Property (QT) for 3-manifold groups

Han, Suzhen¹ ; Nguyen, Hoang Thanh² ; Yang, Wenyuan³

¹School of Mathematics, Hunan University, Changsha, China
²Department of Mathematics, FPT University, Da Nang, Vietnam
³Beijing International Center for Mathematical Research, Peking University, Beijing, China

Algebraic and Geometric Topology, Tome 25 (2025) no. 1, pp. 107-159

Résumé

According to Bestvina, Bromberg and Fujiwara, a finitely generated group is said to have property (QT) if it acts isometrically on a finite product of quasitrees so that orbital maps are quasi-isometric embeddings. We prove that the fundamental group π1(M) of a compact, connected, orientable 3-manifold M has property (QT) if and only if no summand in the sphere-disc decomposition of M supports either Sol or Nil geometry. In particular, all compact, orientable, irreducible 3-manifold groups with nontrivial torus decomposition and not supporting Sol geometry have property (QT). In the course of our study, we establish property (QT) for the class of Croke–Kleiner admissible groups and for relatively hyperbolic groups under natural assumptions on the peripheral subgroups.

Détail

DOI : 10.2140/agt.2025.25.107

Keywords: property (QT), admissible groups, 3-manifold groups

Bibliographie

[1] C Abbott, S H Balasubramanya, D Osin, Hyperbolic structures on groups, Algebr. Geom. Topol. 19 (2019) 1747 | DOI

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

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

[4] J Bajpai, Omnipotence of surface groups, master’s thesis, McGill University (2007)

[5] M Bestvina, K Bromberg, K Fujiwara, Constructing group actions on quasi-trees and applications to mapping class groups, Publ. Math. Inst. Hautes Études Sci. 122 (2015) 1 | DOI

[6] M Bestvina, K Bromberg, K Fujiwara, Proper actions on finite products of quasi-trees, Ann. H. Lebesgue 4 (2021) 685 | DOI

[7] M Bestvina, K Bromberg, K Fujiwara, A Sisto, Acylindrical actions on projection complexes, Enseign. Math. 65 (2019) 1 | DOI

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

[9] B H Bowditch, Tight geodesics in the curve complex, Invent. Math. 171 (2008) 281 | DOI

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

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

[12] D Burago, Y Burago, S Ivanov, A course in metric geometry, 33, Amer. Math. Soc. (2001) | DOI

[13] R G Burns, On finitely generated subgroups of free products, J. Austral. Math. Soc. 12 (1971) 358 | DOI

[14] J O Button, Groups acting purely loxodromically on products of hyperbolic graphs, preprint (2020)

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

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

[17] P E Caprace, Y Cornulier, N Monod, R Tessera, Amenable hyperbolic groups, J. Eur. Math. Soc. 17 (2015) 2903 | DOI

[18] G R Conner, Discreteness properties of translation numbers in solvable groups, J. Group Theory 3 (2000) 77 | DOI

[19] Y De Cornulier, Dimension of asymptotic cones of Lie groups, J. Topol. 1 (2008) 342 | DOI

[20] C B Croke, B Kleiner, Spaces with nonpositive curvature and their ideal boundaries, Topology 39 (2000) 549 | DOI

[21] C B Croke, B Kleiner, The geodesic flow of a nonpositively curved graph manifold, Geom. Funct. Anal. 12 (2002) 479 | DOI

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

[23] A Dranishnikov, T Januszkiewicz, Every Coxeter group acts amenably on a compact space, Topology Proc. 24 (1999) 135

[24] C Druţu, M Kapovich, Geometric group theory, 63, Amer. Math. Soc. (2018) | DOI

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

[26] T Foertsch, A Lytchak, The de Rham decomposition theorem for metric spaces, Geom. Funct. Anal. 18 (2008) 120 | DOI

[27] V Gerasimov, L Potyagailo, Quasiconvexity in relatively hyperbolic groups, J. Reine Angew. Math. 710 (2016) 95 | DOI

[28] M F Hagen, P Przytycki, Cocompactly cubulated graph manifolds, Israel J. Math. 207 (2015) 377 | DOI

[29] M Hagen, J Russell, A Sisto, D Spriano, Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms, Ann. Inst. Fourier (Grenoble) (2024) | DOI

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

[31] E Hamilton, Abelian subgroup separability of Haken 3-manifolds and closed hyperbolic n-orbifolds, Proc. Lond. Math. Soc. 83 (2001) 626 | DOI

[32] M Hull, D Osin, Induced quasicocycles on groups with hyperbolically embedded subgroups, Algebr. Geom. Topol. 13 (2013) 2635 | DOI

[33] M Kapovich, B Leeb, Actions of discrete groups on nonpositively curved spaces, Math. Ann. 306 (1996) 341 | DOI

[34] M Kapovich, B Leeb, 3-manifold groups and nonpositive curvature, Geom. Funct. Anal. 8 (1998) 841 | DOI

[35] B Leeb, 3-manifolds with(out) metrics of nonpositive curvature, Invent. Math. 122 (1995) 277 | DOI

[36] Y Liu, A characterization of virtually embedded subsurfaces in 3-manifolds, Trans. Amer. Math. Soc. 369 (2017) 1237 | DOI

[37] J M Mackay, A Sisto, Embedding relatively hyperbolic groups in products of trees, Algebr. Geom. Topol. 13 (2013) 2261 | DOI

[38] J F Manning, Geometry of pseudocharacters, Geom. Topol. 9 (2005) 1147 | DOI

[39] J F Manning, Quasi-actions on trees and property (QFA), J. Lond. Math. Soc. 73 (2006) 84 | DOI

[40] H A Masur, Y N Minsky, Geometry of the complex of curves, II : Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902 | DOI

[41] H T Nguyen, H Sun, Subgroup distortion of 3-manifold groups, Trans. Amer. Math. Soc. 373 (2020) 6683 | DOI

[42] H T Nguyen, H C Tran, W Yang, Quasiconvexity in 3-manifold groups, Math. Ann. 381 (2021) 405 | DOI

[43] H T Nguyen, W Yang, Croke–Kleiner admissible groups : property (QT) and quasiconvexity, Michigan Math. J. 73 (2023) 971 | DOI

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

[45] G Paulik, Gluing spaces and analysis, PhD thesis, Rheinische Friedrich-Wilhelms-Universität Bonn (2004)

[46] P Przytycki, D T Wise, Mixed 3-manifolds are virtually special, J. Amer. Math. Soc. 31 (2018) 319 | DOI

[47] A W Reid, Profinite rigidity, from: "Proceedings of the International Congress of Mathematicians, II" (editors B Sirakov, P N d Souza, M Viana), World Sci. (2018) 1193

[48] A Sisto, Projections and relative hyperbolicity, Enseign. Math. 59 (2013) 165 | DOI

[49] H Sun, A characterization on separable subgroups of 3-manifold groups, J. Topol. 13 (2020) 187 | DOI

[50] H Sun, All finitely generated 3-manifold groups are Grothendieck rigid, Groups Geom. Dyn. 17 (2023) 385 | DOI

[51] J Tidmore, Cocompact cubulations of mixed 3-manifolds, Groups Geom. Dyn. 12 (2018) 1429 | DOI

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

[53] D T Wise, The structure of groups with a quasiconvex hierarchy, 209, Princeton Univ. Press (2021)

[54] W Y Yang, Statistically convex-cocompact actions of groups with contracting elements, Int. Math. Res. Not. 2019 (2019) 7259 | DOI