be non-Archimedean.
It follows from Proposition 6.7 that β is indeed a barycenter according to Definition 5.1 and from Lemma 6.6 that it is compatible with the crossratio [ · , · , · , · ] ρ defined in § 7.1.
It is now easy to see that the Lagrangians ϕ(x i ) ⊂ F 2n are defined over L ∩ F, as a result we can represent them by X i Id n with X i ∈ Sym n (L ∩ F), which implies that det R(ϕ(x 1 ), ϕ(x 2 ), ϕ(x 3 ), ϕ(x 4 )) ∈ (L ∩ F) × . We conclude using [Lan02, XII §4 Proposition 12] which says that the index of Λ in v((L ∩ F) × ) is at most (8n)!. This concludes the proof.
In particular, if Q(ρ) has discrete valuation, we can assume, up to rescaling the valuation, that the crossratio [x 1 , x 2 , x 3 , x 4 ] ρ is integer valued. Theorem 1.4 is therefore a direct application of Proposition 4.12.

Examples of maximal framed representations
In this section we collect several interesting examples of maximal framed representations over non-Archimedean real closed fields. its framing is defined on ∂H 2 , and Theorem 1.2 applies.
This construction is closely related to asymptotic cones, as we now recall.
Denoting by d the Sp(2n, R)-invariant Riemannian distance on the Siegel nspace, we say that a sequence of scales (λ k ) k∈N ∈ (R >0 ) N is adapted (to the sequence (ρ k ) k∈N ) if for one, and hence every, finite generating set S ⊂ Γ (14) lim ω max γ∈S d ρ k (γ)ıId n , ıId n λ k < +∞ .
We obtain then an action ω ρ λ : Γ → Isom( ω X λ ) , on the asymptotic cone ω S λ of the sequence of pointed metric spaces given by (S n R , ıId n , d λ k ). If we set σ := (e −λ k ) k 1 ∈ R ω , then the asymptotic cone ω X λ can be identified with the metric space B R ω,σ n and, under this identification, ω ρ λ corresponds to ρ ω,σ (see for example [Par12]). SL(2, F). Let F be a real closed field with an order compatible non-Archimedean valuation, and let T F ⊃ B F 1 be the R-tree associated to SL 2 (F). Then P 1 (F) identifies with a subset of ∂ ∞ T F and the restriction to P 1 (F) of the crossratio of ∂ ∞ T F is the standard crossratio [ · , · , · , · ] F in P 1 (F).
8.3. Unipotent representations of the thrice punctured sphere. Let Γ < PSL(2, R) be the (unique up to conjugation) lattice such that Γ \H 2 is the thrice punctured sphere. Then Γ admits a presentation Γ = c 1 , c 2 , c 3 : c 3 c 2 c 1 , where c 1 , c 2 , c 3 are parabolic elements representing the three inequivalent cusps of Γ . Already in this elementary example we are able to illustrate interesting features. For every α ∈ R we construct maximal framed representations ρ α : Γ → Sp(4, H(R)), where H(R) is the Hahn field with exponents R (see Example 6.2(4)), that have the following properties: (1) for α 1/2 the corresponding length functions γ → L(ρ α (γ)) are not proportional and hence the corresponding currents µ ρ α are distinct in the space of projectivized currents; (2) for α ∈ Q the associated geodesic current µ ρ α is a multicurve.
To this end we use the explicit coordinates obtained by Strubel on the set of Sp(2n, R)-conjugacy classes of maximal representations of Γ into Sp(2n, R).
Moreover every maximal representation of Γ into Sp(2n, R) is conjugate to a ρ X for X ∈ R and Sp(2n, R)-conjugacy classes of maximal representations correspond to O(n)-conjugacy classes in R for the diagonal conjugation action of O(n).
For the computation of the length function L it is not difficult to see that if g ∈ Sp(4, F), where F is real closed non-Archimedean with an order compatible valuation, then L(g) = −v(T (g)) .