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On the dynamics of the linear action of SL(n, Z)

Published online by Cambridge University Press:  17 April 2009

Grant Cairns
Affiliation:
Department of Mathematics, La Trobe University, Melbourne Vic 3086, Australia, e-mail: G.Cairns@latrobe.edu.au, A.Nielsen@latrobe.edu.au
Anthony Nielsen
Affiliation:
Department of Mathematics, La Trobe University, Melbourne Vic 3086, Australia, e-mail: G.Cairns@latrobe.edu.au, A.Nielsen@latrobe.edu.au
Corresponding
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Extract

Using Moore's ergodicity theorem, S.G. Dani and S. Raghavan proved that the linear action of SL(n, ℤ) on ℝn is topologically (n − l)-transitive; that is, topologically transitive on the Cartesian product of n − 1 copies of ℝn. In this paper, we give a more direct proof, using the prime number theorem. Further, using the congruence subgroup theorem, we generalise the result to arbitrary finite index subgroups of SL(n, ℤ).

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Bass, H., Lazard, M. and Serre, J.-P., ‘Sous-groupes d'indice fini dans SL(n, Z)’, Bull. Amer. Math. Soc. 70 (1964), 385392.CrossRefGoogle Scholar
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[5]Jameson, G.J.O., The prime number theorem (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
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[7]Sierpiński, W., Elementary theory of numbers, Monografie Matematyczne, Tom 42 (Państwowe Wydawnictwo Naukowe, Warsaw, 1964).Google Scholar

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