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

  • Grant Cairns (a1) and Anthony Nielsen (a1)

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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).

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References

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[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.
[2]Dani, S.G. and Raghavan, S., ‘Orbits of Euclidean frames under discrete linear groups’, Israel J. Math. 36 (1980), 300320.
[3]Hobby, D. and Silberger, D.M., ‘Quotients of primes’, Amer. Math. Monthly 100 (1993), 5052.
[4]Humphreys, J.E., Arithmetic groups, Lecture Notes in Mathematics 789 (Springer-Verlag, Berlin, 1980).
[5]Jameson, G.J.O., The prime number theorem (Cambridge University Press, Cambridge, 2003).
[6]Mennicke, J.M., ‘Finite factor groups of the unimodular group’, Ann. of Math. (2) 81 (1965), 3137.
[7]Sierpiński, W., Elementary theory of numbers, Monografie Matematyczne, Tom 42 (Państwowe Wydawnictwo Naukowe, Warsaw, 1964).
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