Positive crossratios, barycenters, trees and applications to maximal representations

Abstract

Let Σ := Γ \H be a finite volume hyperbolic surface. We associate to any Γ -invariant positive crossratio defined on a dense subset of ∂H a canonical geodesic current µ. This produces a Liouville current µ ρ for every maximal framed representation ρ : Γ → Sp(2n, F) for a real closed field F, which we prove being a weighted multicurve as soon as the field generated by matrix coefficients of ρ has discrete valuation. When µ is a measured lamination, for every Γ -action on a metric space X admitting a compatible barycenter, we construct an isometric embedding in X of the R-tree T(µ) associated to µ. This applies, for instance, to the ρ(Γ )-action on a natural metric space associated to Sp(2n, F).

The real spectrum compactification of character varieties: characterizations and applications

“…Let now as in [18] H Γ → [0, ∞) by the formula [ξ 1 , ξ 2 , ξ 3 , ξ 4 ] := log b 2 cr k (φ(ξ 1 ), φ(ξ 2 ), φ(ξ 3 ), φ(ξ 4 ))cr k (φ(ξ 3 ), φ(ξ 4 ), φ(ξ 1 ), φ(ξ 2 )) then we obviously have: (41) [ξ 1 , ξ 2 , ξ 3 , ξ 4 ] = [ξ 3 , ξ 4 , ξ 1 , ξ 2 ] for all (ξ 1 , ξ 2 , ξ 3 , ξ 4 ) ∈ H…”

“…RSp cl ; denote by L ρ the field generated by the matrix coefficients of ρ, and assume that Λ 0 := v(L ρ ) is a discrete subgroup of R. The same argument as in [18,Theorem 7.3] ensure that for all (ξ 1 , ξ 2 , ξ 3 , ξ 4 ) ∈ H…”

“…An element g ∈ G(F) is called I-proximal over F if it admits an attracting fixed point in F I (F).Let Pr(I, F) denote the subset of G(F) consisting of I-proximal elements. Then(18) Pr(I,F) = {g ∈ G(F)| α(J F (g)) > 1 ∀α ∈ I}.As a result, Pr(I, F) is semialgebraic and coincides with the F-extension Pr(I, K) F of Pr(I, K). The following follows then from Proposition 4.9 and the transfer principle: Let G be a semisimple connected algebraic group defined over a real closed field K; F ⊃ K real closed;…”

“…It remains to show the continuity of PC. As in[18, Section 4.3], let CR(H Γ ) be the topological vector space of cross-ratios on H Γ with the topology of pointwise convergence, and CR + (H Γ ) the closed convex cone of positive ones.To everyρ ∈ R red Γ (Γ , G F ) representing a closed point in (ΞT) RSp clwe can associate a positive cross-ratio[ • , • , • , • ] ρ,b ∈ CR + (H Γ )where we indicate the dependence on ρ and the big element b; this cross-ratio doesn't vanish identically since its periods give the length function ℓ k ([ρ]). Thus we obtain a well defined map:(ΞT) RSp cl → P(CR + (H Γ )) := R >0 CR + (H Γ ) \ {0} .The map PC is the composition of this map with the mapP(CR + (H Γ )) → P(C(Σ))which to a positive cross-ratio associates its geodesic current.…”

Weyl chamber length compactification of the ${\rm PSL}(2,\mathbb R)\times{\rm PSL}(2,\mathbb R)$ maximal character variety

“…Maximal representations share many features with the subset of the character variety consisting of Hitchin representations, which are defined for G real split, and whose character variety we denote by Ξ Hit (Γ, G). In [BIPP19,BIPP21] we used geodesic currents to study the Weyl chamber length compactification X WL (Γ, G) of X (Γ, G), where X (Γ, G) denotes either the Hitchin or the maximal character variety. We showed that, as soon as the group G has higher rank, the mapping class group admits a non-empty open domain of discontinuity for its action on the boundary ∂ WL X (Γ, G), the so-called positive systole subset; moreover for a dense subset of boundary points, the associated length function can be computed as intersection with a weighted multicurve.…”