Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-18T01:44:40.917Z Has data issue: false hasContentIssue false
  • English
  • Français

Growth tightness for groups with contracting elements

Published online by Cambridge University Press:  30 July 2014

WEN-YUAN YANG
WEN-YUAN YANG*
Affiliation:
Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, P.R.China Le Département de Mathématiques de la Faculté des Sciences d'Orsay, Université Paris-Sud 11, France. e-mail: wyang@math.pku.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish growth tightness for a class of groups acting geometrically on a geodesic metric space and containing a contracting element. As a consequence, any group with non-trivial Floyd boundary are proven to be growth tight with respect to word metrics. In particular, all non-elementary relatively hyperbolic group are growth tight. This generalizes previous works of Arzhantseva-Lysenok and Sambusetti. Another interesting consequence is that CAT(0) groups with rank-1 elements are growth tight with respect to CAT(0)-metric.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 2 , September 2014 , pp. 297 - 319
Copyright
Copyright © Cambridge Philosophical Society 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Arzhantseva, G. and Lysenok, I.Growth tightness for word hyperbolic groups. Math. Z. 241 (2002), no. 3, 597–611.CrossRefGoogle Scholar
[2]Bestvina, M., Bromberg, K. and Fujiwara, K. Constructing group actions on quasi-trees and applications to mapping class groups (2010), arXiv:1006.1939.Google Scholar
[3]Bestvina, M. and Feighn, M.A combination theorem for negatively curved groups. J. Diff. Geom. 35 (1992), 85–101.Google Scholar
[4]Bestvina, M. and Fujiwara, K.A characterization of higher rank symmetric spaces via bounded cohomology. Geom. Funct. Anal. 19 (2009), 11–40.CrossRefGoogle Scholar
[5]Bowditch, B.Convergence groups and configuration spaces, Geometric Group Theory Down Under (Cossey, J., Miller, C.F., Neumann, W.D., Shapiro, M., eds.), pp. 23–54, de Gruyter, 1999.Google Scholar
[6]Dahmani, F., Guirardel, V. and Osin, D. Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces. arXiv:1111.7048, (2011).Google Scholar
[7]Dal'bo, F., Peigné, M., Picaud, J.C. and Sambusetti, A.On the growth of quotients of kleinian groups. Ergodic Theory Dynam. Systems 31 (2011), no. 3, 835–851.Google Scholar
[8]Drutu, C. and Sapir, M.Tree-graded spaces and asymptotic cones of groups. Topology 44 (2005), no. 5, 959–1058, with an appendix by D. Osin and M. Sapir.Google Scholar
[9]Floyd, W.Group completions and limit sets of kleinian groups. Invent. Math. 57 (1980), 205–218.CrossRefGoogle Scholar
[10]Gerasimov, V.Floyd maps for relatively hyperbolic groups. Geom. Funct. Anal. (2012), no. 22, 1361–1399.CrossRefGoogle Scholar
[11]Gerasimov, V. and Potyagailo, L. Quasiconvexity in the relatively hyperbolic groups. arXiv:1103.1211 (2011).Google Scholar
[12]Gerasimov, V. and Potyagailo, L. Non-finitely generated relatively hyperbolic groups and Floyd quasiconvexity. arXiv:1008.3470 to appear in Groups, Geometry and Dynamics.Google Scholar
[13]Gerasimov, V. and Potyagailo, L. Quasi-isometries and Floyd boundaries of relatively hyperbolic groups. arXiv:0908.0705 to appear in J. Eur. Math. Soc.Google Scholar
[14]Ghys, E. and de la Harpe, P.Sur les groupes hyperboliques d'après Mikhael Gromov. Prog. Math. (Birkaüser, 1990).Google Scholar
[15]Grigorchuk, R. and de la Harpe, P.On problems related to growth, entropy and spectrum in group theory. J. Control System. 3 (1997), no. 1, 51–89.CrossRefGoogle Scholar
[16]Karlsson, A.Free subgroups of groups with non-trivial Floyd boundary. Comm. Algebra. 31 (2003), 5361–5376.CrossRefGoogle Scholar
[17]Karlsson, A. and Noskov, G.Some groups having only elementary actions on metric spaces with hyperbolic boundaries. Geom. Dedicata 104 (2004), 119–137.Google Scholar
[18]Olshanskii, A., Osin, D. and Sapir, M.Lacunary hyperbolic groups. Geom. Topol. 13 (2009), no. 4, 2051–2140, with an appendix by Michael Kapovich and Bruce Kleiner.CrossRefGoogle Scholar
[19]Sabourau, S.Growth of quotients of groups acting by isometries on Gromov hyperbolic spaces. J. Mod. Dyn. 7 (2013), no. 2, 269–290.Google Scholar
[20]Sambusetti, A.Growth tightness of free and amalgamated products. Ann. Sci. École Norm. Sup. série 35 (2002), no. 4, 477–488.Google Scholar
[21]Sambusetti, A.Growth tightness of surfaces groups. Expo. Math. 20 (2002), 335–363.CrossRefGoogle Scholar
[22]Sambusetti, A.Asymptotic properties of coverings in negative curvature. Geom. & Topol. (2008), no. 1, 617–637.CrossRefGoogle Scholar
[23]Sisto, A. Contracting elements and random walk. arXiv:1112.2666 (2011).Google Scholar
[24]Yang, W. Patterson-Sullivan measures and growth of relatively hyperbolic groups. Preprint (2013).Google Scholar
[25]Yang, W.Peripheral structures of relatively hyperbolic groups. J. Reine Angew. Math. 689 (2014), 101–135.Google Scholar