|
Comparison theorems and orbit counting in hyperbolic geometry
Author(s):
Mark
Pollicott;
Richard
Sharp
Journal:
Trans. Amer. Math. Soc.
350
(1998),
473-499.
MSC (1991):
Primary 20F32, 22E40, 58E40;
Secondary 11F72, 20F10, 58F20
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
In this article we address an interesting problem in hyperbolic geometry. This is the problem of comparing different quantities associated to the fundamental group of a hyperbolic manifold (e.g. word length, displacement in the universal cover, etc.) asymptotically. Our method involves a mixture of ideas from both ``thermodynamic'' ergodic theory and the automaton associated to strongly Markov groups.
References:
- 1.
- R.Adler and L.Flatto, Geodesic flows, interval maps and symbolic dynamics, Bull. Amer. Math. Soc. 25 (1991), 229-334. MR 92b:58172
- 2.
- S. Agmon, Complex Variable Tauberians, Trans. Amer. Math. Soc. 74 (1953), 444-481. MR 14:869a
- 3.
- L. Bers, Uniformization, moduli and Kleinian groups, Bull. London Math. Soc. 4 (1972), 257-300. MR 52:5955
- 4.
- M. Bourdon, Actions quasi-convexes d'un groupe hyperbolique, flot géodésique, PhD thesis (Orsay) 1993.
- 5.
- J.Cannon, The combinatorial structure of co-compact discrete hyperbolic groups, Geometriae Dedicata 16 (1984), 123-148. MR 86j:20032
- 6.
- M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics, vol. 1441, Springer, Berlin, 1990. MR 92f:57003
- 7.
- H.Delange, Généralisation du théorème de Ikehara, Annales Scientifique de Ecole Normale Supérieure 71 (1954), 213-244. MR 16:921e
- 8.
- E. Ghys and P. de la Harpe, Sur les groupes hyperboliques d'après Mikhael Gromov, Birkhauser, Boston, 1990. MR 92f:53050
- 9.
- M. Gromov, Hyperbolic groups, Essays in group theory, M.S.R.I. publication vol. 8, MSRI, Berkeley, 1987. MR 89e:20070
- 10.
- T. Kato, Perturbation theory of linear operators, Springer, 1966. MR 34:3324
- 11.
- S. Lalley, Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits, Acta Math. 163 (1989), 1-55. MR 91c:58112
- 12.
- R. Mazzeo and R. Melrose, Meromorphic extension of the resolvent on complete spaces of asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), 260-310. MR 89c:58133
- 13.
- J. Milnor, A note on curvature and the fundamental group, J. Diff. Geom. 2 (1967), 1-7. MR 38:636
- 14.
- R. Nussbaum, The radius of the essential spectrum, Duke Math. J. 37 (1970), 473-478. MR 41:9028
- 15.
- W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Asterisque 187-88 (1990), 1-268. MR 92f:58141
- 16.
- S. Patterson, On a lattice-point problem in hyperbolic space and related questions in spectral theory, Arkiv för Mat. 26 (1988), 167-172. MR 89g:11093
- 17.
- M. Pollicott and R. Sharp, Orbit counting for some discrete groups acting on simply connected manifolds with negative curvature, Invent. Math. 117 (1994), 275-302. MR 96a:58147
- 18.
- D. Ruelle, Thermodynamic Formalism, Addison Wesley, New York, 1978. MR 80g:82017
- 19.
- E. Seneta, Non-negative matrices, George Allen and Unwin, London, 1973. MR 52:10773
- 20.
- C. Series, The infinite word problem and limit sets of fuchsian groups, Ergod. Th. and Dynam. Sys. 1 (1981), 337-360. MR 84d:30084
- 21.
- C. Series, Geometrical Markov coding of geodesics on surfaces of constant negative curvature, Ergod. Th. and Dynam. Sys. 6 (1986), 601-625. MR 88k:58130
- 22.
- D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Publ. Math. (IHES) 50 (1979), 171-202. MR 81b:58031
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(1991):
20F32, 22E40, 58E40,
11F72, 20F10, 58F20
Retrieve articles in all Journals with MSC
(1991):
20F32, 22E40, 58E40,
11F72, 20F10, 58F20
Additional Information:
Mark
Pollicott
Affiliation:
Department of Mathematics, University of Warwick, Coventry, CV4 7AL, U.K.
Address at time of publication:
Department of Mathematics, University of Manchester, Oxford Road, Man- chester, M13 9PL, U.K.
Email:
mp@ma.man.ac.uk
Richard
Sharp
Affiliation:
Mathematical Institute, 24-29 St. Giles, Oxford, OX1 3LB, U.K.
Address at time of publication:
Department of Mathematics, University of Manchester, Oxford Road, Man- chester, M13 9PL, U.K.
Email:
sharp@ma.man.ac.uk
DOI:
10.1090/S0002-9947-98-01756-5
PII:
S 0002-9947(98)01756-5
Keywords:
Strongly Markov,
hyperbolic group,
Kleinian group,
orbit counting function,
negative curvature,
Poincar\'{e} series
Received by editor(s):
May 23, 1995
Additional Notes:
The first author was supported by The Royal Society through a University Research Fellowship. The second author was supported by the UK SERC under grant number GR/G51930 held at Queen Mary and Westfield College.
Copyright of article:
Copyright
1998,
American Mathematical Society
|
Comments: webmaster@ams.org