Thick groups have trivial Floyd boundary
HTML articles powered by AMS MathViewer
- by Ivan Levcovitz PDF
- Proc. Amer. Math. Soc. 148 (2020), 513-521 Request permission
Abstract:
We prove that thick groups (and more generally thick graphs) have trivial Floyd boundary. This shows that a wide class of finitely generated groups that are non-relatively hyperbolic have trivial Floyd boundary. In addition to giving new examples, our result provides a common proof and framework for many of the known results in the literature.References
- Jason Behrstock and Cornelia Druţu, Divergence, thick groups, and short conjugators, Illinois J. Math. 58 (2014), no. 4, 939–980. MR 3421592
- 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
- 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
- Jason Behrstock, Mark F. Hagen, and Alessandro Sisto, Thickness, relative hyperbolicity, and randomness in Coxeter groups, Algebr. Geom. Topol. 17 (2017), no. 2, 705–740. With an appendix written jointly with Pierre-Emmanuel Caprace. MR 3623669, DOI 10.2140/agt.2017.17.705
- 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
- Cornelia Druţu, Shahar Mozes, and Mark Sapir, Divergence in lattices in semisimple Lie groups and graphs of groups, Trans. Amer. Math. Soc. 362 (2010), no. 5, 2451–2505. MR 2584607, DOI 10.1090/S0002-9947-09-04882-X
- William J. Floyd, Group completions and limit sets of Kleinian groups, Invent. Math. 57 (1980), no. 3, 205–218. MR 568933, DOI 10.1007/BF01418926
- Victor Gerasimov, Floyd maps for relatively hyperbolic groups, Geom. Funct. Anal. 22 (2012), no. 5, 1361–1399. MR 2989436, DOI 10.1007/s00039-012-0175-6
- Ilya Gekhtman, Victor Gerasimov, Leonid Potyagailo, and Wenyuan Yang, Martin boundary covers Floyd boundary, preprint, arXiv:1708.02133, 2017.
- Victor Gerasimov and Leonid Potyagailo, Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 6, 2115–2137. MR 3120738, DOI 10.4171/JEMS/417
- M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
- Mark Hagen, A remark on thickness of free-by-cyclic groups, https://www.wescac.net/research.html, 2019, to appear in Illinois J. Math.
- Anders Karlsson, Free subgroups of groups with nontrivial Floyd boundary, Comm. Algebra 31 (2003), no. 11, 5361–5376. MR 2005231, DOI 10.1081/AGB-120023961
- Anders Karlsson and Guennadi A. Noskov, Some groups having only elementary actions on metric spaces with hyperbolic boundaries, Geom. Dedicata 104 (2004), 119–137. MR 2043957, DOI 10.1023/B:GEOM.0000022949.67521.0c
- Alexander Yu. Ol′shanskii, Denis V. Osin, and Mark V. Sapir, Lacunary hyperbolic groups, Geom. Topol. 13 (2009), no. 4, 2051–2140. With an appendix by Michael Kapovich and Bruce Kleiner. MR 2507115, DOI 10.2140/gt.2009.13.2051
- Bin Sun, A dynamical characterization of acylindrically hyperbolic groups, preprint, arXiv:1707.04587.
- Pekka Tukia, A remark on a paper by Floyd, Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 11, Springer, New York, 1988, pp. 165–172. MR 955838, DOI 10.1007/978-1-4613-9611-6_{1}1
- Wen-yuan Yang, Growth tightness for groups with contracting elements, Math. Proc. Cambridge Philos. Soc. 157 (2014), no. 2, 297–319. MR 3254594, DOI 10.1017/S0305004114000322
Additional Information
- Ivan Levcovitz
- Affiliation: Department of Mathematics, Technion–Israel Institute of Technology, Haifa 3200003, Israel
- MR Author ID: 1267306
- Received by editor(s): December 13, 2018
- Received by editor(s) in revised form: June 4, 2019
- Published electronically: August 28, 2019
- Communicated by: David Futer
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 513-521
- MSC (2010): Primary 20F65; Secondary 57M07
- DOI: https://doi.org/10.1090/proc/14745
- MathSciNet review: 4052190