Twisted Patterson-Sullivan densities on nilpotent covers for Anosov subgroups
Abstract
Let Γ<SL(d,ℝ) be a torsion-free, Zariski-dense Borel-Anosov subgroup, and let φ be a linear functional positive on the limit cone of Γ. For every χ∈ Hom(Γ, ℝ), we construct twisted (φ,χ)-conformal densities and prove their existence, uniqueness and ergodicity. For every non-trivial normal subgroup Γ0lhdΓ with nilpotent quotient, we classify the ergodic φ-conformal densities of Γ0 in terms of characters of Γ/Γ0. Up to scaling, they are naturally parametrized by Hom(Γ/Γ0,ℝ), and their dimensions are precisely the critical exponents of the corresponding twisted Poincaré series. We also prove a shadow lemma for twisted densities of normal subgroups, with estimates uniform over densities with fixed parameters, and a twisted Hopf-Tsuji-Sullivan theorem.
Related
- Brauer character degrees and nilpotent subgroups
- Finite groups with simple or p-nilpotent maximal subgroups
- Wielandt subgroups of nilpotent residuals and Fitting-height bounds
- Formalization of Sullivan's No Wandering Domains Theorem in Lean
- The Calabi-Yau and Sullivan conjectures for minimal surfaces with symmetries
BibTeX
@misc{gongopadhyay2026twisted,
title = {{Twisted Patterson-Sullivan densities on nilpotent covers for Anosov subgroups}},
author = {Krishnendu Gongopadhyay and Neelanjan Mondal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.27125},
eprint = {2609.27125},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27125},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Twisted Patterson-Sullivan densities on nilpotent covers for Anosov subgroups
AU - Krishnendu Gongopadhyay
AU - Neelanjan Mondal
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.27125
UR - https://arxiv.org/abs/2609.27125
AB - Let Γ<SL(d,ℝ) be a torsion-free, Zariski-dense Borel-Anosov subgroup, and let φ be a linear functional positive on the limit cone of Γ. For every χ∈ Hom(Γ, ℝ), we construct twisted (φ,χ)-conformal densities and prove their existence, uniqueness and ergodicity. For every non-trivial normal subgroup Γ_0lhdΓ with nilpotent quotient, we classify the ergodic φ-conformal densities of Γ_0 in terms of characters of Γ/Γ_0. Up to scaling, they are naturally parametrized by Hom(Γ/Γ_0,ℝ), and their dimensions are precisely the critical exponents of the corresponding twisted Poincaré series. We also prove a shadow lemma for twisted densities of normal subgroups, with estimates uniform over densities with fixed parameters, and a twisted Hopf-Tsuji-Sullivan theorem.
ER -