# zbMATH — the first resource for mathematics

Semigroups containing proximal linear maps. (English) Zbl 0845.22004
Let $$V$$ be a finite dimensional vector space. A linear automorphism $$g$$ of $$V$$ is called a proximal element if $$g$$ has a unique eigenvalue $$\lambda = \lambda (g)$$ of maximal absolute value and the corresponding weight space $$V_n = \{v \in V; (g - \lambda E)^n v = 0$$ for some $$n\}$$ is one-dimensional. Goldsheid and Margulis noticed in 1981 that if a subgroup $$G$$ of $$GL(V)$$ contains a proximal element then so does every Zariski dense subsemigroup $$H$$ of $$G$$, provided $$V$$, considered as a $$G$$-module, is strongly irreducible. The authors show further that if $$H$$ is a strongly irreducible subsemigroup of $$GL(V)$$ and if the algebraic closure $$G$$ of $$H$$ contains a proximal element, then there is an $$\varepsilon > 0$$ and for every $$r \geq 1$$ there is a finite subset $$M$$ of $$H$$ such that for every $$g \in GL(V)$$ there is an $$r \in M$$ such that $$\gamma g$$ is $$(r, \varepsilon)$$-proximal. Thus, under the above hypotheses, the subsemigroup $$H$$ provides a rich supply of proximal elements. Extensions and refinements of the above result are obtained in the following directions: a quantitative version of proximality; reducible representations; several eigenvalues of maximal modules. The authors also point out that proximal elements play a crucial role in the dynamics of linear maps because of their simple structure. Generalizations of linear proximal maps to Lyapunov filtrations and to several irreducible summands are obtained.

##### MSC:
 22A15 Structure of topological semigroups
Full Text:
##### References:
  [A] L. Auslander,The structure of compact locally affine manifolds, Topology3 (1964), 131–139. · Zbl 0136.43102 · doi:10.1016/0040-9383(64)90012-6  [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. · Zbl 0186.33202  [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. · Zbl 0773.57008  [DG] T. Drunm and W. Goldman,Complete flat Lorentz 3-manifolds with free fundamental group, International Journal of Mathematics1 (1990), 149–161. · Zbl 0704.57026 · doi:10.1142/S0129167X90000101  [FG] D. Fried and W. M. Goldman,Three-dimensional affine crystallographic groups, Advances in Mathematics47 (1983), 1–49. · Zbl 0571.57030 · doi:10.1016/0001-8708(83)90053-1  [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. · Zbl 0546.53039  [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. · Zbl 0706.57022 · doi:10.1016/0393-0440(88)90017-4  [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. · Zbl 0677.60007 · doi:10.1007/BF02764859  [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. · Zbl 0611.57023 · doi:10.1007/BF01104978  [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. · Zbl 0457.20046 · doi:10.1016/0021-8693(81)90123-X  [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. · Zbl 0364.55001 · doi:10.1016/0001-8708(77)90004-4  [P] G. Prasad, $$\mathbb{R}$$-regular elements in Zariski-dense subgroups, Oxford Quarterly Journal of Mathematics, to appear. · Zbl 0828.22010  [PR] G. Prasad and M. S. Raghunathan,Cartan subgroups and lattices in semi-simple groups, Annals of Mathematics96 (1972), 296–317. · Zbl 0245.22013 · doi:10.2307/1970790  [Ra] M. S. Raghunathan,Discrete Subgroups of Lie Groups, Springer, Berlin, 1972. · Zbl 0254.22005  [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. · Zbl 0236.20032 · doi:10.1016/0021-8693(72)90058-0  [To] G. Tomanov,The virtual solvability of the fundamental group of a generalized Lorentz space form, Journal of Differential Geometry32 (1990), 539–547. · Zbl 0681.57027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.