On strongly quasiconvex subgroups

Tran, Hung¹

¹ Department of Mathematics, The University of Georgia, Athens, GA, United States

Geometry & topology, Tome 23 (2019) no. 3, pp. 1173-1235.

Résumé

We develop a theory of strongly quasiconvex subgroups of an arbitrary finitely generated group. Strong quasiconvexity generalizes quasiconvexity in hyperbolic groups and is preserved under quasi-isometry. We show that strongly quasiconvex subgroups are also more reflective of the ambient group’s geometry than the stable subgroups defined by Durham and Taylor, while still having many properties analogous to those of quasiconvex subgroups of hyperbolic groups. We characterize strongly quasiconvex subgroups in terms of the lower relative divergence of ambient groups with respect to them.

We also study strong quasiconvexity and stability in relatively hyperbolic groups, right-angled Coxeter groups, and right-angled Artin groups. We give complete descriptions of strong quasiconvexity and stability in relatively hyperbolic groups and we characterize strongly quasiconvex special subgroups and stable special subgroups of two-dimensional right-angled Coxeter groups. In the case of right-angled Artin groups, we prove that the two notions of strong quasiconvexity and stability are equivalent when the right-angled Artin group is one-ended and the subgroups have infinite index. We also characterize nontrivial strongly quasiconvex subgroups of infinite index (ie nontrivial stable subgroups) in right-angled Artin groups by quadratic lower relative divergence, expanding the work of Koberda, Mangahas, and Taylor on characterizing purely loxodromic subgroups of right-angled Artin groups.

Détail

DOI : 10.2140/gt.2019.23.1173

Classification : 20F67, 20F65
Keywords: strongly quasiconvex subgroups, stable subgroups, strong quasiconvexity, stability

Bibliographie

[1] C Abbott, M G Durham, J Behrstock, Largest acylindrical actions and stability in hierarchically hyperbolic groups, preprint (2017)

[2] Y Antolín, M Mj, A Sisto, S J Taylor, Intersection properties of stable subgroups and bounded cohomology, preprint (2016)

[3] T Aougab, M G Durham, S J Taylor, Pulling back stability with applications to Out(Fn) and relatively hyperbolic groups, J. Lond. Math. Soc. 96 (2017) 565 | DOI

[4] G N Arzhantseva, C H Cashen, D Gruber, D Hume, Characterizations of Morse quasi-geodesics via superlinear divergence and sublinear contraction, Doc. Math. 22 (2017) 1193

[5] J Behrstock, A counterexample to questions about boundaries, stability, and commensurability, preprint (2017)

[6] J Behrstock, R Charney, Divergence and quasimorphisms of right-angled Artin groups, Math. Ann. 352 (2012) 339 | DOI

[7] J Behrstock, M F Hagen, A Sisto, Hierarchically hyperbolic spaces, I : Curve complexes for cubical groups, Geom. Topol. 21 (2017) 1731 | DOI

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

[9] R Charney, H Sultan, Contracting boundaries of CAT(0) spaces, J. Topol. 8 (2015) 93 | DOI

[10] M Cordes, Morse boundaries of proper geodesic metric spaces, Groups Geom. Dyn. 11 (2017) 1281 | DOI

[11] M Cordes, D Hume, Stability and the Morse boundary, J. Lond. Math. Soc. 95 (2017) 963 | DOI

[12] F Dahmani, Combination of convergence groups, Geom. Topol. 7 (2003) 933 | DOI

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

[14] P Dani, A Thomas, Divergence in right-angled Coxeter groups, Trans. Amer. Math. Soc. 367 (2015) 3549 | DOI

[15] M W Davis, The geometry and topology of Coxeter groups, 32, Princeton Univ. Press (2008)

[16] C Druţu, M Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005) 959 | DOI

[17] M G Durham, S J Taylor, Convex cocompactness and stability in mapping class groups, Algebr. Geom. Topol. 15 (2015) 2839 | DOI

[18] B Farb, L Mosher, Convex cocompact subgroups of mapping class groups, Geom. Topol. 6 (2002) 91 | DOI

[19] A Genevois, Hyperbolicities in CAT(0) cube complexes, preprint (2017)

[20] R Gitik, M Mitra, E Rips, M Sageev, Widths of subgroups, Trans. Amer. Math. Soc. 350 (1998) 321 | DOI

[21] M Gromov, Asymptotic invariants of infinite groups, from: "Geometric group theory, II" (editors G A Niblo, M A Roller), London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press (1993) 1

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

[23] G C Hruska, D T Wise, Packing subgroups in relatively hyperbolic groups, Geom. Topol. 13 (2009) 1945 | DOI

[24] I Kapovich, H Short, Greenberg’s theorem for quasiconvex subgroups of word hyperbolic groups, Canad. J. Math. 48 (1996) 1224 | DOI

[25] H Kim, Stable subgroups and Morse subgroups in mapping class groups, preprint (2017)

[26] T Koberda, J Mangahas, S J Taylor, The geometry of purely loxodromic subgroups of right-angled Artin groups, Trans. Amer. Math. Soc. 369 (2017) 8179 | DOI

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

[28] D Osin, Acylindrically hyperbolic groups, Trans. Amer. Math. Soc. 368 (2016) 851 | DOI

[29] M Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. 71 (1995) 585 | DOI

[30] M Sageev, Codimension-1 subgroups and splittings of groups, J. Algebra 189 (1997) 377 | DOI

[31] H Short, Quasiconvexity and a theorem of Howson’s, from: "Group theory from a geometrical viewpoint" (editors É Ghys, A Haefliger, A Verjovsky), World Sci. (1991) 168

[32] A Sisto, Quasi-convexity of hyperbolically embedded subgroups, Math. Z. 283 (2016) 649 | DOI

[33] H C Tran, Relative divergence of finitely generated groups, Algebr. Geom. Topol. 15 (2015) 1717 | DOI

[34] H C Tran, Divergence of Morse geodesics, Geom. Dedicata 180 (2016) 385 | DOI

[35] H C Tran, Malnormality and join-free subgroups in right-angled Coxeter groups, preprint (2018)