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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Splittings and the asymptotic topology of the lamplighter group


Author: Panos Papasoglu
Journal: Trans. Amer. Math. Soc. 364 (2012), 3861-3873
MSC (2010): Primary 20F65, 20E08
Published electronically: January 25, 2012
MathSciNet review: 2901237
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Abstract: It is known that splittings of one-ended finitely presented groups over 2-ended groups can be characterized geometrically. Here we show that this characterization does not extend to all finitely generated groups, by showing that the lamplighter group is coarsely separated by quasi-lines.

It is also known that virtual surface groups are characterized in the class of one-ended finitely presented groups by the property that their Cayley graphs are coarsely separated by quasi-circles. Answering a question of Kleiner we show that the Cayley graph of the lamplighter group is coarsely separated by quasi-circles. It follows that the quasi-circle characterization of virtual surface groups does not extend to the finitely generated case.


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Additional Information

Panos Papasoglu
Affiliation: Mathematical Institute, University of Oxford, 24-29 St. Giles’, Oxford, OX1 3LB, United Kingdom
Email: papazoglou@maths.ox.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05578-4
Received by editor(s): September 19, 2010
Received by editor(s) in revised form: December 30, 2010, February 1, 2011, and February 4, 2011
Published electronically: January 25, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.