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. doi:10.1007/s00039-013-0228-5. 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:

URL	URL Type	Description
http://dx.doi.org/10.1007/s00039-013-0228-5	DOI	Article
http://link.springer.com/article/10.1007%2Fs00039-013-0228-5	Publisher	Article
http://arxiv.org/abs/1205.5747	arXiv	Discussion 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 Agency	Grant Number
NSF	DMS-1201463

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Classification Code:

Mathematics Subject Classification (2000): Primary 20H10

DOI:

10.1007/s00039-013-0228-5

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

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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:

09 Nov 2021 23:44

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