Constructing group actions on quasi-trees and applications to mapping class groups

Bestvina, Mladen; Bromberg, Ken; Fujiwara, Koji

doi:10.1007/s10240-014-0067-4

Bestvina, Mladen ¹ ; Bromberg, Ken ¹ ; Fujiwara, Koji ²

¹ Department of Mathematics, University of Utah 84112 Salt Lake City UT USA
² Department of Mathematics, Kyoto University 606-8502 Kyoto Japan

Publications Mathématiques de l'IHÉS, Tome 122 (2015), pp. 1-64.

Résumé

A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general construction of many actions of groups on quasi-trees. The groups we can handle include non-elementary (relatively) hyperbolic groups, CAT(0) groups with rank 1 elements, mapping class groups and Out(F_n). As an application, we show that mapping class groups act on finite products of δ-hyperbolic spaces so that orbit maps are quasi-isometric embeddings. We prove that mapping class groups have finite asymptotic dimension.

DOI : 10.1007/s10240-014-0067-4

Mots-clés : Asymptotic Dimension, Cayley Graph, Mapping Class Group, Hyperbolic Group, Cayley Tree

Affiliations des auteurs :

Bestvina, Mladen ¹ ; Bromberg, Ken ¹ ; Fujiwara, Koji ²

¹ Department of Mathematics, University of Utah 84112 Salt Lake City UT USA
² Department of Mathematics, Kyoto University 606-8502 Kyoto Japan

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Bestvina, Mladen; Bromberg, Ken; Fujiwara, Koji. Constructing group actions on quasi-trees and applications to mapping class groups. Publications Mathématiques de l'IHÉS, Tome 122 (2015), pp. 1-64. doi : 10.1007/s10240-014-0067-4. https://www.numdam.org/articles/10.1007/s10240-014-0067-4/

