Abstract
A linear automorphism of a finite dimensional real vector spaceV is calledproximal if it has a unique eigenvalue—counting multiplicities—of maximal modulus. Goldsheid and Margulis have shown that if a subgroupG of GL(V) contains a proximal element then so does every Zariski dense subsemigroupH ofG, providedV considered as aG-module is strongly irreducible. We here show thatH contains a finite subsetM such that for everyg∈GL(V) at least one of the elements γg, γ∈M, is proximal. We also give extensions and refinements of this result in the following directions: a quantitative version of proximality, reducible representations, several eigenvalues of maximal modulus.
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References
[A] L. Auslander,The structure of compact locally affine manifolds, Topology3 (1964), 131–139.
[BL] Y. Benoist and F. Labourie,Sur les difféomorphismes d'Anosov affines à feuilletages stables et instables différentiables, Preprint.
[B] A. Borel,Introduction aux groupes arithmétiques, Hermann, Paris, 1969.
[BT] A. Borel and J. Tits,Groupes réductifs, Publications de Mathématiques de l'IHES27 (1965), 55–151.
[D] T. Drumm,Fundamental polyhedra for Margulis space-times, Doctoral dissertation University of Maryland, 1990.
[DG] T. Drunm and W. Goldman,Complete flat Lorentz 3-manifolds with free fundamental group, International Journal of Mathematics1 (1990), 149–161.
[FG] D. Fried and W. M. Goldman,Three-dimensional affine crystallographic groups, Advances in Mathematics47 (1983), 1–49.
[F] H. Furstenberg,Boundary theory and stochastic processes on homogeneous spaces, Proceedings of Symposia in Pure Mathematics (Williamstown, MA, 1972), American Mathematical Society, 1973, pp. 193–229.
[GK] W. M. Goldman and Y. Kamishima,The fundamental group of a compact flat Lorentz space form is virtually polycyclic, Journal of Differential Geometry19 (1984), 233–240.
[GM] I. Ya. Goldsheid and G. A. Margulis,Lyapunov exponents of a product of random matrices, Uspekhi matematicheskikh Nauk44 (1989), 13–60. English translation in Russian Mathematics Surveys44 (1989), 11–71.
[G] I. Ya. Goldsheid,Quasi-projective transformations, Preprint 220, SFB 237 Bochum, 1994.
[GrM] F. Grünewald and G. A. Margulis,Transitive and quasitransitive actions of affine groups preserving a generalized Lorentz-structure, Journal of Geometry and Physics5 (1988), 493–531.
[GR] Y. Guivar'ch and A. Raugi,Propriétés de contraction d'un semigroupe de matrices inversibles. Coefficients de Liapunoff d'un produit de matrices aléatoires indépendantes, Israel Journal of Mathematics65 (1989), 165–196.
[Ma1] G. A. Margulis,Free properly discontinuous groups of affine transformations, Doklady Akademii Nauk SSSR272 (1983), 937–940.
[Ma2] G. A. Margulis,Complete affine locally flat manifolds with a free fundamental group, Journal of Soviet Mathematics134 (1987), 129–134, translated from Zapiski NS LOMI134 (1984), 190–205.
[Ma3] G. A. Margulis,On the Zariski closure of the linear part of a properly discontinuous group of affine transformations, Preprint (1991).
[Ma4] G. A. Margulis,Discrete Subgroups of Semisimple Lie Groups, Springer, Berlin, 1991.
[MS] G. A. Margulis and G. A. Soifer,Maximal subgroups of infinite index in finitely generated linear groups, Journal of Algebra69 (1981), 1–23.
[Me] G. Mess,Flat Lorentz spacetimes, Preprint, February 1990.
[Mi] J. Milnor,On fundamental groups of complete affinely flat manifolds, Advances in Mathematics25 (1977), 178–187.
[P] G. Prasad, ℝ-regular elements in Zariski-dense subgroups, Oxford Quarterly Journal of Mathematics, to appear.
[PR] G. Prasad and M. S. Raghunathan,Cartan subgroups and lattices in semi-simple groups, Annals of Mathematics96 (1972), 296–317.
[Ra] M. S. Raghunathan,Discrete Subgroups of Lie Groups, Springer, Berlin, 1972.
[S] G. A. Soifer,Affine discontinuous groups, inAlgebra and Analysis (L. A. Bokut', M. Hazewinkel and Yu. Reshetnyak, eds.), Proceedings of the First Siberian School, Lake Baikal, American Mathematical Society, 1992.
[Ti] J. Tits,Free subgroups in linear groups, Journal of Algebra20 (1972), 250–270.
[To] G. Tomanov,The virtual solvability of the fundamental group of a generalized Lorentz space form, Journal of Differential Geometry32 (1990), 539–547.
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Partially supported by NSF grant DMS 9204-720.
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Abels, H., Margulis, G.A. & Soifer, G.A. Semigroups containing proximal linear maps. Israel J. Math. 91, 1–30 (1995). https://doi.org/10.1007/BF02761637
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DOI: https://doi.org/10.1007/BF02761637