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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Conical limit points and the Cannon-Thurston map
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by Woojin Jeon, Ilya Kapovich, Christopher Leininger and Ken’ichi Ohshika PDF
Conform. Geom. Dyn. 20 (2016), 58-80 Request permission

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 $|i^{-1}(z)|=1$. 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
  • MR Author ID: 909248
  • 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
  • MR Author ID: 688414
  • Email: clein@math.uiuc.edu
  • Ken’ichi Ohshika
  • Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
  • MR Author ID: 215829
  • Email: ohshika@math.sci.osaka-u.ac.jp
  • 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.
  • © Copyright 2016 American Mathematical Society
  • 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
  • MathSciNet review: 3488025