The geometry of purely loxodromic subgroups of right-angled Artin groups

Koberda, Thomas; Mangahas, Johanna; Taylor, Samuel

doi:10.1090/tran/6933

The geometry of purely loxodromic subgroups of right-angled Artin groups
HTML articles powered by AMS MathViewer

by Thomas Koberda, Johanna Mangahas and Samuel J. Taylor PDF

Trans. Amer. Math. Soc. 369 (2017), 8179-8208 Request permission

Abstract:

We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group $A(\Gamma )$ fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups $\mathrm {Mod}(S)$. In particular, such subgroups are quasiconvex in $A(\Gamma )$. In addition, we identify a milder condition for a finitely generated subgroup of $A(\Gamma )$ that guarantees it is free, undistorted, and retains finite generation when intersected with $A(\Lambda )$ for subgraphs $\Lambda$ of $\Gamma$. These results have applications to both the study of convex cocompactness in $\mathrm {Mod}(S)$ and the way in which certain groups can embed in right-angled Artin groups.

References

Ian Agol, The virtual Haken conjecture, Doc. Math. 18 (2013), 1045–1087. With an appendix by Agol, Daniel Groves, and Jason Manning. MR 3104553
Jason Behrstock and Ruth Charney, Divergence and quasimorphisms of right-angled Artin groups, Math. Ann. 352 (2012), no. 2, 339–356. MR 2874959, DOI 10.1007/s00208-011-0641-8
Jason A. Behrstock, Asymptotic geometry of the mapping class group and Teichmüller space, Geom. Topol. 10 (2006), 1523–1578. MR 2255505, DOI 10.2140/gt.2006.10.1523
Mladen Bestvina and Koji Fujiwara, A characterization of higher rank symmetric spaces via bounded cohomology, Geom. Funct. Anal. 19 (2009), no. 1, 11–40. MR 2507218, DOI 10.1007/s00039-009-0717-8
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
Jason Behrstock, Mark Hagen, and Alessandro Sisto, Hierarchically hyperbolic spaces I: curve complexes for cubical groups, preprint.
Matt T. Clay, Christopher J. Leininger, and Johanna Mangahas, The geometry of right-angled Artin subgroups of mapping class groups, Groups Geom. Dyn. 6 (2012), no. 2, 249–278. MR 2914860, DOI 10.4171/GGD/157
Ruth Charney and Luis Paris, Convexity of parabolic subgroups in Artin groups, Bull. Lond. Math. Soc. 46 (2014), no. 6, 1248–1255. MR 3291260, DOI 10.1112/blms/bdu077
John Crisp and Bert Wiest, Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups, Algebr. Geom. Topol. 4 (2004), 439–472. MR 2077673, DOI 10.2140/agt.2004.4.439
F. Dahmani, V. Guirardel, and D. Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Amer. Math. Soc. 245 (2017), no. 1156, v+152. MR 3589159, DOI 10.1090/memo/1156
Spencer Dowdall, Richard P. Kent IV, and Christopher J. Leininger, Pseudo-Anosov subgroups of fibered 3-manifold groups, Groups Geom. Dyn. 8 (2014), no. 4, 1247–1282. MR 3314946, DOI 10.4171/GGD/302
Carl Droms, Graph groups, coherence, and three-manifolds, J. Algebra 106 (1987), no. 2, 484–489. MR 880971, DOI 10.1016/0021-8693(87)90010-X
Matthew Gentry Durham and Samuel J. Taylor, Convex cocompactness and stability in mapping class groups, Algebr. Geom. Topol. 15 (2015), no. 5, 2839–2859. MR 3426695, DOI 10.2140/agt.2015.15.2839
Benson Farb and Lee Mosher, Convex cocompact subgroups of mapping class groups, Geom. Topol. 6 (2002), 91–152. MR 1914566, DOI 10.2140/gt.2002.6.91
Frédéric Haglund, Finite index subgroups of graph products, Geom. Dedicata 135 (2008), 167–209. MR 2413337, DOI 10.1007/s10711-008-9270-0
Ursula Hamenstadt, Word hyperbolic extensions of surface groups, preprint (2005), arXiv:math/0505244.
Susan Hermiller and John Meier, Algorithms and geometry for graph products of groups, J. Algebra 171 (1995), no. 1, 230–257. MR 1314099, DOI 10.1006/jabr.1995.1010
Mark F. Hagen and Daniel T. Wise, Cubulating hyperbolic free-by-cyclic groups: the irreducible case, Duke Math. J. 165 (2016), no. 9, 1753–1813. MR 3513573, DOI 10.1215/00127094-3450752
