PreprintCC BY-NC-ND 4.0arXiv
On Line-Hyperplane Structures for Hitchin Representations
Abstract
Let n ≥ 4 and ρ: π1S → PSL(n,ℝ) be a Hitchin representation. We study the topology of a cocompact domain of discontinuity Ωρ in the flag manifold ℱ1,n−1 of line-hyperplane pairs in ℝn defined by Guichard-Wienhard. In particular, we lift Ω to its (ℤ2× ℤ2)-cover Ω̂ in the Stiefel manifold V2(ℝn). The domain Ω̂ is highly connected and smoothly fibers over hyperbolic space ℍ2 with unknown fiber F̂, a closed (2n−5)-manifold. We determine the homeomorphism type of F̂ as well as its diffeomorphism type up to connected sum with a homotopy sphere. The classification of F̂ involves computing both its homology and a certain differential topology invariant needed to invoke Wall's classification of highly connected odd-dimensional manifolds.
Related
- Non-Hitchin Borel Anosov representations from surface groups to SL3kℝ
- Length-convexity and domination for Hitchin representations
- Entropy and domination for quasi-Hitchin representations
- Branched real projective structures on surfaces and geometrisation of representations
- On derived equivalences of exact structures on the category of representations k[x]/(xn)
BibTeX
@misc{evans2026linehyperplane,
title = {{On Line-Hyperplane Structures for Hitchin Representations}},
author = {Parker Evans and Andrea Tamburelli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31567},
eprint = {2609.31567},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31567},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On Line-Hyperplane Structures for Hitchin Representations
AU - Parker Evans
AU - Andrea Tamburelli
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31567
UR - https://arxiv.org/abs/2609.31567
AB - Let n ≥ 4 and ρ: π_1S → PSL(n,ℝ) be a Hitchin representation. We study the topology of a cocompact domain of discontinuity Ω_ρ in the flag manifold ℱ_1,n−1 of line-hyperplane pairs in ℝ^n defined by Guichard-Wienhard. In particular, we lift Ω to its (ℤ_2× ℤ_2)-cover Ω̂ in the Stiefel manifold V_2(ℝ^n). The domain Ω̂ is highly connected and smoothly fibers over hyperbolic space ℍ^2 with unknown fiber F̂, a closed (2n−5)-manifold. We determine the homeomorphism type of F̂ as well as its diffeomorphism type up to connected sum with a homotopy sphere. The classification of F̂ involves computing both its homology and a certain differential topology invariant needed to invoke Wall's classification of highly connected odd-dimensional manifolds.
ER -