Boundaries of Dehn fillings

Groves, Daniel¹ ; Manning, Jason Fox² ; Sisto, Alessandro³

¹ Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, United States
² Department of Mathematics, Cornell University, Ithaca, NY, United States
³ Department of Mathematics, ETH Zürich, Zürich, Switzerland

Geometry & topology, Tome 23 (2019) no. 6, pp. 2929-3002.

Résumé

We begin an investigation into the behavior of Bowditch and Gromov boundaries under the operation of Dehn filling. In particular, we show many Dehn fillings of a toral relatively hyperbolic group with $2$–sphere boundary are hyperbolic with $2$–sphere boundary. As an application, we show that the Cannon conjecture implies a relatively hyperbolic version of the Cannon conjecture.

Détail

DOI : 10.2140/gt.2019.23.2929

Classification : 20F67, 57M07
Keywords: relatively hyperbolic groups, boundaries, Dehn filling

Bibliographie

[1] I Agol, The virtual Haken conjecture, Doc. Math. 18 (2013) 1045

[2] I Agol, D Groves, J F Manning, Residual finiteness, QCERF and fillings of hyperbolic groups, Geom. Topol. 13 (2009) 1043 | DOI

[3] I Agol, D Groves, J F Manning, An alternate proof of Wise’s malnormal special quotient theorem, Forum Math. Pi 4 (2016) | DOI

[4] M Bestvina, M Feighn, Stable actions of groups on real trees, Invent. Math. 121 (1995) 287 | DOI

[5] S A Bleiler, C D Hodgson, Spherical space forms and Dehn filling, Topology 35 (1996) 809 | DOI

[6] M Bonk, B Kleiner, Quasi-hyperbolic planes in hyperbolic groups, Proc. Amer. Math. Soc. 133 (2005) 2491 | DOI

[7] B H Bowditch, A topological characterisation of hyperbolic groups, J. Amer. Math. Soc. 11 (1998) 643 | DOI

[8] B H Bowditch, Boundaries of geometrically finite groups, Math. Z. 230 (1999) 509 | DOI

[9] B H Bowditch, Connectedness properties of limit sets, Trans. Amer. Math. Soc. 351 (1999) 3673 | DOI

[10] B H Bowditch, Peripheral splittings of groups, Trans. Amer. Math. Soc. 353 (2001) 4057 | DOI

[11] B H Bowditch, Relatively hyperbolic groups, Internat. J. Algebra Comput. 22 (2012) 1250016, 66 | DOI

[12] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, 319, Springer (1999) | DOI

[13] S Buyalo, V Schroeder, Elements of asymptotic geometry, Eur. Math. Soc. (2007) | DOI

[14] J W Cannon, The theory of negatively curved spaces and groups, from: "Ergodic theory, symbolic dynamics, and hyperbolic spaces" (editors T Bedford, M Keane, C Series), Oxford Univ. Press (1991) 315

[15] J W Cannon, D Cooper, A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three, Trans. Amer. Math. Soc. 330 (1992) 419 | DOI

[16] J W Cannon, E L Swenson, Recognizing constant curvature discrete groups in dimension 3, Trans. Amer. Math. Soc. 350 (1998) 809 | DOI

[17] C H Chen, Warped products of metric spaces of curvature bounded from above, Trans. Amer. Math. Soc. 351 (1999) 4727 | DOI

[18] S Claytor, Topological immersion of Peanian continua in a spherical surface, Ann. of Math. 35 (1934) 809 | DOI

[19] R Coulon, On the geometry of Burnside quotients of torsion free hyperbolic groups, Internat. J. Algebra Comput. 24 (2014) 251 | DOI

[20] F Dahmani, V Guirardel, Recognizing a relatively hyperbolic group by its Dehn fillings, Duke Math. J. 167 (2018) 2189 | DOI

[21] F Dahmani, V Guirardel, D Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, 1156, Amer. Math. Soc. (2017) | DOI

[22] T Delzant, M Gromov, Courbure mésoscopique et théorie de la toute petite simplification, J. Topol. 1 (2008) 804 | DOI

[23] M G Durham, The coarse geometry of the Teichmuller metric: a quasiisometry model and the actions of finite groups, PhD thesis, University of Illinois at Chicago (2014)

[24] H Freudenthal, Über die Enden topologischer Räume und Gruppen, Math. Z. 33 (1931) 692 | DOI

[25] K Fujiwara, J F Manning, CAT(0) and CAT(−1) fillings of hyperbolic manifolds, J. Differential Geom. 85 (2010) 229 | DOI

[26] V Gerasimov, Floyd maps for relatively hyperbolic groups, Geom. Funct. Anal. 22 (2012) 1361 | DOI

[27] V Gerasimov, L Potyagailo, Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups, J. Eur. Math. Soc. 15 (2013) 2115 | DOI

[28] B W Groff, Quasi-isometries, boundaries and JSJ-decompositions of relatively hyperbolic groups, J. Topol. Anal. 5 (2013) 451 | DOI

[29] D Groves, J F Manning, Dehn filling in relatively hyperbolic groups, Israel J. Math. 168 (2008) 317 | DOI

[30] D Groves, J F Manning, Dehn fillings and elementary splittings, Trans. Amer. Math. Soc. 370 (2018) 3017 | DOI

[31] G C Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr. Geom. Topol. 10 (2010) 1807 | DOI

[32] S V Ivanov, Gromov–Hausdorff convergence and volumes of manifolds, Algebra i Analiz 9 (1997) 65

[33] M Kapovich, Problems on boundaries of groups and Kleinian groups, online document (2007)

[34] M Kapovich, B Kleiner, Hyperbolic groups with low-dimensional boundary, Ann. Sci. École Norm. Sup. 33 (2000) 647 | DOI

[35] J M Mackay, Existence of quasi-arcs, Proc. Amer. Math. Soc. 136 (2008) 3975 | DOI

[36] J M Mackay, A Sisto, Quasi-hyperbolic planes in relatively hyperbolic groups, preprint (2011)

[37] J F Manning, The Bowditch boundary of (G,H) when G is hyperbolic, preprint (2015)

[38] G J Martin, R K Skora, Group actions of the 2–sphere, Amer. J. Math. 111 (1989) 387 | DOI

[39] Y Matsuda, S I Oguni, S Yamagata, Blowing up and down compacta with geometrically finite convergence actions of a group, preprint (2012)

[40] E E Moise, Geometric topology in dimensions 2 and 3, 47, Springer (1977) | DOI

[41] L Mosher, M Sageev, Nonmanifold hyperbolic groups of high cohomological dimension, preprint (1997)

[42] D V Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, 843, Amer. Math. Soc. (2006) | DOI

[43] D V Osin, Peripheral fillings of relatively hyperbolic groups, Invent. Math. 167 (2007) 295 | DOI

[44] I Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc. 106 (1963) 259 | DOI

[45] G A Swarup, On the cut point conjecture, Electron. Res. Announc. Amer. Math. Soc. 2 (1996) 98 | DOI

[46] W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979)

[47] H C Tran, Relations between various boundaries of relatively hyperbolic groups, Internat. J. Algebra Comput. 23 (2013) 1551 | DOI

[48] J Väisälä, Gromov hyperbolic spaces, Expo. Math. 23 (2005) 187 | DOI

[49] A Yaman, A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math. 566 (2004) 41 | DOI