Geometric finiteness in paracomplex hyperbolic spaces
Abstract
We develop a framework for studying discrete subgroups of PGL(d+1,ℝ) via the paracomplex hyperbolic space ℍτd, a rank-$1$ pseudo-Riemannian symmetric space. We characterize projective transverse, relatively Anosov, and Anosov subgroups in terms of properly discontinuous, geometrically finite, and convex-cocompact actions respectively on their weak hulls, which are canonical flow spaces in the spacelike unit tangent bundle of ℍτd. A key ingredient is the construction of a Busemann-type horofunction on the spacelike unit tangent bundle with the properties needed to describe cuspidal geometry. We further prove for relatively Anosov subgroups that the geodesic flows on their weak hulls are uniformly hyperbolic, giving a relative analogue of the Axiom A property.
Related
- Finiteness of Anosov and pseudo-Anosov flows on hyperbolic 3-manifolds
- Finiteness of closed arithmetic hyperbolic surface bundles
- Finiteness of open substacks with proper moduli spaces
- Geometric Detection and Finiteness of Crystalline Local Systems
- Finiteness and boundedness of positive monotone Hamiltonian GKM3 spaces
BibTeX
@misc{wang2026geometric,
title = {{Geometric finiteness in paracomplex hyperbolic spaces}},
author = {Tianqi Wang and Zhufeng Yao and Tengren Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.07628},
eprint = {2609.07628},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07628},
}
RIS
TY - UNPB
TI - Geometric finiteness in paracomplex hyperbolic spaces
AU - Tianqi Wang
AU - Zhufeng Yao
AU - Tengren Zhang
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2609.07628
UR - https://arxiv.org/abs/2609.07628
AB - We develop a framework for studying discrete subgroups of PGL(d+1,ℝ) via the paracomplex hyperbolic space ℍ_τ^d, a rank-$1$ pseudo-Riemannian symmetric space. We characterize projective transverse, relatively Anosov, and Anosov subgroups in terms of properly discontinuous, geometrically finite, and convex-cocompact actions respectively on their weak hulls, which are canonical flow spaces in the spacelike unit tangent bundle of ℍ_τ^d. A key ingredient is the construction of a Busemann-type horofunction on the spacelike unit tangent bundle with the properties needed to describe cuspidal geometry. We further prove for relatively Anosov subgroups that the geodesic flows on their weak hulls are uniformly hyperbolic, giving a relative analogue of the Axiom A property.
ER -