×
Home

# Geometrically finite groups, Khintchine-type theorems and Hausdorff dimension

## Extract

1·1. Groups of the first kind. In [11], Patterson proved a hyperbolic space analogue of Khintchine's theorem on simultaneous Diophantine approximation. In order to state Patterson's theorem, some notation and terminology are needed. Let ‖x‖ denote the usual Euclidean norm of a vector x in k+1, k + 1-dimensional Euclidean space, and let be the unit ball model of k + 1-dimensional hyperbolic space with Poincaré metric ρ. A non-elementary geometrically finite group G acting on Bk + 1 is a discrete subgroup of Möb (Bk+l), the group of orientation preserving Mobius transformations preserving Bk + 1, for which there exists some convex fundamental polyhedron with finitely many faces. Since G is non-elementary, the limit set L(G) of G – the set of limit points in the unit sphere Sk of any orbit of G in Bk+1 – is uncountable. The group G is said to be of the first kind if L(G) = Sk and of the second kind otherwise.

## References

Hide All
[1]Cassels, J. W. S.. An introduction to Diophantine approximation (Cambridge University Press, 1957).
[2]Dodson, M. M.. Hausdorff dimension, lower order and Khintchine's theorem in metric Diophantine approximation. J. reine angew. Math. 432 (1992), 6976.
[3]Dodson, M. M., Melián, M., Pestana, D. and Velani, S. L.. Patterson measure and ubiquity. Ann. Acad. Sci. Fenn. 20: 1 (1995), 3760.
[4]Dodson, M. M., Rynne, B. P. and Vickers, J. A. G.. Diophantine approximation and a lower bound for Hausdorff dimension. Mathematika 37 (1990), 5973.
[5]Falconer, K. J.. Fractal geometry – mathematical foundations and applications (J. Wiley, 1990).
[6]Hill, R. and Velani, S. L.. Ergodic theory of shrinking targets. Inventiones math. 119 (1995), 175198.
[7]Hill, R. and Velani, S. L.. The Jarník–Besicovitch theorem for geometrically finite Kleinian groups, preprint (1994).
[8]Melián, M. V. and Pestana, D.. Geodesic excursions into cusps in finite volume hyperbolic manifolds. Michigan Math. J. 40 (1993), 7793.
[9]Melián, M. V. and Velani, S. L.. Geodesic excursions into cusps infinite volume hyperbolic manifolds. Mathematica Gottingensis 45, preprint.
[10]Patterson, S. J.. Lectures on measures on limit sets of Kleinian groups; in Analytic and geometrical aspects of hyperbolic space (Epstein, D. B. A., ed). LMS 111 (Cambridge University Press, 1987) 281323.
[11]Patterson, S. J.. Metric Diophantine approximation of quadratic forms. In Number theory and dynamical systems (Dodson, M. M. and Vickers, J. A. G., eds.). LMS 134 (Cambridge University Press, 1989), 3748.
[12]Sprindžuk, V. G.. Metric theory of Diophantine approximation. Translated by Silverman, R. A. (V. H. Winston & Sons, 1979).
[13]Stratmann, B.. Finer structure of limit sets of Kleinian groups. Math. Proc. Cam. Phil. Soc. 116 (1994), 5768.
[14]Stratmann, B.. Fractal Dimensions for the Jarník limit sets of geometrically finite Kleinian groups; the semi-classical approach. Mathematica Gottingensis 8, preprint.
[15]Stratmann, B. and Velani, S. L.. The Patterson measure for geometrically finite groups with parabolic elements, new and old. Proc. London Math. Soc. (3) 71 (1995), 197220.
[16]Sullivan, D.. Entropy, Hausdorff measures old and new, and the limit set of geometrically finite Kleinian groups. Acta Math. 153 (1984), 259277.
[17]Velani, S. L.. Diophantine approximation and Hausdorff dimension in Fuchsian groups. Math. Proc. Cam. Phil. Soc. 113 (1993), 343354.
[18]Velani, S. L.. An application of metric Diophantine approximation in hyperbolic space to quadratic forms. Publicacions mathematiques 38 (1994), 175185.
Recommend this journal

Mathematical Proceedings of the Cambridge Philosophical Society
• ISSN: 0305-0041
• EISSN: 1469-8064
• URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Who would you like to send this to? *

×

## Metrics

### Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 0 *