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ISSN 1088-6850(e) ISSN 0002-9947(p)

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
Posted: January 25, 2012
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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.

References

1. Brian H. Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998), no. 2, 145–186. MR 1638764 (99g:20069), http://dx.doi.org/10.1007/BF02392898
2. Brian H. Bowditch, Planar groups and the Seifert conjecture, J. Reine Angew. Math. 576 (2004), 11–62. MR 2099199 (2005h:57033), http://dx.doi.org/10.1515/crll.2004.084
3. Sean Cleary and Jennifer Taback, Dead end words in lamplighter groups and other wreath products, Q. J. Math. 56 (2005), no. 2, 165–178. MR 2143495 (2006h:20055), http://dx.doi.org/10.1093/qmath/hah030
4. A. N. Dranishnikov, Asymptotic topology, Uspekhi Mat. Nauk 55 (2000), no. 6(336), 71–116 (Russian, with Russian summary); English transl., Russian Math. Surveys 55 (2000), no. 6, 1085–1129. MR 1840358 (2002j:55002), http://dx.doi.org/10.1070/rm2000v055n06ABEH000334
5. Benson Farb and Richard Schwartz, The large-scale geometry of Hilbert modular groups, J. Differential Geom. 44 (1996), no. 3, 435–478. MR 1431001 (98f:22018)
6. Thanos Gentimis, Asymptotic dimension of finitely presented groups, Proc. Amer. Math. Soc. 136 (2008), no. 12, 4103–4110. MR 2431020 (2009e:20095), http://dx.doi.org/10.1090/S0002-9939-08-08973-9
7. M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544 (95m:20041)
8. Tadeusz Januszkiewicz and Jacek Światkowski, Filling invariants of systolic complexes and groups, Geom. Topol. 11 (2007), 727–758. MR 2302501 (2008d:20079), http://dx.doi.org/10.2140/gt.2007.11.727
9. Michael Kapovich and Bruce Kleiner, Coarse Alexander duality and duality groups, J. Differential Geom. 69 (2005), no. 2, 279–352. 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. Panos Papasoglu, Quasi-isometry invariance of group splittings, Ann. of Math. (2) 161 (2005), no. 2, 759–830. MR 2153400 (2006d:20076), http://dx.doi.org/10.4007/annals.2005.161.759
13. P. Papasoglu, JSJ decompositions, in Problems in Geometric group theory, http:// aimath.org/pggt.
14. John R. Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. (2) 88 (1968), 312–334. MR 0228573 (37 #4153)

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