Cores for quasiconvex actions

Sageev, Michah; Wise, Daniel

doi:10.1090/S0002-9939-2015-12297-6

Cores for quasiconvex actions
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by Michah Sageev and Daniel T. Wise

Proc. Amer. Math. Soc. 143 (2015), 2731-2741

DOI: https://doi.org/10.1090/S0002-9939-2015-12297-6

Published electronically: February 26, 2015

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Abstract:

We prove that any full relatively quasiconvex subgroup of a relatively hyperbolic group acting on a CAT(0) cube complex has a convex cocompact core. We give an application towards separability of quasiconvex subgroups of the fundamental group of a special cube complex.

References

I. Agol, D. D. Long, and A. W. Reid, The Bianchi groups are separable on geometrically finite subgroups, Ann. of Math. (2) 153 (2001), no. 3, 599–621. MR 1836283, DOI 10.2307/2661363
B. H. Bowditch, Relatively hyperbolic groups, Internat. J. Algebra Comput. 22 (2012), no. 3, 1250016, 66. MR 2922380, DOI 10.1142/S0218196712500166
Cornelia Druţu and Mark Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005), no. 5, 959–1058. With an appendix by Denis Osin and Mark Sapir. MR 2153979, DOI 10.1016/j.top.2005.03.003
Eric Chesebro, Jason DeBlois, and Henry Wilton, Some virtually special hyperbolic 3-manifold groups, (2009), 1–51.
B. Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998), no. 5, 810–840. MR 1650094, DOI 10.1007/s000390050075
Frédéric Haglund, Finite index subgroups of graph products, Geom. Dedicata 135 (2008), 167–209. MR 2413337, DOI 10.1007/s10711-008-9270-0
Frédéric Haglund and Daniel T. Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008), no. 5, 1551–1620. MR 2377497, DOI 10.1007/s00039-007-0629-4
G. Christopher Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr. Geom. Topol. 10 (2010), no. 3, 1807–1856. MR 2684983, DOI 10.2140/agt.2010.10.1807
G. C. Hruska and Daniel T. Wise, Finiteness properties of cubulated groups, Compos. Math. 150 (2014), no. 3, 453–506. MR 3187627, DOI 10.1112/S0010437X13007112
Tim Hsu and Daniel T. Wise, Cubulating malnormal amalgams, Inventiones Mathematicae, 199 (2015), no. 2, 293-331.
Eduardo Martínez-Pedroza, Combination of quasiconvex subgroups of relatively hyperbolic groups, Groups Geom. Dyn. 3 (2009), no. 2, 317–342. MR 2486802, DOI 10.4171/GGD/59
Eduardo Martínez-Pedroza and Daniel T. Wise, Relative quasiconvexity using fine hyperbolic graphs, Algebr. Geom. Topol. 11 (2011), no. 1, 477–501. MR 2783235, DOI 10.2140/agt.2011.11.477
Lee Mosher, Geometry of cubulated $3$-manifolds, Topology 34 (1995), no. 4, 789–814. MR 1362788, DOI 10.1016/0040-9383(94)00050-6
Michah Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. (3) 71 (1995), no. 3, 585–617. MR 1347406, DOI 10.1112/plms/s3-71.3.585
Peter Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), no. 3, 555–565. MR 494062, DOI 10.1112/jlms/s2-17.3.555
Daniel T. Wise, The structure of groups with a quasiconvex hierarchy, Available at http://www.math.mcgill.ca/wise/papers, pp. 1–189. Submitted.
Daniel T. Wise, Subgroup separability of the figure 8 knot group, Topology 45 (2006), no. 3, 421–463. MR 2218750, DOI 10.1016/j.top.2005.06.004