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Relative cubulations and groups with a 2-sphere boundary

  • Eduard Einstein (a1) and Daniel Groves (a2)


We introduce a new kind of action of a relatively hyperbolic group on a $\text{CAT}(0)$ cube complex, called a relatively geometric action. We provide an application to characterize finite-volume Kleinian groups in terms of actions on cube complexes, analogous to the results of Markovic and Haïssinsky in the closed case.



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The second author was partially supported by the National Science Foundation, DMS-1507067.



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[Ago13]Agol, I., The virtual Haken conjecture, Doc. Math. 18 (2013), 10451087; with an appendix by Agol, Daniel Groves, and Jason Manning.
[AGM09]Agol, I., Groves, D. and Manning, J. F., Residual finiteness, QCERF and fillings of hyperbolic groups, Geom. Topol. 13 (2009), 10431073.10.2140/gt.2009.13.1043
[AGM16]Agol, I., Groves, D. and Manning, J. F., An alternate proof of Wise’s malnormal special quotient theorem, Forum Math. Pi 4 (2016), E1.10.1017/fmp.2015.8
[BW12]Bergeron, N. and Wise, D. T., A boundary criterion for cubulation, Amer. J. Math. 134 (2012), 843859.10.1353/ajm.2012.0020
[Can91]Cannon, J. W., The theory of negatively curved spaces and groups, in Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989) (Oxford University Press, New York, 1991), 315369.
[CS98]Cannon, J. W. and Swenson, E. L., Recognizing constant curvature discrete groups in dimension 3, Trans. Amer. Math. Soc. 350 (1998), 809849.10.1090/S0002-9947-98-02107-2
[CC07]Charney, R. and Crisp, J., Relative hyperbolicity and Artin groups, Geom. Dedicata 129 (2007), 113.10.1007/s10711-007-9178-0
[CF19]Cooper, D. and Futer, D., Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3-manifolds, Geom. Topol. 23 (2019), 241298.10.2140/gt.2019.23.241
[GM08]Groves, D. and Manning, J. F., Dehn filling in relatively hyperbolic groups, Israel J. Math. 168 (2008), 317429.10.1007/s11856-008-1070-6
[GM17]Groves, D. and Manning, J. F., Quasiconvexity and Dehn filling, Amer. J. Math., to appear. Preprint (2017), arXiv:1708.07968.
[GM18]Groves, D. and Manning, J. F., Hyperbolic groups acting improperly, Preprint (2018), arXiv:1808.02325.
[GMS19]Groves, D., Manning, J. F. and Sisto, A., Boundaries of Dehn fillings, Geom. Topol. 23 (2019), 29293002.10.2140/gt.2019.23.2929
[HW12]Haglund, F. and Wise, D. T., A combination theorem for special cube complexes, Ann. of Math. (2) 176 (2012), 14271482.10.4007/annals.2012.176.3.2
[Haï15]Haïssinsky, P., Hyperbolic groups with planar boundaries, Invent. Math. 201 (2015), 239307.10.1007/s00222-014-0552-x
[Hru10]Hruska, G. C., Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr. Geom. Topol. 10 (2010), 18071856.10.2140/agt.2010.10.1807
[HW14]Hruska, G. C. and Wise, D. T., Finiteness properties of cubulated groups, Compos. Math. 150 (2014), 453506.10.1112/S0010437X13007112
[HW15]Hsu, T. and Wise, D. T., Cubulating malnormal amalgams, Invent. Math. 199 (2015), 293331.10.1007/s00222-014-0513-4
[KM12]Kahn, J. and Markovic, V., Immersing almost geodesic surfaces in a closed hyperbolic three manifold, Ann. of Math. (2) 175 (2012), 11271190.10.4007/annals.2012.175.3.4
[Kap07]Kapovich, M., Problems on boundaries of groups and Kleinian groups,∼kapovich/EPR/problems.pdf, 2007.
[Mar13]Markovic, V., Criterion for Cannon’s conjecture, Geom. Funct. Anal. 23 (2013), 10351061.10.1007/s00039-013-0228-5
[Sag97]Sageev, M., Codimension-1 subgroups and splittings of groups, J. Algebra 189 (1997), 377389.10.1006/jabr.1996.6884
[SW15]Sageev, M. and Wise, D. T., Cores for quasiconvex actions, Proc. Amer. Math. Soc. 143 (2015), 27312741.10.1090/S0002-9939-2015-12297-6
[Wis]Wise, D. T., The structure of groups with a quasiconvex hierarchy, Ann. of Math. Stud., to appear.
[Yam04]Yaman, A., A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math. 566 (2004), 4189.
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Relative cubulations and groups with a 2-sphere boundary

  • Eduard Einstein (a1) and Daniel Groves (a2)


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