Quasiconvexity of virtual joins and separability of products in relatively hyperbolic groups

Minasyan, Ashot¹ ; Mineh, Lawk²

¹School of Mathematical Sciences, University of Southampton, Southampton, United Kingdom
²Mathematical Institute, University of Bonn, Bonn, Germany

Algebraic and Geometric Topology, Tome 25 (2025) no. 1, pp. 399-488

Résumé

A relatively hyperbolic group G is said to be QCERF if all finitely generated relatively quasiconvex subgroups are closed in the profinite topology on G.

Assume that G is a QCERF relatively hyperbolic group with double coset separable (eg virtually polycyclic) peripheral subgroups. Given any two finitely generated relatively quasiconvex subgroups Q,R ≤ G we prove the existence of finite-index subgroups Q′≤fQ and R′≤fR such that the join 〈Q′,R′〉 is again relatively quasiconvex in G. We then show that, under the minimal necessary hypotheses on the peripheral subgroups, products of finitely generated relatively quasiconvex subgroups are closed in the profinite topology on G. From this we obtain the separability of products of finitely generated subgroups for several classes of groups, including limit groups, Kleinian groups and balanced fundamental groups of finite graphs of free groups with cyclic edge groups.

Détail

DOI : 10.2140/agt.2025.25.399

Keywords: relatively hyperbolic groups, relatively quasiconvex subgroups, virtual joins, double coset separability, product separability, limit groups, Kleinian groups

Bibliographie

[1] I Agol, Tameness of hyperbolic 3-manifolds, preprint (2004)

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

[3] E Alibegović, A combination theorem for relatively hyperbolic groups, Bull. London Math. Soc. 37 (2005) 459 | DOI

[4] R B J T Allenby, R J Gregorac, On locally extended residually finite groups, from: "Conference on group theory" (editors R W Gatterdam, K W Weston), Lecture Notes in Math. 319, Springer (1973) 9 | DOI

[5] M Baker, D Cooper, A combination theorem for convex hyperbolic manifolds, with applications to surfaces in 3-manifolds, J. Topol. 1 (2008) 603 | DOI

[6] H Bigdely, D T Wise, Quasiconvexity and relatively hyperbolic groups that split, Michigan Math. J. 62 (2013) 387 | DOI

[7] B H Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993) 245 | DOI

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

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

[10] R Brooks, Circle packings and co-compact extensions of Kleinian groups, Invent. Math. 86 (1986) 461 | DOI

[11] D Calegari, D Gabai, Shrinkwrapping and the taming of hyperbolic 3-manifolds, J. Amer. Math. Soc. 19 (2006) 385 | DOI

[12] R D Canary, A covering theorem for hyperbolic 3-manifolds and its applications, Topology 35 (1996) 751 | DOI

[13] T Coulbois, Propriétés de Ribes–Zalesskii, topologie profinie, produit libre et généralisations, PhD thesis, Université Paris VII (2000)

[14] T Coulbois, Free product, profinite topology and finitely generated subgroups, Internat. J. Algebra Comput. 11 (2001) 171 | DOI

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

[16] W Dicks, M J Dunwoody, Groups acting on graphs, 17, Cambridge Univ. Press (1989)

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

[18] E Einstein, D Groves, Relatively geometric actions on CAT(0) cube complexes, J. Lond. Math. Soc. 105 (2022) 691 | DOI

[19] E Einstein, T Ng, Relative cubulation of small cancellation free products, preprint (2021)

[20] B Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998) 810 | DOI

[21] R Gitik, Graphs and separability properties of groups, J. Algebra 188 (1997) 125 | DOI

[22] R Gitik, On the profinite topology on negatively curved groups, J. Algebra 219 (1999) 80 | DOI

[23] R Gitik, Ping-pong on negatively curved groups, J. Algebra 217 (1999) 65 | DOI

[24] R Gitik, E Rips, On separability properties of groups, Internat. J. Algebra Comput. 5 (1995) 703 | DOI

[25] M Gromov, Hyperbolic groups, from: "Essays in group theory" (editor S M Gersten), Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 | DOI

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

[27] D Groves, J F Manning, Specializing cubulated relatively hyperbolic groups, J. Topol. 15 (2022) 398 | DOI

