Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-06-12T09:35:30.197Z Has data issue: false hasContentIssue false
  • English
  • Français

Growth of quasiconvex subgroups

Published online by Cambridge University Press:  28 June 2018

FRANÇOIS DAHMANI ,
DAVID FUTER  and
DANIEL T. WISE
FRANÇOIS DAHMANI
Affiliation:
Université Grenoble Alpes, Institut Fourier (UMR 5582), 100 Rue des Maths, CS 40700. F-38 058 Grenoble, Cedex 9, France. e-mail: françois.dahmani@univ.grenoble.alpes.fr
DAVID FUTER
Affiliation:
Dept. of Mathematics, Temple University, Philadelphia, PA 19122U.S.A. e-mail: dfuter@gmail.com
DANIEL T. WISE
Affiliation:
Dept. of Math. & Stats. McGill University Montreal, QC, CanadaH3A 0B9 e-mail: daniel.wise@mcgil.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron–Frobenius theory.

We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions use the notion of growth tightness and the work of Dahmani, Guirardel and Osin on rotating families.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 3 , November 2019 , pp. 505 - 530
Copyright
Copyright © Cambridge Philosophical Society 2018 

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

[ABC+91] Alonso, J. M., Brady, T., Cooper, D., Ferlini, V., Lustig, M., Mihalik, M. L., Shapiro, M. and Short, H.. Notes on word hyperbolic groups. In: Ghys, É., Haefliger, A. and Verjovsky, A., editors, Group theory from a geometrical viewpoint (Trieste, 1990), pages 3–63 (World Science Publishing, River Edge, NJ, 1991). Edited by H. Short.Google Scholar
[ACT15] Arzhantseva, G. N., Cashen, C. H. and Tao, J. Growth tight actions. Pacific J. Math. 278 (1) (2015), 1–49.Google Scholar
[AL02] Arzhantseva, G. N. and Lysenok, I. G. Growth tightness for word hyperbolic groups. Math. Z. 241 (3) (2002), 597–611.Google Scholar
[Arz01] Arzhantseva, G. N. On quasiconvex subgroups of word hyperbolic groups. Geom. Dedicata 87 (1–3) (2001), 191–208.Google Scholar
[BF02] Bestvina, M. and Fujiwara, K. Bounded cohomology of subgroups of mapping class groups. Geom. Topol. 6: (electronic) (2002), 69–89.Google Scholar
[BF09] Bestvina, M. and Fujiwara, K. A characterisation of higher rank symmetric spaces via bounded cohomology. Geometric and Functional Analysis 19 (1) (2009), 11–40.Google Scholar
[BHS14] Behrstock, J., Hagen, M. F. and Sisto, A. Hierarchically hyperbolic spaces I: curve complexes for cubical groups. Geom. Topol. 21 (3) (2017), 1731–1804.Google Scholar
[Cal13] Calegari, D. The ergodic theory of hyperbolic groups. In Geometry and topology down under, volume 597 of Contemp. Math. (Amer. Math. Soc., Providence, RI, 2013), pages 15–52.Google Scholar
[Can84] Cannon, J. W. The combinatorial structure of cocompact discrete hyperbolic groups. Geom. Dedicata 16 (2) (1984), 123–148.Google Scholar
[Can91] Cannon, J. W. The theory of negatively curved spaces and groups. In Ergodic theory, symbolic dynamics and hyperbolic spaces (Trieste, 1989), Oxford Sci. Publ. (Oxford Univ. Press, New York, 1991), pages 315–369.Google Scholar
[CDS17] Coulon, R. and Dal'bo, F., Sambusetti, A.. Growth gap in hyperbolic groups and amenability. arXiv:1709.07287 (2017).Google Scholar
[Coo93] Coornaert, M. Mésures de Patterson–Sullivan sur le bord d'un éspace hyperbolique au sens de Gromov. Pacific J. Math. 159 (2) (1993), 241–270.Google Scholar
[Cor90] Corlette, K. Hausdorff dimensions of limit sets. I. Invent. Math. 102 (3) (1990), 521–541.Google Scholar
[CS11] Caprace, P.–E. and Sageev, M.. Rank rigidity for CAT(0) cube complexes. Geometric And Functional Analysis 21 (2011), 851–891. 10.1007/s00039-011-0126-7.Google Scholar
[CSW02] Ceccherini–Silberstein, T. and Woess, W.. Growth and ergodicity of context-free languages. Trans. Amer. Math. Soc. 354 (11) (2002), 4597–4625.Google Scholar
[CSW03] Ceccherini–Silberstein, T. and Woess, W.. Growth-sensitivity of context-free languages. Theoret. Comput. Sci. 307 (1) (2003), 103–116.Google Scholar
