Non-maximal Anosov representations from surface groups to $\mathrm{SO}_0(2,3)$

TL;DR

This paper proves that certain stable Higgs bundle representations of surface groups into SO(2,3) are almost dominated, generalizing Filip's work and answering a question by Collier, Tholozan, and Toulisse.

Contribution

It establishes a new class of non-maximal Anosov representations from surface groups to SO(2,3) via Higgs bundles, extending previous results and addressing open questions.

Findings

Stable α1-cyclic parabolic SO(2,3)-Higgs bundles are almost dominated.

Generalization of Filip's weight 3 variation of Hodge structures.

Answers to open questions on non-maximal Anosov representations.

Abstract

We prove the representation given by a stable $\alpha_1$-cyclic parabolic $\mathrm{SO}_0(2,3)$-Higgs bundle through the non-Abelian Hodge correspondence is $\{\alpha_2\}$-almost dominated. This is a generalization of Filip's result on weight $3$ variation of Hodge structures and answers a question asked by Collier, Tholozan and Toulisse.