Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms
[Structures équivariantes hiérarchiquement hyperboliques pour les groupes de 3-variétés via les quasimorphismes]
Hagen, Mark1 ; Russell, Jacob2 ; Sisto, Alessandro3 ; Spriano, Davide4
1 School of Mathematics, University of Bristol, Bristol (UK)
2 Department of Mathematics, Rice University, Houston, TX (USA)
3 Maxwell Institute and Department of Mathematics, Heriot-Watt University, Edinburgh (UK)
4 Mathematical Institute, University of Oxford, Oxford (UK)
Annales de l'Institut Fourier, Online first, 60 p.

Behrstock, Hagen et Sisto ont classifié les groupes fondamentaux des 3-variétés qui admettent une structure d’espace hyperbolique hiérarchique. Mais ces structures e sont pas toujours équivariantes. Dans cet article, nous classifions les groupes fondamentaux des 3-variétés admettant des structures HHS équivariantes. L’élément clé de notre preuve est que les groupes admissibles introduits par Croke et Kleiner admettent toujours des structures HHS équivariantes. Pour les variétés de graphes non géométriques, cela est contraire à une conjecture de Behrstock, Hagen et Sisto et contraste également avec des résultats sur les structures cubiques sur ces groupes. Nos arguments impliquent la construction de quasimorphismes appropriés sur les pièces de Seifert, afin de construire des actions sur des quasi-lignes.

Behrstock, Hagen and Sisto classified 3-manifold groups admitting a hierarchically hyperbolic space structure. However, these structures were not always equivariant with respect to the group. In this paper, we classify 3-manifold groups admitting equivariant hierarchically hyperbolic structures. The key component of our proof is that the admissible groups introduced by Croke and Kleiner always admit equivariant hierarchically hyperbolic structures. For non-geometric graph manifolds, this is contrary to a conjecture of Behrstock, Hagen and Sisto and also contrasts with results about CAT(0) cubical structures on these groups. Perhaps surprisingly, our arguments involve the construction of suitable quasimorphisms on the Seifert pieces, in order to construct actions on quasi-lines.

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Révisé le :
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DOI : 10.5802/aif.3654
Classification : 20F67, 57K35
Keywords: Hierarchically hyperbolic group, 3-manifold.
Mot clés : groupe hiérarchique hyperbolic, 3-variété.

Hagen, Mark 1 ; Russell, Jacob 2 ; Sisto, Alessandro 3 ; Spriano, Davide 4

1 School of Mathematics, University of Bristol, Bristol (UK)
2 Department of Mathematics, Rice University, Houston, TX (USA)
3 Maxwell Institute and Department of Mathematics, Heriot-Watt University, Edinburgh (UK)
4 Mathematical Institute, University of Oxford, Oxford (UK) 
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Hagen, Mark; Russell, Jacob; Sisto, Alessandro; Spriano, Davide. Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms. Annales de l'Institut Fourier, Online first, 60 p.

[1] Abbott, Carolyn; Balasubramanya, Sahana H.; Osin, Denis Hyperbolic structures on groups, Algebr. Geom. Topol., Volume 19 (2019) no. 4, pp. 1747-1835 | DOI | MR | Zbl

[2] Abbott, Carolyn; Behrstock, Jason; Durham, Matthew G. Largest acylindrical actions and stability in hierarchically hyperbolic groups. With an appendix by Daniel Berlyne and Jacob Russell, Trans. Amer. Math. Soc., Ser. B, Volume 8 (2021), pp. 66-104 | DOI | MR | Zbl

[3] Abbott, Carolyn; Ng, Thomas; Spriano, Davide Hierarchically hyperbolic groups and uniform exponential growth (2019) (with appendix by Radhika Gupta and Harry Petyt, https://arxiv.org/abs/1909.00439)

[4] Aschenbrenner, Matthias; Friedl, Stefan; Wilton, Henry 3-manifold groups, EMS Series of Lectures in Mathematics, European Mathematical Society, 2015, xiv+215 pages | DOI | MR | Zbl

[5] Behrstock, Jason; Hagen, Mark F.; Martin, Alexandre; Sisto, Alessandro A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups (2020) (https://arxiv.org/abs/2005.00567)

[6] Behrstock, Jason; Hagen, Mark F.; Sisto, Alessandro Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups, Proc. Lond. Math. Soc., Volume 114 (2017) no. 5, pp. 890-926 | DOI | MR | Zbl

[7] Behrstock, Jason; Hagen, Mark F.; Sisto, Alessandro Hierarchically hyperbolic spaces I: Curve complexes for cubical groups, Geom. Topol., Volume 21 (2017) no. 3, pp. 1731-1804 | DOI | MR | Zbl

[8] Behrstock, Jason; Hagen, Mark F.; Sisto, Alessandro Hierarchically hyperbolic spaces II: Combination theorems and the distance formula, Pac. J. Math., Volume 299 (2019), pp. 257-338 | DOI | Zbl

