Randomized Geodesic Flow on Hyperbolic Groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915767605264384 |
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| author | Kupffer, Luzie Mj, Mahan Mukherjee, Chiranjib |
| author_facet | Kupffer, Luzie Mj, Mahan Mukherjee, Chiranjib |
| contents | Motivated by Gromov's geodesic flow problem on hyperbolic groups $G$, we develop in this paper an analog using random walks. This leads to a notion of a harmonic analog $Θ$ of the Bowen-Margulis-Sullivan measure on $\partial^2 G$. We provide three different but related constructions of $Θ$: 1) by moving the base-point along a quasigeodesic ray 2) by moving the base-point along random walk trajectories 3) directly as a push-forward under the boundary map to $\partial^2 G$ of a measure inherited from studying all bi-infinite random walk trajectories (with no restriction on base-point) on $G^\mathbb{Z}$. Of these, the third construction is the most involved and needs new techniques. It relies on developing a framework where we can treat bi-infinite random walk trajectories as analogs of bi-infinite geodesics on complete simply connected negatively curved manifolds. Geodesic flow on a hyperbolic group is typically not well-defined due to non-uniqueness of geodesics. We circumvent this problem in the random walk setup by considering \emph{all} trajectories. We thus get a well-defined discrete flow that we call the \emph{randomized geodesic flow}, given by the $\mathbb{Z}-$shift on bi-infinite random walk trajectories. The $\mathbb{Z}-$shift is the random analog of the time one map of the geodesic flow. As an analog of ergodicity of the geodesic flow on a closed negatively curved manifold, we establish ergodicity of the $G$-action on $(\partial^2G, Θ)$. As a consequence of our construction, we prove that the randomized geodesic flow is exponentially mixing of all orders and establish a functional CLT. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14350 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Randomized Geodesic Flow on Hyperbolic Groups Kupffer, Luzie Mj, Mahan Mukherjee, Chiranjib Probability Dynamical Systems Group Theory Geometric Topology Motivated by Gromov's geodesic flow problem on hyperbolic groups $G$, we develop in this paper an analog using random walks. This leads to a notion of a harmonic analog $Θ$ of the Bowen-Margulis-Sullivan measure on $\partial^2 G$. We provide three different but related constructions of $Θ$: 1) by moving the base-point along a quasigeodesic ray 2) by moving the base-point along random walk trajectories 3) directly as a push-forward under the boundary map to $\partial^2 G$ of a measure inherited from studying all bi-infinite random walk trajectories (with no restriction on base-point) on $G^\mathbb{Z}$. Of these, the third construction is the most involved and needs new techniques. It relies on developing a framework where we can treat bi-infinite random walk trajectories as analogs of bi-infinite geodesics on complete simply connected negatively curved manifolds. Geodesic flow on a hyperbolic group is typically not well-defined due to non-uniqueness of geodesics. We circumvent this problem in the random walk setup by considering \emph{all} trajectories. We thus get a well-defined discrete flow that we call the \emph{randomized geodesic flow}, given by the $\mathbb{Z}-$shift on bi-infinite random walk trajectories. The $\mathbb{Z}-$shift is the random analog of the time one map of the geodesic flow. As an analog of ergodicity of the geodesic flow on a closed negatively curved manifold, we establish ergodicity of the $G$-action on $(\partial^2G, Θ)$. As a consequence of our construction, we prove that the randomized geodesic flow is exponentially mixing of all orders and establish a functional CLT. |
| title | Randomized Geodesic Flow on Hyperbolic Groups |
| topic | Probability Dynamical Systems Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2411.14350 |