Stable cubulations, bicombings, and barycenters

Durham, Matthew G¹ ; Minsky, Yair N² ; Sisto, Alessandro³

¹ Department of Mathematics, University of California Riverside, Riverside, CA, United States
² Department of Mathematics, Yale University, New Haven, CT, United States
³ Department of Mathematics, Heriot-Watt University, Edinburgh, United Kingdom

Geometry & topology, Tome 27 (2023) no. 6, pp. 2383-2478.

Résumé

We prove that the hierarchical hulls of finite sets of points in mapping class groups and Teichmüller spaces are stably approximated by CAT(0) cube complexes, strengthening a result of Behrstock, Hagen and Sisto. As applications, we prove that mapping class groups are semihyperbolic and Teichmüller spaces are coarsely equivariantly bicombable, and both admit stable coarse barycenters. Our results apply to the broader class of “colorable” hierarchically hyperbolic spaces and groups.

Détail

DOI : 10.2140/gt.2023.27.2383

Keywords: semihyperbolic, bicombing, mapping class group, hierarchically hyperbolic, cube complexes, Teichmüller space

Bibliographie

[1] J M Alonso, M R Bridson, Semihyperbolic groups, Proc. London Math. Soc. 70 (1995) 56 | DOI

[2] G Baumslag, S M Gersten, M Shapiro, H Short, Automatic groups and amalgams — a survey, from: "Algorithms and classification in combinatorial group theory" (editors G Baumslag, C F Miller III), Math. Sci. Res. Inst. Publ. 23, Springer (1992) 179 | DOI

[3] J A Behrstock, Asymptotic geometry of the mapping class group and Teichmüller space, Geom. Topol. 10 (2006) 1523 | DOI

[4] J Behrstock, M Hagen, A Martin, A Sisto, A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups, preprint (2020)

[5] J Behrstock, M F Hagen, A Sisto, Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups, Proc. Lond. Math. Soc. 114 (2017) 890 | DOI

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

[7] J Behrstock, M Hagen, A Sisto, Hierarchically hyperbolic spaces, II : Combination theorems and the distance formula, Pacific J. Math. 299 (2019) 257 | DOI

[8] J Behrstock, M F Hagen, A Sisto, Quasiflats in hierarchically hyperbolic spaces, Duke Math. J. 170 (2021) 909 | DOI

[9] J Behrstock, B Kleiner, Y Minsky, L Mosher, Geometry and rigidity of mapping class groups, Geom. Topol. 16 (2012) 781 | DOI

[10] J A Behrstock, Y N Minsky, Centroids and the rapid decay property in mapping class groups, J. Lond. Math. Soc. 84 (2011) 765 | DOI

[11] F Berlai, B Robbio, A refined combination theorem for hierarchically hyperbolic groups, Groups Geom. Dyn. 14 (2020) 1127 | DOI

[12] 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

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

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

[15] B H Bowditch, Coarse median spaces and groups, Pacific J. Math. 261 (2013) 53 | DOI

[16] B Bowditch, Convex hulls in coarse median spaces, preprint (2018)

[17] M R Bridson, Semisimple actions of mapping class groups on CAT(0) spaces, from: "Geometry of Riemann surfaces" (editors F P Gardiner, G González-Diez, C Kourouniotis), London Math. Soc. Lecture Note Ser. 368, Cambridge Univ. Press (2010) 1 | DOI

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

[19] P E Caprace, M Sageev, Rank rigidity for CAT(0) cube complexes, Geom. Funct. Anal. 21 (2011) 851 | DOI

[20] J Chalopin, V Chepoi, A Genevois, H Hirai, D Osajda, Helly groups, preprint (2020)

[21] I Chatterji, G Niblo, From wall spaces to CAT(0) cube complexes, Internat. J. Algebra Comput. 15 (2005) 875 | DOI

[22] V Chepoi, Graphs of some CAT(0) complexes, Adv. in Appl. Math. 24 (2000) 125 | DOI

[23] M Chesser, Stable subgroups of the genus 2 handlebody group, Algebr. Geom. Topol. 22 (2022) 919 | DOI

[24] M T Clay, C J Leininger, J Mangahas, The geometry of right-angled Artin subgroups of mapping class groups, Groups Geom. Dyn. 6 (2012) 249 | DOI

[25] S Dowdall, M G Durham, C Leininger, A Sisto, Extensions of Veech groups, I: A hyperbolic action, preprint (2020)

[26] S Dowdall, M G Durham, C J Leininger, A Sisto, Extensions of Veech groups, II: Hierarchical hyperbolicity and quasi-isometric rigidity, preprint (2021)

