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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
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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 

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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