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Conical limit points and the Cannon-Thurston map


Authors: Woojin Jeon, Ilya Kapovich, Christopher Leininger and Ken’ichi Ohshika
Journal: Conform. Geom. Dyn. 20 (2016), 58-80
MSC (2010): Primary 20F65; Secondary 30F40, 57M60, 37Exx, 37Fxx
DOI: https://doi.org/10.1090/ecgd/294
Published electronically: March 18, 2016
MathSciNet review: 3488025
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Abstract: Let $ G$ be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space $ Z$ so that there exists a continuous $ G$-equivariant map $ i:\partial G\to Z$, which we call a Cannon-Thurston map. We obtain two characterizations (a dynamical one and a geometric one) of conical limit points in $ Z$ in terms of their pre-images under the Cannon-Thurston map $ i$. As an application we prove, under the extra assumption that the action of $ G$ on $ Z$ has no accidental parabolics, that if the map $ i$ is not injective, then there exists a non-conical limit point $ z\in Z$ with $ \vert i^{-1}(z)\vert=1$. This result applies to most natural contexts where the Cannon-Thurston map is known to exist, including subgroups of word-hyperbolic groups and Kleinian representations of surface groups. As another application, we prove that if $ G$ is a non-elementary torsion-free word-hyperbolic group, then there exists $ x\in \partial G$ such that $ x$ is not a ``controlled concentration point'' for the action of $ G$ on $ \partial G$.


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Additional Information

Woojin Jeon
Affiliation: School of Mathematics, KIAS, Hoegiro 87, Dongdaemun-gu, Seoul, 130-722, Korea
Email: jwoojin@kias.re.kr

Ilya Kapovich
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801
Email: kapovich@math.uiuc.edu

Christopher Leininger
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801
Email: clein@math.uiuc.edu

Ken’ichi Ohshika
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Email: ohshika@math.sci.osaka-u.ac.jp

DOI: https://doi.org/10.1090/ecgd/294
Keywords: Convergence groups, Cannon-Thurston map, conical limit points, Kleinian groups
Received by editor(s): May 7, 2015
Received by editor(s) in revised form: January 29, 2016
Published electronically: March 18, 2016
Additional Notes: The second author was partially supported by Collaboration Grant no. 279836 from the Simons Foundation and by NSF grant DMS-1405146. The third author was partially supported by NSF grants DMS-1207183 and DMS-1510034. The last author was partially supported by JSPS Grants-in-Aid 70183225.
Article copyright: © Copyright 2016 American Mathematical Society