April  2013, 7(2): 269-290. doi: 10.3934/jmd.2013.7.269

Growth of quotients of groups acting by isometries on Gromov-hyperbolic spaces

1. 

Université Paris-Est, Laboratoire d’Analyse et Mathématiques Appliquées (UMR 8050), UPEC, UPEMLV, CNRS, F-94010, Créteil, France

Received  December 2012 Published  September 2013

We show that every group $G$ with no cyclic subgroup of finite index that acts properly and cocompactly by isometries on a proper geodesic Gromov-hyperbolic space $X$ is growth-tight. In other words, the exponential growth rate of $G$ for the geometric (pseudo)-distance induced by $X$ is greater than the exponential growth rate of any of its quotients by an infinite normal subgroup. This result unifies and extends previous works of Arzhantseva-Lysenok and Sambusetti using a geometric approach.
Citation: Stéphane Sabourau. Growth of quotients of groups acting by isometries on Gromov-hyperbolic spaces. Journal of Modern Dynamics, 2013, 7 (2) : 269-290. doi: 10.3934/jmd.2013.7.269
References:
[1]

G. N. Arzhantseva and I. G. Lysenok, Growth tightness for word-hyperbolic groups, Math. Z., 241 (2002), 597-611. doi: 10.1007/s00209-002-0434-6.

[2]

M. Bonk and O. Schramm, Embeddings of Gromov-hyperbolic spaces, Geom. Funct. Anal., 10 (2000), 266-306. doi: 10.1007/s000390050009.

[3]

T. Ceccherini-Silberstein and F. Scarabotti, Random walks, entropy and hopfianity of free groups, in "Random Walks and Geometry'' (Proceedings of the workshop held in Vienna, June 18-July 13, 2001) (ed. Vadim A. Kaimanovich in collaboration with Klaus Schmidt and Wolfgang Woess), Walter de Gruyter, Berlin, (2004), 413-419.

[4]

M. Coornaert, Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov, Pacific J. Math., 159 (1993), 241-270. doi: 10.2140/pjm.1993.159.241.

[5]

M. Coornaert, T. Delzant and A. Papadopoulos, "Géométrie et Théorie des Groupes. Les Groupes Hyperboliques de Gromov,'' Lecture Notes in Mathematics, 1441, Springer-Verlag, Berlin, 1990.

[6]

R. Coulon, Growth of periodic quotients of hyperbolic groups,, \arXiv{1211.4271}., (). 

[7]

F. Dal'Bo, J.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel J. Math., 118 (2000), 109-124. doi: 10.1007/BF02803518.

[8]

F. Dal'Bo, M. Peigné, J.-C. Picaud and A. Sambusetti, On the growth of quotients of Kleinian groups, Ergodic Theory Dynam. Systems, 31 (2011), 835-851. doi: 10.1017/S0143385710000131.

[9]

É. Ghys and P. de la Harpe, eds., "Sur les Groupes Hyperboliques d'après Mikhael Gromov,'' Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988, Progress in Mathematics, 83, Birkhäuser Boston, Inc., Boston, MA, 1990.

[10]

R. Grigorchuk and P. de la Harpe, On problems related to growth, entropy, and spectrum in group theory, J. Dynam. Control Systems, 3 (1997), 51-89. doi: 10.1007/BF02471762.

[11]

M. Gromov, Hyperbolic groups, in "Essays in Group Theory,'' Math. Sci. Res. Inst. Publ., 8, Springer, New York, (1987), 75-263. doi: 10.1007/978-1-4613-9586-7_3.

[12]

G. Margulis, "Discrete Subgroups of Semisimple Lie Groups,'' Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17, Springer-Verlag, Berlin, 1991.

[13]

A. Sambusetti, On minimal growth in group theory and Riemannian geometry, in "Differential Geometry, Valencia, 2001,'' World Sci. Publ., River Edge, NJ, (2002), 268-280.

[14]

A. Sambusetti, Growth tightness of surface groups, Expo. Math., 20 (2002), 345-363. doi: 10.1016/S0723-0869(02)80012-3.

[15]

A. Sambusetti, Growth tightness of free and amalgamated products, Ann. Sci. École Norm. Sup. (4), 35 (2002), 477-488. doi: 10.1016/S0012-9593(02)01101-1.

[16]

A. Sambusetti, Growth tightness of negatively curved manifolds, C. R. Math. Acad. Sci. Paris, 336 (2003), 487-491. doi: 10.1016/S1631-073X(03)00086-4.

[17]

A. Sambusetti, Growth tightness in group theory and Riemannian geometry, in "Recent Advances in Geometry and Topology,'' Cluj Univ. Press, Cluj-Napoca, (2004), 341-352.

[18]

A. Sambusetti, Asymptotic properties of coverings in negative curvature, Geom. Topol., 12 (2008), 617-637. doi: 10.2140/gt.2008.12.617.