[9] Behrstock, Jason; Hagen, Mark F.; Sisto, Alessandro Quasiflats in hierarchically hyperbolic spaces, Duke Math. J., Volume 170 (2021) no. 5, pp. 909-996 | DOI | MR | Zbl

[10] Berlai, Federico; Robbio, Bruno A refined combination theorem for hierarchically hyperbolic groups, Groups Geom. Dyn., Volume 14 (2020) no. 4, pp. 1127-1203 | DOI | MR | Zbl

[11] Berlyne, Daniel Hierarchical Hyperbolicity of Graph Products and Graph Braid Groups, Ph. D. Thesis, City University of New York (2021) (https://www.proquest.com/docview/2543702948)

[12] Berlyne, Daniel; Russell, Jacob Hierarchical hyperbolicity of graph products, Groups Geom. Dyn., Volume 16 (2022) no. 2, pp. 523-580 | DOI | MR | Zbl

[13] Bestvina, Mladen; Bromberg, Ken; Fujiwara, Koji Constructing group actions on quasi-trees and applications to mapping class groups, Publ. Math., Inst. Hautes Étud. Sci., Volume 122 (2015), pp. 1-64 | DOI | Numdam | MR | Zbl

[14] Bigdely, Hadi; Wise, Daniel T. Quasiconvexity and relatively hyperbolic groups that split, Mich. Math. J., Volume 62 (2013) no. 2, pp. 387-406 | DOI | MR | Zbl

[15] Bowditch, Brian H. Relatively hyperbolic groups, Int. J. Algebra Comput., Volume 22 (2012) no. 3, 1250016, 66 pages | DOI | MR | Zbl

[16] Caprace, Pierre-Emmanuel; Cornulier, Yves; Monod, Nicolas; Tessera, Romain Amenable hyperbolic groups, J. Eur. Math. Soc., Volume 17 (2015) no. 11, pp. 2903-2947 | DOI | MR | Zbl

[17] Charney, Ruth; Cordes, Matthew; Sisto, Alessandro Complete topological descriptions of certain Morse boundaries, Groups Geom. Dyn., Volume 17 (2023) no. 1, pp. 157-184 | DOI | MR | Zbl

[18] Charney, Ruth; Crisp, John Relative hyperbolicity and Artin groups, Geom. Dedicata, Volume 129 (2007), pp. 1-13 | DOI | MR | Zbl

[19] Chatterji, Indira; Niblo, Graham From wall spaces to CAT (0) cube complexes, Int. J. Algebra Comput., Volume 15 (2005) no. 5-6, pp. 875-885 | DOI | MR | Zbl

[20] Croke, Christopher B.; Kleiner, Bruce The geodesic flow of a nonpositively curved graph manifold, Geom. Funct. Anal., Volume 12 (2002) no. 3, pp. 479-545 | DOI | MR | Zbl

[21] Dahmani, François Combination of convergence groups, Geom. Topol., Volume 7 (2003), pp. 933-963 | DOI | MR | Zbl

[22] Dowdall, Spencer; Durham, Matthew G.; Leininger, Christopher J.; Sisto, Alessandro Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity (2020) (https://arxiv.org/abs/2111.00685)

[23] Durham, Matthew G.; Hagen, Mark F.; Sisto, Alessandro Boundaries and automorphisms of hierarchically hyperbolic spaces, Geom. Topol., Volume 21 (2017) no. 6, pp. 3659-3758 | DOI | MR | Zbl

[24] Durham, Matthew G.; Hagen, Mark F.; Sisto, Alessandro Correction to the article Boundaries and automorphisms of hierarchically hyperbolic spaces, Geom. Topol., Volume 24 (2020) no. 2, pp. 1051-1073 | DOI | MR | Zbl

[25] Durham, Matthew G.; Minsky, Yair N.; Sisto, Alessandro Stable cubulations, bicombings, and barycenters (2020) (https://arxiv.org/abs/2009.13647)

[26] Epstein, David B. A.; Fujiwara, Koji The second bounded cohomology of word-hyperbolic groups, Topology, Volume 36 (1997) no. 6, pp. 1275-1289 | DOI | MR | Zbl

[27] Haettel, Thomas; Hoda, Nima; Petyt, Harry The coarse Helly property, hierarchical hyperbolicity, and semihyperbolicity (2020) (https://arxiv.org/abs/2009.14053)

[28] Hagen, Mark F. Cocompactly cubulated crystallographic groups, J. Lond. Math. Soc., Volume 90 (2014) no. 1, pp. 140-166 | DOI | MR | Zbl

[29] Hagen, Mark F.; Martin, Alexandre; Sisto, Alessandro Extra-large type artin groups are hierarchically hyperbolic (2021) (https://arxiv.org/abs/2109.04387)

[30] Hagen, Mark F.; Petyt, Harry Projection complexes and quasimedian maps, Algebr. Geom. Topol., Volume 22 (2022) no. 7, pp. 3277-3304 | DOI | MR | Zbl

