Rigidity Of Actions Research Articles

Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups, II

AbstractLet $M\stackrel {\rho _0}{\curvearrowleft }S$ be a $C^\infty $ locally free action of a connected simply connected solvable Lie group S on a closed manifold M. Roughly speaking, $\rho _0$ is parameter rigid if any $C^\infty $ locally free action of S on M having the same orbits as $\rho _0$ is $C^\infty $ conjugate to $\rho _0$ . In this paper we prove two types of result on parameter rigidity.First let G be a connected semisimple Lie group with finite center of real rank at least $2$ without compact factors nor simple factors locally isomorphic to $\mathop {\mathrm {SO}}\nolimits _0(n,1)(n\,{\geq}\, 2)$ or $\mathop {\mathrm {SU}}\nolimits (n,1)(n\geq 2)$ , and let $\Gamma $ be an irreducible cocompact lattice in G. Let $G=KAN$ be an Iwasawa decomposition. We prove that the action $\Gamma \backslash G\curvearrowleft AN$ by right multiplication is parameter rigid. One of the three main ingredients of the proof is the rigidity theorems of Pansu, and Kleiner and Leeb on the quasi-isometries of Riemannian symmetric spaces of non-compact type.Secondly we show that if $M\stackrel {\rho _0}{\curvearrowleft }S$ is parameter rigid, then the zeroth and first cohomology of the orbit foliation of $\rho _0$ with certain coefficients must vanish. This is a partial converse to the results in the author’s [Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups. Geom. Topol. 21(1) (2017), 157–191], where we saw sufficient conditions for parameter rigidity in terms of vanishing of the first cohomology with various coefficients.

Rigidity of Actions with Extreme Deviation from Multiple Mixing

We introduce a class of systems, including Ledrappier’s example, which do not have multiple mixing. A classification of such systems for 2D lattice actions is constructed.

Local rigidity of higher rank non-abelian action on torus

In this paper, we show local smooth rigidity for higher rank ergodic nilpotent action by toral automorphisms. In former papers all examples for actions enjoying the local smooth rigidity phenomenon are higher rank and have no rank-one factors. In this paper we give examples of smooth rigidity of actions having rank-one factors. The method is a generalization of the KAM (Kolmogorov–Arnold–Moser) iterative scheme.

Faithful actions of locally compact quantum groups on classical spaces

We construct examples of locally compact quantum groups coming from bicrossed product construction, including non-Kac ones, which can faithfully and ergodically act on connected classical (noncompact) smooth manifolds. However, none of these actions can be isometric in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009), leading to the conjecture that the result obtained by Goswami and Joardar (Rigidity of action of compact quantum groups on compact, connected manifolds, 2013. arXiv:1309.1294) about nonexistence of genuine quantum isometry of classical compact connected Riemannian manifolds may hold in the noncompact case as well.

Poincaré inequalities and rigidity for actions on Banach spaces

The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group G on a reflexive Banach space X has a fixed point. This last property is a strong version of Kazhdan's property (T) and is equivalent to the fact that H^1(G,\pi)=0 for every isometric representation \pi of G on X . The condition is expressed in terms of p -Poincar\'{e} constants and we provide examples of groups, which satisfy such conditions and for which H^1(G,\pi) vanishes for every isometric representation \pi on an L_p space for some p>2 . Our methods allow to estimate such a p explicitly and yield several interesting applications. In particular, we obtain quantitative estimates for vanishing of 1-cohomology with coefficients in uniformly bounded representations on a Hilbert space. We also give lower bounds on the conformal dimension of the boundary of a hyperbolic group in the Gromov density model.

Property (T) and rigidity for actions on Banach spaces

We study property (T) and the fixed point property for actions on $L^p$ and other Banach spaces. We show that property (T) holds when $L^2$ is replaced by $L^p$ (and even a subspace/quotient of $L^p$), and that in fact it is independent of $1\leq p<\infty$. We show that the fixed point property for $L^p$ follows from property (T) when $1<p< 2+\e$. For simple Lie groups and their lattices, we prove that the fixed point property for $L^p$ holds for any $1< p<\infty$ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces.

Infinitesimal rigidity of boundary lattice actions

In Part 1 we describe a duality method for calculating twisted cocycles. In Part 2 we use our method to prove various results on cohomological rigidity of higher-rank cocompact lattice actions. In Part 3 we use the results of Parts 1 and 2 to prove infinitesimal rigidity of the actions of cocompact lattices on the maximal boundaries of some non-compact type symmetric spaces.

Anosov automorphisms for nilmanifolds and rigidity of group actions

AbstractWe obtain the density of Lyapunov exponents for maximal abelian ℝ-split group in kth tensor product representation of a subgroup Γ ⊂ SL(n, ℤ) of finite index under certain conditions. Anosov and Cartan actions of such groups associated with irreducible representations of SL(n, ℝ) are also classified. Examples of rigidity of actions on nilmanifolds are discussed.

A Management Viewpoint

Widespread collective bargaining in the United States originated with the National War Labor Board of 1918. Later legislation made participation mandatory for employers. Since 1935 government intervention in collective bargaining has been continued by each administration to carry out its na tional purposes. The real choice appears to be among alter nate degrees and forms of government regulation, not between laissez faire and regulation. Measured by the criterion of in dustrial peace alone, comparatively unregulated collective bar gaining has functioned effectively. Strike activity has declined over the past three decades both in time lost and in workers involved. The contribution of government intervention to such industrial peace is not clear. Intervention experiences have been varied and somewhat contradictory. It may be necessary, however, to be able to invoke government interven tion in certain emergency disputes. It appears desirable to change the labor disputes section of the Taft-Hartley Labor- Management Relations Act of 1947 from the current rigidity of action to an elastic choice-of-procedures approach. Un regulated collective bargaining has not functioned effectively in safeguarding internal democracy within unions, in checking excessive and illegal use of power, and in the promotion of full employment. Legislative change is necessary to meet some of the specific problems in each of these three areas.