Uniformizing non-proper Gromov Hyperbolic Spaces
Abstract
In this paper, we extend a large part of the uniformization theory of Bonk-Heinonen-Koskela [Asterisque 2001] to length spaces that are not necessarily proper or geodesic. Among other things, we show that there is a one-to-one correspondence between the quasiisometry classes of complete roughly starlike Gromov hyperbolic spaces and the quasisimilarity classes of bounded uniform spaces, which provides an affirmative solution to an open question of Bonk-Heinonen-Koskela. Our approach relies crucially on the work of Väisälä [Expo. Math. 2005], who investigated in depth Gromov hyperbolic spaces that are not necessarily proper or geodesic. One key new ingredient is to use the so-called (quasihyperbolic) (c,μ)-quasigeodesic as a suitable substitute for quasihyperbolic geodesic.
Related
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BibTeX
@misc{gao2026uniformizing,
title = {{Uniformizing non-proper Gromov Hyperbolic Spaces}},
author = {Shu-Jing Gao and Chang-Yu Guo and Miao He},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29670},
eprint = {2608.29670},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29670},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Uniformizing non-proper Gromov Hyperbolic Spaces
AU - Shu-Jing Gao
AU - Chang-Yu Guo
AU - Miao He
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29670
UR - https://arxiv.org/abs/2608.29670
AB - In this paper, we extend a large part of the uniformization theory of Bonk-Heinonen-Koskela [Asterisque 2001] to length spaces that are not necessarily proper or geodesic. Among other things, we show that there is a one-to-one correspondence between the quasiisometry classes of complete roughly starlike Gromov hyperbolic spaces and the quasisimilarity classes of bounded uniform spaces, which provides an affirmative solution to an open question of Bonk-Heinonen-Koskela. Our approach relies crucially on the work of Väisälä [Expo. Math. 2005], who investigated in depth Gromov hyperbolic spaces that are not necessarily proper or geodesic. One key new ingredient is to use the so-called (quasihyperbolic) (c,μ)-quasigeodesic as a suitable substitute for quasihyperbolic geodesic.
ER -