Splittings and the asymptotic topology of the lamplighter group
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- by Panos Papasoglu PDF
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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
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 3861-3873
- MSC (2010): Primary 20F65, 20E08
- DOI: https://doi.org/10.1090/S0002-9947-2012-05578-4
- MathSciNet review: 2901237