Dominating surface-group representations via Fock-Goncharov coordinates

2024/05/24 by Barman, Pabitra, Gupta, Subhojoy
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2405.15378

Abstract

Let S be a punctured surface of negative Euler characteristic. We show that given a generic representation ρ:π₁(S) → PSL_n(ℂ), there exists a positive representation ρ₀:π₁(S) → PSL_n(ℝ) that dominates ρ in the Hilbert length spectrum as well as in the translation length spectrum, for the translation length in the symmetric space \mathbbX_n= PSL_n(ℂ)/PSU(n). Moreover, the ρ₀-lengths of peripheral curves remain unchanged. The dominating representation ρ₀ is explicitly described via Fock-Goncharov coordinates. Our methods are linear-algebraic, and involve weight matrices of weighted planar networks.

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