[27] M G Durham, The augmented marking complex of a surface, J. Lond. Math. Soc. 94 (2016) 933 | DOI

[28] M G Durham, Elliptic actions on Teichmüller space, Groups Geom. Dyn. 13 (2019) 415 | DOI

[29] M G Durham, M F Hagen, A Sisto, Boundaries and automorphisms of hierarchically hyperbolic spaces, Geom. Topol. 21 (2017) 3659 | DOI

[30] M G Durham, M F Hagen, A Sisto, Correction to the article “Boundaries and automorphisms of hierarchically hyperbolic spaces”, Geom. Topol. 24 (2020) 1051 | DOI

[31] D B A Epstein, J W Cannon, D F Holt, S V F Levy, M S Paterson, W P Thurston, Word processing in groups, Jones and Bartlett (1992) | DOI

[32] A Eskin, H Masur, K Rafi, Large-scale rank of Teichmüller space, Duke Math. J. 166 (2017) 1517 | DOI

[33] S M Gersten, H B Short, Rational subgroups of biautomatic groups, Ann. of Math. 134 (1991) 125 | DOI

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

[35] T Haettel, N Hoda, H Petyt, Coarse injectivity, hierarchical hyperbolicity, and semihyperbolicity, preprint (2020)

[36] M Hagen, Non-colorable hierarchically hyperbolic groups, Internat. J. Algebra Comput. 33 (2023) 337 | DOI

[37] M Hagen, A Martin, A Sisto, Extra-large type Artin groups are hierarchically hyperbolic, Math. Ann. (2022) | DOI

[38] M F Hagen, T Susse, On hierarchical hyperbolicity of cubical groups, Israel J. Math. 236 (2020) 45 | DOI

[39] F Haglund, F Paulin, Simplicité de groupes d’automorphismes d’espaces à courbure négative, from: "The Epstein birthday schrift" (editors I Rivin, C Rourke, C Series), Geom. Topol. Monogr. 1, Geom. Topol. (1998) 181 | DOI

[40] U Hamenstädt, Geometry of the mapping class group, II : A biautomatic structure, preprint (2009)

[41] G C Hruska, D T Wise, Finiteness properties of cubulated groups, Compos. Math. 150 (2014) 453 | DOI

[42] H A Masur, Y N Minsky, Geometry of the complex of curves, I : Hyperbolicity, Invent. Math. 138 (1999) 103 | DOI

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

[44] B Miesch, Injective metrics on cube complexes, preprint (2014)

[45] L Mosher, Mapping class groups are automatic, Ann. of Math. 142 (1995) 303 | DOI

[46] G A Niblo, L D Reeves, The geometry of cube complexes and the complexity of their fundamental groups, Topology 37 (1998) 621 | DOI

[47] B Nica, Cubulating spaces with walls, Algebr. Geom. Topol. 4 (2004) 297 | DOI

[48] K Rafi, A combinatorial model for the Teichmüller metric, Geom. Funct. Anal. 17 (2007) 936 | DOI

[49] K Rafi, Hyperbolicity in Teichmüller space, Geom. Topol. 18 (2014) 3025 | DOI

[50] K Rafi, Y Verberne, Geodesics in the mapping class group, Algebr. Geom. Topol. 21 (2021) 2995 | DOI

[51] B Robbio, D Spriano, Hierarchical hyperbolicity of hyperbolic-2–decomposable groups, preprint (2020)

[52] J Russell, Extensions of multicurve stabilizers are hierarchically hyperbolic, preprint (2021)

[53] J Russell, D Spriano, H C Tran, Convexity in hierarchically hyperbolic spaces, Algebr. Geom. Topol. 23 (2023) 1167 | DOI

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

[55] M Sageev, CAT(0) cube complexes and groups, from: "Geometric group theory" (editors M Bestvina, M Sageev, K Vogtmann), IAS/Park City Math. Ser. 21, Amer. Math. Soc. (2014) 7 | DOI

[56] D Spriano, Hyperbolic HHS, II : Graphs of hierarchically hyperbolic groups, preprint (2018)

[57] P A Storm, The Novikov conjecture for mapping class groups as a corollary of Hamenstädt’s theorem, preprint (2005)

[58] J Tao, Linearly bounded conjugator property for mapping class groups, Geom. Funct. Anal. 23 (2013) 415 | DOI

[59] A J Tromba, On a natural algebraic affine connection on the space of almost complex structures and the curvature of Teichmüller space with respect to its Weil–Petersson metric, Manuscripta Math. 56 (1986) 475 | DOI

[60] S A Wolpert, Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85 (1986) 119 | DOI