Dominating surface-group representations via Fock-Goncharov coordinates

TL;DR

This paper proves that for a generic complex surface group representation, there exists a real positive representation that dominates it in length spectra, explicitly constructed using Fock-Goncharov coordinates and linear algebraic methods.

Contribution

It introduces a method to explicitly construct dominating positive representations for generic complex surface group representations using Fock-Goncharov coordinates.

Findings

Existence of a dominating positive representation for generic complex representations.

Explicit construction of the dominating representation via Fock-Goncharov coordinates.

Preservation of peripheral curve lengths in the dominating representation.

Abstract

Let $S$ be a punctured surface of negative Euler characteristic. We show that given a generic representation $\rho:\pi_1(S) \rightarrow \mathrm{PSL}_n(\mathbb{C})$, there exists a positive representation $\rho_0:\pi_1(S) \rightarrow \mathrm{PSL}_n(\mathbb{R})$ that dominates $\rho$ in the Hilbert length spectrum as well as in the translation length spectrum, for the translation length in the symmetric space $\mathbb{X}_n= \mathrm{PSL}_n(\mathbb{C})/\mathrm{PSU}(n)$. Moreover, the $\rho_0$-lengths of peripheral curves remain unchanged. The dominating representation $\rho_0$ is explicitly described via Fock-Goncharov coordinates. Our methods are linear-algebraic, and involve weight matrices of weighted planar networks.