Not all finitely generated groups have universal acylindrical actions
HTML articles powered by AMS MathViewer
- by Carolyn R. Abbott PDF
- Proc. Amer. Math. Soc. 144 (2016), 4151-4155 Request permission
Abstract:
The class of acylindrically hyperbolic groups, which are groups that admit a certain type of non-elementary action on a hyperbolic space, contains many interesting groups such as non-exceptional mapping class groups and $\operatorname {Out}(\mathbb F_n)$ for $n\geq 2$. In such a group, a generalized loxodromic element is one that is loxodromic for some acylindrical action of the group on a hyperbolic space. Osin asks whether every finitely generated group has an acylindrical action on a hyperbolic space for which all generalized loxodromic elements are loxodromic. We answer this question in the negative, using Dunwoody’s example of an inaccessible group as a counterexample.References
- Jason Behrstock, Cornelia Druţu, and Lee Mosher, Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity, Math. Ann. 344 (2009), no. 3, 543–595. MR 2501302, DOI 10.1007/s00208-008-0317-1
- Mladen Bestvina and Koji Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002), 69–89. MR 1914565, DOI 10.2140/gt.2002.6.69
- B. H. Bowditch, Relatively hyperbolic groups, Internat. J. Algebra Comput. 22 (2012), no. 3, 1250016, 66. MR 2922380, DOI 10.1142/S0218196712500166
- Brian H. Bowditch, Tight geodesics in the curve complex, Invent. Math. 171 (2008), no. 2, 281–300. MR 2367021, DOI 10.1007/s00222-007-0081-y
- Martin J. Dunwoody, An inaccessible group, Geometric group theory, Vol. 1 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 181, Cambridge Univ. Press, Cambridge, 1993, pp. 75–78. MR 1238516, DOI 10.1017/CBO9780511661860.007
- M. J. Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985), no. 3, 449–457. MR 807066, DOI 10.1007/BF01388581
- Ursula Hamenstädt, Bounded cohomology and isometry groups of hyperbolic spaces, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 2, 315–349. MR 2390326, DOI 10.4171/JEMS/112
- 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
- Ashot Minasyan and Denis Osin, Acylindrical hyperbolicity of groups acting on trees, Math. Ann. 362 (2015), no. 3-4, 1055–1105. MR 3368093, DOI 10.1007/s00208-014-1138-z
- D. Osin, Acylindrically hyperbolic groups, Trans. Amer. Math. Soc. 368 (2016), no. 2, 851–888. MR 3430352, DOI 10.1090/tran/6343
- Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504
- Alessandro Sisto, Contracting elements and random walks, Preprint, arXiv:1112.2666.
Additional Information
- Carolyn R. Abbott
- Affiliation: Department of Mathematics, University of Wisconsin - Madison, 480 Lincoln Drive, Madison, Wisconsin 53706
- MR Author ID: 1171294
- Email: abbott@math.wisc.edu
- Received by editor(s): June 15, 2015
- Received by editor(s) in revised form: December 17, 2015
- Published electronically: April 27, 2016
- Communicated by: Kevin Whyte
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4151-4155
- MSC (2010): Primary 20F65; Secondary 20F67
- DOI: https://doi.org/10.1090/proc/13101
- MathSciNet review: 3531168