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. doi: 10.1007/s00209-002-0434-6.

[2]

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

[3]

T. Ceccherini-Silberstein and F. Scarabotti, Random walks, entropy and hopfianity of free groups,, in, (2004), 413.

[4]

M. Coornaert, Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov,, Pacific J. Math., 159 (1993), 241. 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 (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. 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. 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, 83 (1988).

[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. doi: 10.1007/BF02471762.

[11]

M. Gromov, Hyperbolic groups,, in, 8 (1987), 75. 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 (1991).

[13]

A. Sambusetti, On minimal growth in group theory and Riemannian geometry,, in, (2002), 268.

[14]

A. Sambusetti, Growth tightness of surface groups,, Expo. Math., 20 (2002), 345. 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. 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. doi: 10.1016/S1631-073X(03)00086-4.

[17]

A. Sambusetti, Growth tightness in group theory and Riemannian geometry,, in, (2004), 341.

[18]

A. Sambusetti, Asymptotic properties of coverings in negative curvature,, Geom. Topol., 12 (2008), 617. 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. 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. 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. doi: 10.1007/s00209-002-0434-6.

[2]

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

[3]

T. Ceccherini-Silberstein and F. Scarabotti, Random walks, entropy and hopfianity of free groups,, in, (2004), 413.

[4]

M. Coornaert, Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov,, Pacific J. Math., 159 (1993), 241. 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 (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. 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. 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, 83 (1988).

[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. doi: 10.1007/BF02471762.

[11]

M. Gromov, Hyperbolic groups,, in, 8 (1987), 75. 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 (1991).

[13]

A. Sambusetti, On minimal growth in group theory and Riemannian geometry,, in, (2002), 268.

[14]

A. Sambusetti, Growth tightness of surface groups,, Expo. Math., 20 (2002), 345. 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. 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. doi: 10.1016/S1631-073X(03)00086-4.

[17]

A. Sambusetti, Growth tightness in group theory and Riemannian geometry,, in, (2004), 341.

[18]

A. Sambusetti, Asymptotic properties of coverings in negative curvature,, Geom. Topol., 12 (2008), 617. 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. 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. 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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