[31] Hagen, Mark F.; Przytycki, Piotr Cocompactly cubulated graph manifolds, Isr. J. Math., Volume 207 (2015) no. 1, pp. 377-394 | DOI | MR | Zbl

[32] Hagen, Mark F.; Susse, Tim On hierarchical hyperbolicity of cubical groups, Isr. J. Math., Volume 236 (2020) no. 1, pp. 45-89 | DOI | MR | Zbl

[33] Hoda, Nima Crystallographic Helly Groups (2020) (https://arxiv.org/abs/2010.07407)

[34] Hughes, Sam Lattices in a product of trees, hierarchically hyperbolic groups and virtual torsion-freeness, Bull. Lond. Math. Soc., Volume 54 (2022) no. 4, pp. 1413-1419 | DOI | MR | Zbl

[35] Hull, Michael; Osin, Denis Induced quasicocycles on groups with hyperbolically embedded subgroups, Algebr. Geom. Topol., Volume 13 (2013) no. 5, pp. 2635-2665 | DOI | MR | Zbl

[36] Kapovich, Ilya; Rafi, Kasra On hyperbolicity of free splitting and free factor complexes, Groups Geom. Dyn., Volume 8 (2014) no. 2, pp. 391-414 | DOI | MR | Zbl

[37] Kapovich, Michael; Leeb, Bernhard 3-manifold groups and nonpositive curvature, Geom. Funct. Anal., Volume 8 (1998) no. 5, pp. 841-852 | DOI | MR | Zbl

[38] Kim, Sang-hyun; Koberda, Thomas Embedability between right-angled Artin groups, Geom. Topol., Volume 17 (2013) no. 1, pp. 493-530 | DOI | MR | Zbl

[39] Manning, Jason Fox Geometry of pseudocharacters, Geom. Topol., Volume 9 (2005), pp. 1147-1185 | DOI | MR | Zbl

[40] Miller, Marissa Stable subgroups of the genus two handlebody group (2020) (https://arxiv.org/abs/2009.05067)

[41] Mineyev, Igor Straightening and bounded cohomology of hyperbolic groups, Geom. Funct. Anal., Volume 11 (2001) no. 4, pp. 807-839 | DOI | MR | Zbl

[42] Nguyen, Hoang Thanh; Qing, Yulan Sublinearly Morse Boundary of CAT(0) admissible groups (2022) (https://arxiv.org/abs/2203.00935)

[43] Petyt, Harry Mapping class groups are quasicubical (2021) (https://arxiv.org/abs/2112.10681)

[44] Petyt, Harry; Spriano, Davide Unbounded domains in hierarchically hyperbolic groups, Groups Geom. Dyn., Volume 17 (2023) no. 2, pp. 479-500 | DOI | MR | Zbl

[45] Robbio, Bruno; Spriano, Davide Hierarchical hyperbolicity of hyperbolic-2-decomposable groups (2020) (https://arxiv.org/abs/2007.13383)

[46] Russell, Jacob Extensions of multicurve stabilizers are hierarchically hyperbolic (2021) (https://arxiv.org/abs/2107.14116)

[47] Russell, Jacob From hierarchical to relative hyperbolicity, Int. Math. Res. Not. (2022) no. 1, pp. 575-624 | DOI | MR | Zbl

[48] Russell, Jacob; Spriano, Davide; Tran, Hung Cong Convexity in hierarchically hyperbolic spaces, Algebr. Geom. Topol., Volume 23 (2023) no. 3, pp. 1167-1248 | DOI | MR | Zbl

[49] Scott, Peter The geometries of 3-manifolds, Bull. Lond. Math. Soc., Volume 15 (1983) no. 5, pp. 401-487 | DOI | MR | Zbl

[50] Scott, Peter; Wall, Terry Topological methods in group theory, Homological group theory (Proc. Sympos., Durham, 1977) (London Mathematical Society Lecture Note Series), Volume 36, Cambridge University Press (1979), pp. 137-203 | DOI | MR | Zbl

[51] Sisto, Alessandro On metric relative hyperbolicity (2012) (https://arxiv.org/abs/1210.8081)

[52] Sisto, Alessandro Projections and relative hyperbolicity, Enseign. Math., Volume 59 (2013) no. 1-2, pp. 165-181 | DOI | MR | Zbl

[53] Sisto, Alessandro What is a hierarchically hyperbolic space?, Beyond hyperbolicity (London Mathematical Society Lecture Note Series), Volume 454, Cambridge University Press, 2019, pp. 117-148 | DOI | MR | Zbl

[54] Spriano, Davide Hyperbolic HHS I: Factor Systems and Quasi-convex subgroups. (2017) (https://arxiv.org/abs/1711.10931)

[55] Vokes, Kate M. Hierarchical hyperbolicity of graphs of multicurves, Algebr. Geom. Topol., Volume 22 (2022) no. 1, pp. 113-151 | DOI | MR | Zbl

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