On fluctuations of the drift in hyperbolic groups

Abstract

Let Gamma be a convex-cocompact group of isometries of a CAT(-1) space X and let Y to X/Gamma be a Galois cover with a word-hyperbolic group of deck transformations. We show that, for almost every geodesic xi with respect to the Bowen-Margulis-Sullivan measure on the lift of X/Gamma, there exist m,sigma > 0 and a standard Brownian motion B(s) such that, for any lambda > 1/4, d(p(g(s,xi)), o) = ms + sigma B(s) + o(s^lambda), where g(s,xi) denotes the geodesic flow acting on the geodesics of Y and p the canonical projection to Y. The result is a consequence of an almost sure invariance principle for random walks on hyperbolic groups with dependent increments. Its proof makes use of a new Ruelle operator theorem for skew products and Martin boundary techniques for random walks with dependent increments.

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BibTeX RIS

BibTeX

@misc{hernandez2026fluctuations,
  title = {{On fluctuations of the drift in hyperbolic groups}},
  author = {Gil José Astudillo Hernandez and Manuel Stadlbauer},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18053},
  eprint = {2609.18053},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18053},
}

RIS

TY  - UNPB
TI  - On fluctuations of the drift in hyperbolic groups
AU  - Gil José Astudillo Hernandez
AU  - Manuel Stadlbauer
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.18053
UR  - https://arxiv.org/abs/2609.18053
AB  - Let Gamma be a convex-cocompact group of isometries of a CAT(-1) space X and let Y to X/Gamma be a Galois cover with a word-hyperbolic group of deck transformations. We show that, for almost every geodesic xi with respect to the Bowen-Margulis-Sullivan measure on the lift of X/Gamma, there exist m,sigma > 0 and a standard Brownian motion B(s) such that, for any lambda > 1/4, d(p(g(s,xi)), o) = ms + sigma B(s) + o(s^lambda), where g(s,xi) denotes the geodesic flow acting on the geodesics of Y and p the canonical projection to Y. The result is a consequence of an almost sure invariance principle for random walks on hyperbolic groups with dependent increments. Its proof makes use of a new Ruelle operator theorem for skew products and Martin boundary techniques for random walks with dependent increments.
ER  -

Abstract

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