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Criterion for Cannon’s Conjecture

Markovic, Vladimir (2013) Criterion for Cannon’s Conjecture. Geometric and Functional Analysis, 23 (3). pp. 1035-1061. ISSN 1016-443X. https://resolver.caltech.edu/CaltechAUTHORS:20130712-090917049

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Abstract

The Cannon Conjecture from the geometric group theory asserts that a word hyperbolic group that acts effectively on its boundary, and whose boundary is homeomorphic to the 2-sphere, is isomorphic to a Kleinian group. We prove the following Criterion for Cannon’s Conjecture: a hyperbolic group G (that acts effectively on its boundary) whose boundary is homeomorphic to the 2-sphere is isomorphic to a Kleinian group if and only if every two points in the boundary of G are separated by a quasi-convex surface subgroup. Thus, the Cannon’s conjecture is reduced to showing that such a group contains “enough” quasi-convex surface subgroups.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/s00039-013-0228-5DOIArticle
http://link.springer.com/article/10.1007%2Fs00039-013-0228-5PublisherArticle
http://arxiv.org/abs/1205.5747arXivDiscussion Paper
Additional Information:© 2013 Springer Basel. Received: May 28, 2012; Revised: December 30, 2012; Accepted: January 7, 2013. Published online April 9, 2013. Vladimir Markovic is supported by the NSF Grant Number DMS-1201463. Ian Agol suggested independently that Theorem 1.1 should be true. In fact, I am grateful to Ian for reading this manuscript and sending me detailed comments that have improved the paper. Also, I would very much like to thank Leonid Potyagailo and Victor Gerasimov for sending me suggestions and corrections. My sincere thanks go to the referee for her/his numerous suggestions and comments that have greatly improved the exposition.
Funders:
Funding AgencyGrant Number
NSFDMS-1201463
Issue or Number:3
Classification Code:Mathematics Subject Classification (2000): Primary 20H10
Record Number:CaltechAUTHORS:20130712-090917049
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20130712-090917049
Official Citation:Markovic, V. Geom. Funct. Anal. (2013) 23: 1035. doi:10.1007/s00039-013-0228-5
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:39334
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:15 Jul 2013 21:55
Last Modified:03 Oct 2019 05:06

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