Dominating surface-group representations via Fock-Goncharov coordinates

Includes bibliographical references

PhD;2023;Mathematics

Let Sg,k be a connected oriented surface of negative Euler characteristic and ρ1, ρ2 : π1(Sg,k) → P SL2(C) be two representations. ρ2 is said to dominate ρ1 if there exists λ ≤ 1 such that `ρ1 (γ) ≤ λ•`ρ2 (γ) for all γ ∈ π1(Sg,k), where `ρ(γ) denotes the translation length of ρ(γ) in H3 . In 2016, Deroin-Tholozan showed that for a closed surface S and a non-Fuchsian representation ρ : π1(Sg,k) → P SL2(C), there exists a Fuchsian representation j : π1(Sg,k) → P SL2(R) that strictly dominates ρ. In 2023, Gupta-Su proved a similar result for punctured surfaces, where the representations lie in the same relative representation variety. Here, we generalize these results to the case of higher-rank representations. For a representation ρ : π1(Sg,k) → P SLn(C), the Hilbert length of a curve γ ∈ π1(Sg,k) for n > 2 is defined as `ρ(γ) := ln λn λ1 , where λn and λ1 are the largest and smallest eigenvalues of ρ(γ) in modulus respectively. We show that for any generic representation ρ : π1(Sg,k) → P SLn(C), there is a Hitchin representation j : π1(Sg,k) → P SLn(R) that dominates ρ in the Hilbert length spectrum. The proof uses Fock-Goncharov coordinates on the moduli space of framed P SLn(C) representation. Weighted planar networks and the Collatz-Wielandt formula for totally positive matrices play a crucial role. Let Xn be the symmetric space of P SLn(C). The translation length of A ∈ P SLn(C) in Xn is given as τ (A) = Xn i=1 log |λi(A)| 2 , where λi(A) are the eigenvalues of A. We show that the same j dominates ρ with respect to the translation length at the origin as well. Lindstr¨om’s Lemma for planar networks and Weyl’s Majorant Theorem are some of the key ingredients of the proof. In both cases, if Sg,k is a punctured surface, then j lies in the same relative representation variety as ρ