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Cannon-Thurston maps and measures
Abstract
We give a short proof of the following extension of a recent result of Gadre, Maher, Pfaff and Uyanki: Consider a closed hyperbolic 3-manifold M which fibers over the circle, with fiber a closed surface S. Any measure class on the ideal boundary of hyperbolic 3-space which is invariant and totally ergodic under the action of the fundamental group of M is singular with respect to the push-forward by a Cannon-Thurston map of a stationary measure on the circle of a random walk with finite first moment on the fundamental group of S.
Related
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BibTeX
@misc{hamenstaedt2026cannonthurston,
title = {{Cannon-Thurston maps and measures}},
author = {Ursula Hamenstaedt},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25378},
eprint = {2609.25378},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25378},
}
RIS
TY - UNPB
TI - Cannon-Thurston maps and measures
AU - Ursula Hamenstaedt
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.25378
UR - https://arxiv.org/abs/2609.25378
AB - We give a short proof of the following extension of a recent result of Gadre, Maher, Pfaff and Uyanki: Consider a closed hyperbolic 3-manifold M which fibers over the circle, with fiber a closed surface S. Any measure class on the ideal boundary of hyperbolic 3-space which is invariant and totally ergodic under the action of the fundamental group of M is singular with respect to the push-forward by a Cannon-Thurston map of a stationary measure on the circle of a random walk with finite first moment on the fundamental group of S.
ER -