Actions Of Semisimple Lie Groups Research Articles

A geometric splitting theorem for actions of semisimple Lie groups

A geometric splitting theorem for actions of semisimple Lie groups

Quantitative multiple mixing

We develop a method for providing quantitative estimates for higher order correlations of group actions. In particular, we establish effective mixing of all orders for actions of semisimple Lie groups as well as semisimple S -algebraic groups and semisimple adele groups. As an application, we deduce existence of approximate configurations in lattices of semisimple groups.

CONFORMAL ACTIONS OF REAL-RANK 1 SIMPLE LIE GROUPS ON PSEUDO-RIEMANNIAN MANIFOLDS

Given a simple Lie group G of rank 1, we consider compact pseudo-Riemannian manifolds (M,g) of signature (p,q) on which G can act conformally. Precisely, we determine the smallest possible value for the index min(p,q) of the metric. When the index is optimal and G non-exceptional, we prove that the metric must be conformally flat, confirming the idea that in a "good" dynamical context, a geometry is determined by its automorphisms group. This completes anterior investigations on pseudo-Riemannian conformal actions of semi-simple Lie groups of maximal real-rank. Combined with these results, we obtain as corollary the list of semi-simple Lie groups without compact factor that can act on compact Lorentzian manifolds. We also derive consequences in CR geometry via the Fefferman fibration.

The Gromov’s centralizer theorem for semisimple Lie group actions

We give a new version of the Gromov’s centralizer theorem in the case of semisimple Lie group actions and arbitrary rigid geometric structures of algebraic type.

Differential contra algebraic invariants: Applications to classical algebraic problems

In this paper we discuss an approach to the study of orbits of actions of semisimple Lie groups in their irreducible complex representations,which is based on differential invariants on the one hand, and on geometry of reductive homogeneous spaces on the other hand. According to the Borel–Weil–Bott theorem, every irreducible representation of semisimple Lie group is isomorphic to the action of this group on the module of holomorphic sections of some one–dimensional bundle over homogeneous space. Using this, we give a complete description of the structure of the field of differential invariants for this action and obtain a criterion which separates regular orbits.

An algorithm for determining actions of semisimple Lie groups on free resolutions

An algorithm for determining actions of semisimple Lie groups on free resolutions

On differential invariants of actions of semisimple Lie groups

In this paper we suggest an approach to the study of orbits of actions of semisimple Lie groups in their irreducible complex representations. This approach is based on differential invariants on the one hand, and on geometry of reductive homogeneous spaces on the other hand. According to Borel–Weil–Bott theorem, every irreducible representation of semisimple Lie group is isomorphic to the action of this group on the module of holomorphic sections of some one-dimensional bundle over homogeneous space. Using this, we give a complete description of the structure of the field of differential invariants for this action and obtain a criterion, which separates regular orbits.

The signature in actions of semisimple Lie groups on pseudo-Riemannian manifolds

We study the relationship between the signature of a semisimple Lie group and a pseudoRiemannian manifold on wich the group acts topologically transitively and isometrically. We also provide a descrip­tion of the bi-invariant pseudo-Riemannian metrics on a semisimple Lie Group over R in terms of the complexification of the Lie algebra associated to the group, and then we utilize it to prove a remark of Gromov.

Actions of semisimple Lie groups preserving a degenerate Riemannian metric

We prove a rigidity of the lightcone in Minkowski space. It is (essentially) the unique space endowed with a lightlike metric and supporting an isometric nonproper action of a semisimple Lie group.

Entropy of Stationary Measures and Bounded Tangential de-Rham Cohomology of Semisimple Lie Group Actions

Let G be a connected noncompact semisimple Lie group with finite center, K a maximal compact subgroup, and X a compact manifold (or more generally, a Borel space) on which G acts. Assume that ν is a μ -stationary measure on X, where μ is an admissible measure on G, and that the G-action is essentially free. We consider the foliation of K\ X with Riemmanian leaves isometric to the symmetric space K\ G, and the associated tangential bounded de-Rham cohomology, which we show is an invariant of the action. We prove both vanishing and nonvanishing results for bounded tangential cohomology, whose range is dictated by the size of the maximal projective factor G/Q of (X, ν). We give examples showing that the results are often best possible. For the proofs we formulate a bounded tangential version of Stokes’ theorem, and establish a bounded tangential version of Poincare’s Lemma. These results are made possible by the structure theory of semisimple Lie groups actions with stationary measure developed in Nevo and Zimmer [Ann of Math. 156, 565--594]. The structure theory assert, in particular, that the G-action is orbit equivalent to an action of a uniquely determined parabolic subgroup Q. The existence of Q allows us to establish Stokes’ and Poincare’s Lemmas, and we show that it is the size of Q (determined by the entropy) which controls the bounded tangential cohomology.

Ergodic Actions of Semisimple Lie Groups on Compact Principal Bundles

Let G = SL(n, ℝ) (or, more generally, let G be a connected, noncompact, simple Lie group). For any compact Lie group K, it is easy to find a compact manifold M, such that there is a volume-preserving, connection-preserving, ergodic action of G on some smooth, principal K-bundle P over M. Can M can be chosen independent of K? We show that if M = H/Λ is a homogeneous space, and the action of G on M is by translations, then P must also be a homogeneous space H′Λ′. Consequently, there is a strong restriction on the groups K that can arise over this particular M.