[28] F Haglund, D T Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008) 1551 | DOI

[29] E Hamilton, H Wilton, P A Zalesskii, Separability of double cosets and conjugacy classes in 3-manifold groups, J. Lond. Math. Soc. 87 (2013) 269 | DOI

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

[31] T Hsu, D T Wise, Cubulating graphs of free groups with cyclic edge groups, Amer. J. Math. 132 (2010) 1153 | DOI

[32] O Kharlampovich, A Myasnikov, Elementary theory of free non-abelian groups, J. Algebra 302 (2006) 451 | DOI

[33] J C Lennox, J S Wilson, On products of subgroups in polycyclic groups, Arch. Math. (Basel) 33 (1979/80) 305 | DOI

[34] G Levitt, On the automorphism group of generalized Baumslag–Solitar groups, Geom. Topol. 11 (2007) 473 | DOI

[35] R C Lyndon, P E Schupp, Combinatorial group theory, 89, Springer (1977) | DOI

[36] J F Manning, E Martínez-Pedroza, Separation of relatively quasiconvex subgroups, Pacific J. Math. 244 (2010) 309 | DOI

[37] E Martínez-Pedroza, Combination of quasiconvex subgroups of relatively hyperbolic groups, Groups Geom. Dyn. 3 (2009) 317 | DOI

[38] E Martínez-Pedroza, A Sisto, Virtual amalgamation of relatively quasiconvex subgroups, Algebr. Geom. Topol. 12 (2012) 1993 | DOI

[39] K Matsuzaki, M Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Univ. Press (1998)

[40] C Mcclellan, Separable at birth: products of full relatively quasi-convex subgroups, PhD thesis, University of Illinois at Chicago (2019)

[41] A Minasyan, On residualizing homomorphisms preserving quasiconvexity, Comm. Algebra 33 (2005) 2423 | DOI

[42] A Minasyan, Some properties of subsets of hyperbolic groups, Comm. Algebra 33 (2005) 909 | DOI

[43] A Minasyan, Separable subsets of GFERF negatively curved groups, J. Algebra 304 (2006) 1090 | DOI

[44] G A Niblo, Separability properties of free groups and surface groups, J. Pure Appl. Algebra 78 (1992) 77 | DOI

[45] D V Osin, Relative Dehn functions of amalgamated products and HNN-extensions, from: "Topological and asymptotic aspects of group theory" (editors R Grigorchuk, M Mihalik, M Sapir, Z Šunik), Contemp. Math. 394, Amer. Math. Soc. (2006) 209 | DOI

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

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

[48] J E Pin, On a conjecture of Rhodes, Semigroup Forum 39 (1989) 1 | DOI

[49] J E Pin, C Reutenauer, A conjecture on the Hall topology for the free group, Bull. London Math. Soc. 23 (1991) 356 | DOI

[50] J G Ratcliffe, Foundations of hyperbolic manifolds, 149, Springer (2006) | DOI

[51] E Reyes, On cubulated relatively hyperbolic groups, Geom. Topol. 27 (2023) 575 | DOI

[52] L Ribes, Profinite graphs and groups, 66, Springer (2017) | DOI

[53] L Ribes, P A Zalesskii, On the profinite topology on a free group, Bull. London Math. Soc. 25 (1993) 37 | DOI

[54] Z Sela, Diophantine geometry over groups, VI : The elementary theory of a free group, Geom. Funct. Anal. 16 (2006) 707 | DOI

[55] J P Serre, Trees, Springer (1980) | DOI

[56] S Shepherd, D J Woodhouse, Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups, J. Reine Angew. Math. 782 (2022) 121 | DOI

[57] 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. Publ. (1991) 168

[58] H Wilton, Hall’s theorem for limit groups, Geom. Funct. Anal. 18 (2008) 271 | DOI

[59] D T Wise, Subgroup separability of graphs of free groups with cyclic edge groups, Q. J. Math. 51 (2000) 107 | DOI

[60] W Yang, Combination of fully quasiconvex subgroups and its applications, preprint (2012)

[61] S You, The product separability of the generalized free product of cyclic groups, J. Lond. Math. Soc. 56 (1997) 91 | DOI