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Hamenstädt, Ursula Asymptotic dimension and the disk graph II, Journal of Topology, Volume 12 (2019) no. 3, p. 675 | DOI:10.1112/topo.12096
Abbott, Carolyn R.; Dahmani, François Property $P_{n a i v e}$ P naive for acylindrically hyperbolic groups, Mathematische Zeitschrift, Volume 291 (2019) no. 1-2, p. 555 | DOI:10.1007/s00209-018-2094-1
Behrstock, Jason; Hagen, Mark; Sisto, Alessandro Hierarchically hyperbolic spaces II: Combination theorems and the distance formula, Pacific Journal of Mathematics, Volume 299 (2019) no. 2, p. 257 | DOI:10.2140/pjm.2019.299.257
Charney, Ruth; Morris-Wright, Rose Artin groups of infinite type: Trivial centers and acylindrical hyperbolicity, Proceedings of the American Mathematical Society, Volume 147 (2019) no. 9, p. 3675 | DOI:10.1090/proc/14503
Yu, G. The Novikov conjecture, Russian Mathematical Surveys, Volume 74 (2019) no. 3, p. 525 | DOI:10.1070/rm9882
Cashen, Christopher; Mackay, John A metrizable topology on the contracting boundary of a group, Transactions of the American Mathematical Society, Volume 372 (2019) no. 3, p. 1555 | DOI:10.1090/tran/7544
Handel, Michael; Mosher, Lee The free splitting complex of a free group, II: Loxodromic outer automorphisms, Transactions of the American Mathematical Society, Volume 372 (2019) no. 6, p. 4053 | DOI:10.1090/tran/7698
Watanabe, Yohsuke Pseudo–Anosov mapping classes from pure mapping classes, Transactions of the American Mathematical Society, Volume 373 (2019) no. 1, p. 419 | DOI:10.1090/tran/7919
Yu, Guoliang Гипотеза Новикова, Успехи математических наук, Volume 74 (2019) no. 3(447), p. 167 | DOI:10.4213/rm9882
Parlier, Hugo; Petri, Bram The Genus of Curve, Pants and Flip Graphs, Discrete Computational Geometry, Volume 59 (2018) no. 1, p. 1 | DOI:10.1007/s00454-017-9922-7
GASTER, JONAH; GREENE, JOSHUA EVAN; VLAMIS, NICHOLAS G. COLORING CURVES ON SURFACES, Forum of Mathematics, Sigma, Volume 6 (2018) | DOI:10.1017/fms.2018.12
Dahmani, François The normal closure of big Dehn twists and plate spinning with rotating families, Geometry Topology, Volume 22 (2018) no. 7, p. 4113 | DOI:10.2140/gt.2018.22.4113
Maher, Joseph; Tiozzo, Giulio Random walks on weakly hyperbolic groups, Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2018 (2018) no. 742, p. 187 | DOI:10.1515/crelle-2015-0076
Gekhtman, Ilya; Taylor, Samuel J.; Tiozzo, Giulio Counting loxodromics for hyperbolic actions, Journal of Topology, Volume 11 (2018) no. 2, p. 379 | DOI:10.1112/topo.12053
Beckhardt, Susan; Goldfarb, Boris Extension properties of asymptotic property C and finite decomposition complexity, Topology and its Applications, Volume 239 (2018), p. 181 | DOI:10.1016/j.topol.2018.02.034
Balasubramanya, Sahana Acylindrical group actions on quasi-trees, Algebraic Geometric Topology, Volume 17 (2017) no. 4, p. 2145 | DOI:10.2140/agt.2017.17.2145
Martin, Alexandre On the acylindrical hyperbolicity of the tame automorphism group of SL2(C), Bulletin of the London Mathematical Society, Volume 49 (2017) no. 5, p. 881 | DOI:10.1112/blms.12071
Eskin, Alex; Masur, Howard; Rafi, Kasra Large-scale rank of Teichmüller space, Duke Mathematical Journal, Volume 166 (2017) no. 8 | DOI:10.1215/00127094-0000006x
Behrstock, Jason; Hagen, Mark; Sisto, Alessandro Hierarchically hyperbolic spaces, I: Curve complexes for cubical groups, Geometry Topology, Volume 21 (2017) no. 3, p. 1731 | DOI:10.2140/gt.2017.21.1731
Sisto, Alessandro Tracking rates of random walks, Israel Journal of Mathematics, Volume 220 (2017) no. 1, p. 1 | DOI:10.1007/s11856-017-1508-9
Cordes, Matthew; Hume, David Stability and the Morse boundary, Journal of the London Mathematical Society, Volume 95 (2017) no. 3, p. 963 | DOI:10.1112/jlms.12042
Gültepe, Funda; Leininger, Christopher An arc graph distance formula for the flip graph, Proceedings of the American Mathematical Society, Volume 145 (2017) no. 7, p. 3179 | DOI:10.1090/proc/13451
Behrstock, Jason; Hagen, Mark F.; Sisto, Alessandro Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups, Proceedings of the London Mathematical Society, Volume 114 (2017) no. 5, p. 890 | DOI:10.1112/plms.12026
Martínez-Pérez, Álvaro Generalized Chordality, Vertex Separators and Hyperbolicity on Graphs, Symmetry, Volume 9 (2017) no. 10, p. 199 | DOI:10.3390/sym9100199
Delzant, Thomas Kähler groups, $R$ R -trees, and holomorphic families of riemann surfaces, Geometric and Functional Analysis, Volume 26 (2016) no. 1, p. 160 | DOI:10.1007/s00039-016-0355-x
Horbez, Camille Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings, Journal of Topology, Volume 9 (2016) no. 2, p. 401 | DOI:10.1112/jtopol/jtv045
Arzhantseva, Goulnara; Cashen, Christopher; Tao, Jing Growth tight actions, Pacific Journal of Mathematics, Volume 278 (2015) no. 1, p. 1 | DOI:10.2140/pjm.2015.278.1
Webb, Richard Combinatorics of tight geodesics and stable lengths, Transactions of the American Mathematical Society, Volume 367 (2015) no. 10, p. 7323 | DOI:10.1090/tran/6301

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