Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-2012-05578-4
Published electronically: January 25, 2012
MathSciNet review: 2901237
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. B.H. Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180, No. 2, pp. 145-186 (1998). MR 1638764 (99g:20069)
  • 2. B.H. Bowditch, Planar groups and the Seifert conjecture, Journal fur die reine und angewandte Mathematik (Crelle's Journal), Volume 2004, Issue 576, 11-62. MR 2099199 (2005h:57033)
  • 3. S. Cleary, J. Taback, Dead end words in lamplighter groups and other wreath products, Quarterly Journal of Mathematics, Vol. 56, Number 2, June 2005, pp. 165-178. MR 2143495 (2006h:20055)
  • 4. A. Dranishnikov, Asymptotic Topology, Russian Math. Surveys 55, (2000), no. 6, 71-116. MR 1840358 (2002j:55002)
  • 5. B. Farb, R. Schwartz, The large-scale geometry of Hilbert modular groups, J. Diff. Geom., vol. 44 (1996) pp. 435-478. MR 1431001 (98f:22018)
  • 6. T. Gentimis Asymptotic dimension of finitely presented groups, Proc. Amer. Math. Soc. 136 (2008), 4103-4110. MR 2431020 (2009e:20095)
  • 7. M. Gromov, Asymptotic invariants of infinite groups, in Geometric group theory (G. Niblo, M. Roller, eds.), LMS Lecture Notes, vol. 182, Cambridge Univ. Press, 1993. MR 1253544 (95m:20041)
  • 8. T. Januszkiewicz, J. Swiatkowski, Filling invariants of systolic complexes and groups, Geometry and Topology 11 (2007), pp. 727-758. MR 2302501 (2008d:20079)
  • 9. M. Kapovich, B. Kleiner, Coarse Alexander duality and duality groups, J. Differential Geom. 69, no. 2, 279-352, 2005. MR 2168506 (2007c:57033)
  • 10. B. Kleiner, in preparation.
  • 11. G. Mess, The Seifert conjecture and groups which are coarse quasiisometric to planes preprint, 1990.
  • 12. P. Papasoglu, Quasi-isometry invariance of group splittings, Annals of Math. vol. 161, no. 2, pp. 759-830 (2005). MR 2153400 (2006d:20076)
  • 13. P. Papasoglu, JSJ decompositions, in Problems in Geometric group theory, http:// aimath.org/pggt.
  • 14. J.R. Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. 88, pp. 312-334 (1968). MR 0228573 (37:4153)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20F65, 20E08

Retrieve articles in all journals with MSC (2010): 20F65, 20E08


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: https://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.

American Mathematical Society