Affine Λ-buildings, ultrapowers of Lie groups and Riemannian symmetric spaces: a short proof of the Margulis conjecture
نویسنده
چکیده
Then Γ acts cocompactly on X and X , and the orbit spaces X/Γ and X /Γ are closed manifolds with contractible universal covers. The only nontrivial homotopy groups are thus the fundamental groups, π1(X/Γ) ∼= Γ ∼= π1(X /Γ). It follows that there is a homotopy equivalence X/Γ X /Γ which can be lifted to a map f : X X . This map f can then be shown to be a quasi-isometry. Recall that a (not necessarily continuous) map f : X −→ X ′ between metric spaces if called an (L,C)-quasiisometry if there exist constants L ≥ 1 and C ≥ 0 such that
منابع مشابه
Affine Λ-buildings, ultrapowers of Lie groups and Riemannian symmetric spaces: an algebraic proof of the Margulis conjecture
Then Γ acts cocompactly on X and X , and the orbit spaces X/Γ and X /Γ are closed manifolds with contractible universal covers. The only nontrivial homotopy groups are thus the fundamental groups, π1(X/Γ) ∼= Γ ∼= π1(X /Γ). It follows that there is a homotopy equivalence X/Γ X /Γ which can be lifted to a map f : X X . This map f can then be shown to be a quasi-isometry. Recall that a (not necess...
متن کاملA reduction of axioms
Jacques Tits introduced the notion of a building as a geometry associated to groups of Lie type in [T1], providing new geometries associated to the exceptional groups of Lie type. In 1972, F. Bruhat and Tits [BT] developed a theory of affine buildings for the purpose of studying groups over fields having a discrete valuation, although their work applied more generally to groups over fields havi...
متن کاملAsymptotic cones and ultrapowers of Lie groups
Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the ’large-scale structure’ of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the completeness of the metric space, now follow rather easily from saturation propertie...
متن کاملRealization of locally extended affine Lie algebras of type $A_1$
Locally extended affine Lie algebras were introduced by Morita and Yoshii in [J. Algebra 301(1) (2006), 59-81] as a natural generalization of extended affine Lie algebras. After that, various generalizations of these Lie algebras have been investigated by others. It is known that a locally extended affine Lie algebra can be recovered from its centerless core, i.e., the ideal generated by weight...
متن کاملJ ul 2 00 7 SUPERRIGIDITY , GENERALIZED HARMONIC MAPS AND UNIFORMLY CONVEX SPACES
We prove several superrigidity results for isometric actions on Busemann non-positively curved uniformly convex metric spaces. In particular we generalize some recent theorems of N. Monod on uniform and certain non-uniform irreducible lattices in products of locally compact groups, and we give a proof of an unpublished result on commensurability superrigidity due to G.A. Margulis. The proofs re...
متن کامل