Infinitely many graph manifolds with unique geometrical piece that admit arbitrarily many Anosov flows
Abstract
Anosov flows have a long and rich history, firstly motivated by the study of geodesic flows in negative curvature surface by Anosov and Sinai. Not every closed manifold admits an Anosov flow for well-known reasons: the fundamental group of a 3-manifold M admitting an Anosov flow must have exponential growth, and M must be universally covered by ℝ3. Nevertheless, there are sufficient mechanisms for constructing distinct Anosov flows on admissible 3-manifolds, such as Dehn-Goodman-Fried surgery or playing with hyperbolic building blocks. A central problem in the field has been to determine the number of Anosov flows that can be supported by a single manifold. The question of whether there exists an infinite set of pairwise non-equivalent Anosov flows on a 3-manifold remains open to this day. However, there are several papers proving the existence of a manifold Mn that admits n pairwise inequivalent Anosov flows for any natural number n. In all known examples, the manifolds Mn are composed of several geometric pieces. In the present paper, we prove the existence of a countable number of graph manifolds Mk,n, k ∈ ℕ with a single geometric piece, each of which admits n pairwise non-equivalent transitive Anosov flows. All previously known constructions of different flows on the same graph manifold were based on gluing geodesic flows. The nature of the flows constructed in this paper is completely different; they are constructed from a single hyperbolic plug, which is a suspension over a Morse-Smale diffeomorphism on a surface.
Related
- Finiteness of Anosov and pseudo-Anosov flows on hyperbolic 3-manifolds
- Construction of Anosov flows on fibered hyperbolic 3-manifolds
- On the Finiteness of Anosov flows on $3$-manifolds via Contact Geometry
- Regularity of Manhattan manifolds and exact dimensionality for relatively Anosov groups
- Contact 3-manifolds with infinitely many rational homology ball Stein fillings
BibTeX
@misc{pochinka2026infinitely,
title = {{Infinitely many graph manifolds with unique geometrical piece that admit arbitrarily many Anosov flows}},
author = {O. Pochinka and V. Shmukler},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27706},
eprint = {2608.27706},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27706},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Infinitely many graph manifolds with unique geometrical piece that admit arbitrarily many Anosov flows
AU - O. Pochinka
AU - V. Shmukler
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27706
UR - https://arxiv.org/abs/2608.27706
AB - Anosov flows have a long and rich history, firstly motivated by the study of geodesic flows in negative curvature surface by Anosov and Sinai. Not every closed manifold admits an Anosov flow for well-known reasons: the fundamental group of a 3-manifold M admitting an Anosov flow must have exponential growth, and M must be universally covered by ℝ^3. Nevertheless, there are sufficient mechanisms for constructing distinct Anosov flows on admissible 3-manifolds, such as Dehn-Goodman-Fried surgery or playing with hyperbolic building blocks. A central problem in the field has been to determine the number of Anosov flows that can be supported by a single manifold. The question of whether there exists an infinite set of pairwise non-equivalent Anosov flows on a 3-manifold remains open to this day. However, there are several papers proving the existence of a manifold M_n that admits n pairwise inequivalent Anosov flows for any natural number n. In all known examples, the manifolds M_n are composed of several geometric pieces. In the present paper, we prove the existence of a countable number of graph manifolds M_k,n, k ∈ ℕ with a single geometric piece, each of which admits n pairwise non-equivalent transitive Anosov flows. All previously known constructions of different flows on the same graph manifold were based on gluing geodesic flows. The nature of the flows constructed in this paper is completely different; they are constructed from a single hyperbolic plug, which is a suspension over a Morse-Smale diffeomorphism on a surface.
ER -