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Length-convexity and domination for Hitchin representations
Abstract
Let S be a closed oriented surface of genus g≥ 2. For any n≥ 3, let Hitn(S) be the Hitchin component and let Hitn(S)′ be the subset of representations that take values in PSp2m(ℝ) if n=2m and PSO(m,m+1) if n=2m+1. In this note we prove that for any Hitchin representation ρ∈ Hitn(S) there is j ∈ Hitn(S)′ such that ρ dominates j in the Hilbert length spectrum; this confirms a conjecture of Lee-Zhang and, independently, Tholozan who had proved this for n=3. The key step is establishing a convexity of the Hilbert length function in the Bonahon-Dreyer parametrization of Hitn(S); this uses a result from the recent work of Barman-Gupta. We also prove these convexity and domination results for translation length functions and their spectra.
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BibTeX
@misc{gupta2026lengthconvexity,
title = {{Length-convexity and domination for Hitchin representations}},
author = {Subhojoy Gupta},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24606},
eprint = {2610.24606},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24606},
}
RIS
TY - UNPB
TI - Length-convexity and domination for Hitchin representations
AU - Subhojoy Gupta
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24606
UR - https://arxiv.org/abs/2610.24606
AB - Let S be a closed oriented surface of genus g≥ 2. For any n≥ 3, let Hit_n(S) be the Hitchin component and let Hit_n(S)^′ be the subset of representations that take values in PSp_2m(ℝ) if n=2m and PSO(m,m+1) if n=2m+1. In this note we prove that for any Hitchin representation ρ∈ Hit_n(S) there is j ∈ Hit_n(S)^′ such that ρ dominates j in the Hilbert length spectrum; this confirms a conjecture of Lee-Zhang and, independently, Tholozan who had proved this for n=3. The key step is establishing a convexity of the Hilbert length function in the Bonahon-Dreyer parametrization of Hit_n(S); this uses a result from the recent work of Barman-Gupta. We also prove these convexity and domination results for translation length functions and their spectra.
ER -