[CT07] Cannon, J. W. and Thurston, W. P. Group invariant Peano curves. Geom. Topol. 11 (2007), 1315–1355.Google Scholar
[DGO17] Dahmani, F., Guirardel, V. and Osin, D. V. Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces. Mem. Amer. Math. Soc. 245 (1156) (2017), v+152.Google Scholar
[DM17] Dahmani, F. and Mj, M. Height, graded relative hyperbolicity and quasiconvexity. J. Éc. Polytech. Math. 4 (2017), 515–556. Corrigendum posted at arXiv:1602.00834.Google Scholar
[ECH+92] Epstein, D. B. A., Cannon, J. W., Holt, D. F., Levy, S. V. F., Paterson, M. S. and Thurston, W. P.. Word Processing in Groups (Jones and Bartlett Publishers, Boston, MA, 1992).Google Scholar
[FS09] Flajolet, P. and Sedgewick, R. Analytic Combinatorics (Cambridge University Press, Cambridge, 2009).Google Scholar
[GdlH97] Grigorchuk, R. and de la Harpe, P.. On problems related to growth, entropy, and spectrum in group theory. J. Dynam. Control Systems 3 (1) (1997), 51–89.Google Scholar
[Gen16] Genevois, A. Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups. Algebr. Geom. Topol. To appear.Google Scholar
[Git99] Gitik, R. Ping-pong on negatively curved groups. J. Algebra 217 (1) (1999), 65–72.Google Scholar
[Gro87] Gromov, M. Hyperbolic groups. In Essays in group theory, volume 8 of Math. Sci. Res. Inst. Publ. (Springer, New York, 1987), pages 75–263.Google Scholar
[GS91] Gersten, S. and Short, H. Rational subgroups of biautomatic groups. Ann. of Math. (2) 134 (1) (1991), 125–158.Google Scholar
[GVL96] Golub, G. H. and Van Loan, C. F. Matrix computations. Johns Hopkins Studies in the Mathematical Sciences (Johns Hopkins University Press, Baltimore, MD, third edition, 1996).Google Scholar
[Hag14] Hagen, M. F. Weak hyperbolicity of cube complexes and quasi-arboreal groups. J. Topol. 7 (2) (2014), 385–418.Google Scholar
[Hru10] Hruska, G. C. Relative hyperbolicity and relative quasiconvexity for countable groups. Algebr. Geom. Topol. 10 (3) (2010), 1807–1856.Google Scholar
[Hua17] Huang, J. Nearest point projection in CAT(0) cube complexes (2017). Preprint.Google Scholar
[HW09] Hruska, G. C. and Wise, D. T. Packing subgroups in relatively hyperbolic groups. Geom. Topol. 13 (4) (2009), 1945–1988.Google Scholar
[Lea13] Leary, I. J. A metric Kan–Thurston theorem. J. Topol. 6 (1) (2013), 251–284.Google Scholar
[LW] Li, J. and Wise, D. T. No growth gaps for special cube complexes. Pages 1–13. Submitted.Google Scholar
[Min88] Minc, H. Nonnegative matrices. Wiley-Interscience Series in Discrete Mathematics and Optimisation (John Wiley & Sons, Inc., New York, 1988). A Wiley-Interscience Publication.Google Scholar
[MPS12] Martínez–Pedroza, E. and Sisto, A.. Virtual amalgamation of relatively quasiconvex subgroups. Algebr. Geom. Topol. 12 (4) (2012), 1993–2002.Google Scholar
[MYJ15] Matsuzaki, K., Yabuki, Y. and Jaerisch, J. Normaliser, divergence type and Patterson measure for discrete groups of the Gromov hyperbolic space. arXiv:1511.02664 (2015).Google Scholar
[NR98] Niblo, G. A. and Reeves, L. D. The geometry of cube complexes and the complexity of their fundamental groups. Topology 37 (3) (1998), 621–633.Google Scholar
[Osi16] Osin, D. V. Acylindrically hyperbolic groups. Trans. Amer. Math. Soc. 368 (2) (2016), 851–888.Google Scholar
[Rip82] Rips, E. Subgroups of small cancellation groups. Bull. London Math. Soc. 14 (1) (1982), 45–47.Google Scholar
[Sam02] Sambusetti, A. Growth tightness of free and amalgamated products. Ann. Sci. École Norm. Sup. (4) 35 (4) (2002), 477–488.Google Scholar
[Wis12] Wise, D. T. From riches to raags: 3-manifolds, right-angled Artin groups and cubical geometry Volume 117 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences (Washington, DC, 2012).Google Scholar
[Yan14] Yang, W.–Y.. Growth tightness for groups with contracting elements. Math. Proc. Camb. Phil. Soc. 157 (2) (2014), 297–319.Google Scholar