PreprintCC BY 4.0arXiv
A surface-group action on real-hyperbolic space with no Cannon--Thurston map
Abstract
We present discrete, faithful and purely loxodromic representations of a closed surface group of genus 431, and of a free group, into the group of isometries of a finite-dimensional real-hyperbolic space for which no Cannon--Thurston map exists. The construction combines three ingredients: the small-cancellation strategy employed by Baker--Riley to construct a subgroup of an ambient hyperbolic group with no Cannon--Thurston map, the realization of these groups by octagon complexes, and a theorem of Desgroseilliers--Haglund that supplies a convex cocompact faithful representation of the ambient hyperbolic group on a real-hyperbolic space. These examples admit ray Cannon--Thurston maps, as defined by Halder, Mahan Mj and Sardar.
Related
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- An example of a non-sofic hyperbolic group
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BibTeX
@misc{dular2026surfacegroup,
title = {{A surface-group action on real-hyperbolic space with no Cannon--Thurston map}},
author = {Bruno Dular},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03855},
eprint = {2611.03855},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03855},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A surface-group action on real-hyperbolic space with no Cannon--Thurston map
AU - Bruno Dular
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03855
UR - https://arxiv.org/abs/2611.03855
AB - We present discrete, faithful and purely loxodromic representations of a closed surface group of genus 431, and of a free group, into the group of isometries of a finite-dimensional real-hyperbolic space for which no Cannon--Thurston map exists. The construction combines three ingredients: the small-cancellation strategy employed by Baker--Riley to construct a subgroup of an ambient hyperbolic group with no Cannon--Thurston map, the realization of these groups by octagon complexes, and a theorem of Desgroseilliers--Haglund that supplies a convex cocompact faithful representation of the ambient hyperbolic group on a real-hyperbolic space. These examples admit ray Cannon--Thurston maps, as defined by Halder, Mahan Mj and Sardar.
ER -