[19]

A. Shukhov, On the dependence of the growth exponent on the length of the defining relation, Math. Notes, 65 (1999), 510-515. doi: 10.1007/BF02675367.

[20]

A. Talambutsa, Attainability of the index of exponential growth in free products of cyclic groups, Math. Notes, 78 (2005), 569-572. doi: 10.1007/s11006-005-0156-2.

[21]

W. Yang, Growth tightness of groups with nontrivial Floyd boundary,, \arXiv{1301.5623}., (). 

show all references

References:
[1]

G. N. Arzhantseva and I. G. Lysenok, Growth tightness for word-hyperbolic groups, Math. Z., 241 (2002), 597-611. doi: 10.1007/s00209-002-0434-6.

[2]

M. Bonk and O. Schramm, Embeddings of Gromov-hyperbolic spaces, Geom. Funct. Anal., 10 (2000), 266-306. doi: 10.1007/s000390050009.

[3]

T. Ceccherini-Silberstein and F. Scarabotti, Random walks, entropy and hopfianity of free groups, in "Random Walks and Geometry'' (Proceedings of the workshop held in Vienna, June 18-July 13, 2001) (ed. Vadim A. Kaimanovich in collaboration with Klaus Schmidt and Wolfgang Woess), Walter de Gruyter, Berlin, (2004), 413-419.

[4]

M. Coornaert, Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov, Pacific J. Math., 159 (1993), 241-270. doi: 10.2140/pjm.1993.159.241.

[5]

M. Coornaert, T. Delzant and A. Papadopoulos, "Géométrie et Théorie des Groupes. Les Groupes Hyperboliques de Gromov,'' Lecture Notes in Mathematics, 1441, Springer-Verlag, Berlin, 1990.

[6]

R. Coulon, Growth of periodic quotients of hyperbolic groups,, \arXiv{1211.4271}., (). 

[7]

F. Dal'Bo, J.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel J. Math., 118 (2000), 109-124. doi: 10.1007/BF02803518.

[8]

F. Dal'Bo, M. Peigné, J.-C. Picaud and A. Sambusetti, On the growth of quotients of Kleinian groups, Ergodic Theory Dynam. Systems, 31 (2011), 835-851. doi: 10.1017/S0143385710000131.

[9]

É. Ghys and P. de la Harpe, eds., "Sur les Groupes Hyperboliques d'après Mikhael Gromov,'' Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988, Progress in Mathematics, 83, Birkhäuser Boston, Inc., Boston, MA, 1990.

[10]

R. Grigorchuk and P. de la Harpe, On problems related to growth, entropy, and spectrum in group theory, J. Dynam. Control Systems, 3 (1997), 51-89. doi: 10.1007/BF02471762.

[11]

M. Gromov, Hyperbolic groups, in "Essays in Group Theory,'' Math. Sci. Res. Inst. Publ., 8, Springer, New York, (1987), 75-263. doi: 10.1007/978-1-4613-9586-7_3.

[12]

G. Margulis, "Discrete Subgroups of Semisimple Lie Groups,'' Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17, Springer-Verlag, Berlin, 1991.

[13]

A. Sambusetti, On minimal growth in group theory and Riemannian geometry, in "Differential Geometry, Valencia, 2001,'' World Sci. Publ., River Edge, NJ, (2002), 268-280.

[14]

A. Sambusetti, Growth tightness of surface groups, Expo. Math., 20 (2002), 345-363. doi: 10.1016/S0723-0869(02)80012-3.

[15]

A. Sambusetti, Growth tightness of free and amalgamated products, Ann. Sci. École Norm. Sup. (4), 35 (2002), 477-488. doi: 10.1016/S0012-9593(02)01101-1.

[16]

A. Sambusetti, Growth tightness of negatively curved manifolds, C. R. Math. Acad. Sci. Paris, 336 (2003), 487-491. doi: 10.1016/S1631-073X(03)00086-4.

[17]

A. Sambusetti, Growth tightness in group theory and Riemannian geometry, in "Recent Advances in Geometry and Topology,'' Cluj Univ. Press, Cluj-Napoca, (2004), 341-352.

[18]

A. Sambusetti, Asymptotic properties of coverings in negative curvature, Geom. Topol., 12 (2008), 617-637. doi: 10.2140/gt.2008.12.617.

[19]

A. Shukhov, On the dependence of the growth exponent on the length of the defining relation, Math. Notes, 65 (1999), 510-515. doi: 10.1007/BF02675367.

[20]

A. Talambutsa, Attainability of the index of exponential growth in free products of cyclic groups, Math. Notes, 78 (2005), 569-572. doi: 10.1007/s11006-005-0156-2.

[21]

W. Yang, Growth tightness of groups with nontrivial Floyd boundary,, \arXiv{1301.5623}., (). 

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