Rigidity at infinity for the Borel function of the tetrahedral reflection lattice

Abstract

Let Γ be a non-uniform lattice of P SL(2, C). To every representation ρ : Γ → P SL(n, C) it is possible to associate a numerical invariant β n (ρ), called Borel invariant, which is constant on the P SL(n, C)-conjugancy class of the representation ρ and hence defines a function on the character variety X(Γ, P SL(n, C)). This function is continuous with respect to the topology of the pointwise convergence and it satisfies a strong rigidity property: it holds |β n (ρ)| ≤ n+13 Vol(Γ\H 3 ) for every representation ρ : Γ → P SL(n, C) and we have the equality if and only if the representation ρ is conjugated either to π n • i or to π n • i, where i : Γ → P SL(2, C) is the standard lattice embedding and π n : P SL(2, C) → P SL(n, C) is the irreducible representation.We partially answer to a conjecture proposed in [Gui17] when Γ the reflection group associated to a regular ideal tetrahedron. More precisely let Γ 0 < P SL(2, C) be a torsion-free subgroup of Γ and let ρ k : Γ 0 → P SL(n, C) be a sequence of asymptotically maximal representations, that is lim k→∞ β n (ρ k ) = n+13 Vol(Γ 0 \H 3 ). Assume there exists a measurable map ϕ k : P 1 (C) → F (n, C) which is ρ kequivariant. We prove that there must exists a sequence (g k ) k∈N where g k ∈ P SL(n, C) such that lim k→∞ g k ρ k (γ)g −1 k = (π n • i)(γ), for every γ ∈ Γ 0 .ϑ * (B n )(F, G) = B n ((H 0 , w 0 ), . . . , (H 3 , w 3