Non-tame Lorentz actions of semisimple Lie groups

We show that if G is a connected semisimple Lie group with finite center, and if G admits a locally faithful, non-tame action by isometries of a connected Lorentz manifold, then $\mathfrak{g}$ has an ideal which is Lie algebra isomorphic to $\mathfrak{sl}_2(\mathbb{R})$ . We also analyze the collection of connected Lie groups G admitting a free action by isometries of a connected Lorentz manifold such that the action is properly ergodic with respect to the Lorentz volume form.

A Structure Theorem for Actions of Semisimple Lie Groups

We consider a connected semisimple Lie group G with finite center, an admissible probability measure , on G, and an ergodic (G, ,u)-space (X, v). We first note (Lemma 0.1) that (X, v) has a unique maximal projective factor of the form (G/Q, vo): where Q is a parabolic subgroup of G, and then prove: 1. Theorem 1. If every noncompact simple factor of G has real rank at least two, then the maximal projective factor is nontrivial, unless v is a G

Arithmetic structure of fundamental groups and actions of semisimple Lie groups

If a simple Lie group acts on a space M with a finite invariant measure, we investigate the arithmetic properties of the fundamental group, and the arithmetic structure of the action.

Rigidity of Furstenberg entropy for semisimple Lie group actions

We consider the action of a semi-simple Lie group G on a compact manifold (and more generally a Borel space) X , with a measure ν stationary under a probability measure μ on G . We first establish some properties of the fundamental invariant associated with a ( G, μ) -space ( X, ν) , namely the Furstenberg entropy [? ], given by h μ(X,ν)= ∫ G ∫ X − log dg −1ν dν (x) dν(x) dμ(g). We then prove that when ( X, ν) is a P -mixing ( G, μ) -space [? ], and R -rank ( G)= r≥2 , the value of the Furstenberg entropy must coincide with one of the 2 r values h μ ( G/ Q, ν 0) , where Q⊂ G is a parabolic subgroup. We also construct counterexamples to show that this conclusion fails for both non- P -mixing actions and actions of groups with R -rank 1 . We also characterize amenable actions with a stationary measure as the actions having the maximal possible value of the Furstenberg entropy. We give applications to geometric rigidity for actions with low Furstenberg entropy, to orbit equivalence and to the cohomology of actions with stationary measure.

Homogenous projective factors for actions of semi-simple Lie groups

We analyze the structure of a continuous (or Borel) action of a connected semi-simple Lie group G with finite center and real rank at least 2 on a compact metric (or Borel) space X, using the existence of a stationary measure as the basic tool. The main result has the following corollary: Let P be a minimal parabolic subgroup of G, and K a maximal compact subgroup. Let λ be a P-invariant probability measure on X, and assume the P-action on (X,λ) is mixing. Then either λ is invariant under G, or there exists a proper parabolic subgroup Q⊂G, and a measurable G-equivariant factor map ϕ:(X,ν)→(G/Q,m), where ν=∫ K kλdk and m is the K-invariant measure on G/Q. Furthermore, The extension has relatively G-invariant measure, namely (X,ν) is induced from a (mixing) probability measure preserving action of Q.

Fundamental groups of compact manifolds and symmetric geometry of noncompact type

We introduce the notion of geometrical engagement for actions of semisimple Lie groups and their lattices as a concept closely related to Zimmer's topological engagement condition. Our notion is a geometrical criterion in the sense that it makes use of Riemannian distances. However, it can be used together with the foliated harmonic map techniques introduced in [8] to establish foliated geometric superrigidity results for both actions and geometric objects. In particular, we improve the applications of the main theorem in [9] to consider nonpositively curved compact manifolds (not necessarily with strictly negative curvature). We also establish topological restrictions for Riemannian manifolds whose universal cover have a suitable symmetric de Rham factor (Theorem B), as well as geometric obstructions for nonpositively curved compact manifolds to have fundamental groups isomorphic to certain groups build out of cocompact lattices in higher rank simple Lie groups (Corollary 4.5).

Examples of continuous semisimple lie group actions not equivalent to smooth ones

Examples of continuous semisimple lie group actions not equivalent to smooth ones

Isotropy of semisimple group actions on manifolds with geometric structure

We prove results concerning isotropy of noncompact semisimple Lie group actions that preserve pseudo-Riemannian or affine structures on compact or finite volume manifolds. For example, if a Lie group G acts locally faithfully on a connected compact complete affine manifold M and preserves the affine structure, then the connected component of the identity of each isotropy subgroup is a compact extension of a solvable group. This is a consequence of the result that nontrivial affine actions of connected groups locally isomorphic to SL 2 [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] on such manifolds cannot have fixed points. We also prove that if G is connected semisimple without compact factors and acts isometrically and topologically transitively on a connected finite volume pseudo-Riemannian M , then the action must be (everywhere) locally free. This answers affirmatively a special case of a conjecture by G. D'Ambra and M. Gromov. With only the assumption that no factor of G acts trivially, local freeness is also assured when M is compact and complete.

On Livšic’s theorem, superrigidity, and Anosov actions of semisimple Lie groups

We prove a generalization of Livsic’s Theorem on the vanishing of the cohomology of certain types of dynamical systems. As a consequence, we strengthen a result due to Zimmer concerning algebraic hulls of Anosov actions of semisimple Lie groups. Combining this with Topological Superrigidity, we find a Holder geometric structure for multiplicity free Anosov actions.