Mark F. Hagen and Daniel T. Wise, Cubulating hyperbolic free-by-cyclic groups: the general case, Geom. Funct. Anal. 25 (2015), no. 1, 134–179. MR 3320891, DOI 10.1007/s00039-015-0314-y
Sang-hyun Kim, Co-contractions of graphs and right-angled Artin groups, Algebr. Geom. Topol. 8 (2008), no. 2, 849–868. MR 2443098, DOI 10.2140/agt.2008.8.849
Sang-hyun Kim and Thomas Koberda, Embedability between right-angled Artin groups, Geom. Topol. 17 (2013), no. 1, 493–530. MR 3039768, DOI 10.2140/gt.2013.17.493
Sang-Hyun Kim and Thomas Koberda, The geometry of the curve graph of a right-angled Artin group, Internat. J. Algebra Comput. 24 (2014), no. 2, 121–169. MR 3192368, DOI 10.1142/S021819671450009X
Sang-Hyun Kim and Thomas Koberda, An obstruction to embedding right-angled Artin groups in mapping class groups, Int. Math. Res. Not. IMRN 14 (2014), 3912–3918. MR 3239092, DOI 10.1093/imrn/rnt064
Richard P. Kent IV and Christopher J. Leininger, Subgroups of mapping class groups from the geometrical viewpoint, In the tradition of Ahlfors-Bers. IV, Contemp. Math., vol. 432, Amer. Math. Soc., Providence, RI, 2007, pp. 119–141. MR 2342811, DOI 10.1090/conm/432/08306
Richard P. Kent IV and Christopher J. Leininger, Shadows of mapping class groups: capturing convex cocompactness, Geom. Funct. Anal. 18 (2008), no. 4, 1270–1325. MR 2465691, DOI 10.1007/s00039-008-0680-9
Richard P. Kent IV, Christopher J. Leininger, and Saul Schleimer, Trees and mapping class groups, J. Reine Angew. Math. 637 (2009), 1–21. MR 2599078, DOI 10.1515/CRELLE.2009.087
Jeremy Kahn and Vladimir Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold, Ann. of Math. (2) 175 (2012), no. 3, 1127–1190. MR 2912704, DOI 10.4007/annals.2012.175.3.4
Thomas Koberda, Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups, Geom. Funct. Anal. 22 (2012), no. 6, 1541–1590. MR 3000498, DOI 10.1007/s00039-012-0198-z
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition. MR 1812024, DOI 10.1007/978-3-642-61896-3
Joseph Maher, Random walks on the mapping class group, Duke Math. J. 156 (2011), no. 3, 429–468. MR 2772067, DOI 10.1215/00127094-2010-216
K. A. Mihaĭlova, The occurrence problem for free products of groups, Mat. Sb. (N.S.) 75 (117) (1968), 199–210 (Russian). MR 0222179
Yair N. Minsky, Quasi-projections in Teichmüller space, J. Reine Angew. Math. 473 (1996), 121–136. MR 1390685, DOI 10.1515/crll.1995.473.121
Howard A. Masur and Yair N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103–149. MR 1714338, DOI 10.1007/s002220050343
Johanna Mangahas and Samuel J. Taylor, Convex cocompactness in mapping class groups via quasiconvexity in right-angled Artin groups, Proc. Lond. Math. Soc. (3) 112 (2016), no. 5, 855–881. MR 3502022, DOI 10.1112/plms/pdw009
D. Osin, Acylindrically hyperbolic groups, Trans. Amer. Math. Soc. 368 (2016), no. 2, 851–888. MR 3430352, DOI 10.1090/tran/6343
Igor Rivin, Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms, Duke Math. J. 142 (2008), no. 2, 353–379. MR 2401624, DOI 10.1215/00127094-2008-009
Michah Sageev, $\rm CAT(0)$ cube complexes and groups, Geometric group theory, IAS/Park City Math. Ser., vol. 21, Amer. Math. Soc., Providence, RI, 2014, pp. 7–54. MR 3329724, DOI 10.1090/pcms/021/02
Mark Sapir, A Higman embedding preserving asphericity, J. Amer. Math. Soc. 27 (2014), no. 1, 1–42. MR 3110794, DOI 10.1090/S0894-0347-2013-00776-6
Petra Schwer, Lecture notes on cat(0) cubical complexes, Preprint.
Herman Servatius, Automorphisms of graph groups, J. Algebra 126 (1989), no. 1, 34–60. MR 1023285, DOI 10.1016/0021-8693(89)90319-0
Alessandro Sisto, Exponential triples, Electron. J. Combin. 18 (2011), no. 1, Paper 147, 17. MR 2817797
Samuel J. Taylor and Giulio Tiozzo, Random extensions of free groups and surface groups are hyperbolic, Int. Math. Res. Not. IMRN 1 (2016), 294–310. MR 3514064, DOI 10.1093/imrn/rnv138
Daniel T. Wise, The structure of groups with a quasiconvex hierarchy.