Rigidity Of Group Actions Research Articles

Measure-theoretic equicontinuity and rigidity of group actions

Let ( G , X ) be a G-system, which means that X is a compact Hausdorff space and G is an infinite topological group continuously acting on X, and let μ be a G-invariant measure of ( G , X ) . In this paper, we introduce the concepts of rigidity, uniform rigidity and μ-Ω-equicontinuity of ( G , X ) with respect to an infinite sequence Ω of G and the notions of μ-Ω-equicontinuity and μ-Ω-mean-equicontinuity of a function f ∈ L 2 ( μ ) with respect to an infinite sequence Ω of G. Then we give some equivalent conditions for f ∈ L 2 ( μ ) and ( G , X ) to be rigid, respectively. In addition, if G is commutative and X satisfies the first axiom of countability, we present some equivalent conditions for ( G , X ) to be uniformly rigid.

Entropy, Lyapunov exponents, and rigidity of group actions

This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled Workshop for young researchers: Groups acting on manifolds held in Teresopolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory-particularly Lya-punov exponents and metric entropy-as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [114] and recent the work of the author with D. Fisher, S. Hurtado, F. Rodriguez Hertz, and Z. Wang on actions of lattices in higher-rank semisimple Lie groups on manifolds [31, 36]. We give complete proofs of these results and present sufficient background in smooth ergodic theory needed for the proofs. A unifying theme in this text is the use of metric entropy and its relation to the geometry of conditional measures along foliations as a mechanism to verify invariance of measures. i

Rigidity of group actions on homogeneous spaces, III

Consider homogeneous G/H and G/F, for an S-algebraic group G. A lattice {\Gamma} acts on the left strictly conservatively. The following rigidity results are obtained: morphisms, factors and joinings defined apriori only in the measurable category are in fact algebraically constrained. Arguing in an elementary fashion we manage to classify all the measurable {\Phi} commuting with the {\Gamma}-action: assuming ergodicity, we find they are algebraically defined.

Topologie

This conference is one of the few occasions where researchers from many different areas in algebraic and geometric topology are able to meet and exchange ideas. Accordingly, the program covered a wide range of new developments in such fields as geometric group theory, rigidity of group actions, knot theory, and stable and unstable homotopy theory. More specifically, we discussed progress on problems such as the Farrell-Jones conjecture, the Levine conjecture in grope cobordism of knots and Rosenberg’s conjecture about homotopy invariance of negative algebraic K-theory, to mention just a few subjects with a name attached. One of the highlights was a series of four talks on the solution of Arf-Kervaire invariant problem by Mike Hill and Doug Ravenel, reporting on their joint work with Mike Hopkins.

Rigidity of group actions on solvable Lie groups

We establish analogs of the three Bieberbach theorems for a lattice \(\Gamma\) in a semidirect product \(\mathsf{K}\rtimes\mathsf{K}\) where \(\mathsf{S}\) is a connected, simply connected solvable Lie group and \(\mathsf{K}\) is a compact subgroup of its automorphism group. We first prove that the action of \(\Gamma\) on \(\mathsf{S}\) is metrically equivalent to an action of \(\Gamma\) on a supersolvable Lie group. The latter is shown to be determined by \(\Gamma\) itself up to an affine diffeomorphism. Then we characterize these lattices algebraically as polycrystallographic groups. Furthermore, we realize any polycrystallographic group \(\Gamma\) as a lattice in a semidirect product \(\mathsf{S}\rtimes\mathsf{F}\) with \(\mathsf{F}\) being a finite group whose order is bounded by a constant only depending on the dimension of \(\mathsf{S}\). This generalization of the first Bieberbach theorem is used to obtain a partial generalization of the third one as well. Finally we show for any torsion free closed subgroup \(\Upsilon \subset \mathsf{K}\rtimes\mathsf{K}\) that the quotient \(\mathsf{S}/\Upsilon\) is the total space of a vector bundle over a compact manifold B, where B is the quotient of a solvable Lie group by a torsion free polycrystallographic group.

Rigidity of group actions

Rigidity of group actions

Nonstationary normal forms and rigidity of group actions

We develop a proper “nonstationary” generalization of the classical theory of normal forms for local contractions. In particular, it is shown under some assumptions that the centralizer of a contraction in an extension is a particular Lie group, determined by the spectrum of the linear part of the contractions. We show that most homogeneous Anosov actions of higher rank abelian groups are locally C∞ rigid (up to an automorphism). This result is the main part in the proof of local C∞ rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the actions of cocompact lattices on Furstenberg boundaries, in particular projective spaces, and (ii) the actions by automorphisms of tori and nilmanifolds. The main new technical ingredient in the proofs is the centralizer result

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.

Rigidity of Group Actions on Locally Finite Trees

Rigidity of Group Actions on Locally Finite Trees

Problems on rigidity of group actions and cocycles

AbstractA conference on the interaction of ergodic theory, differential geometry and the theory of Lie Groups was held at the Mathematical Sciences Research Institute from May 24 to June 1, 1984. This is a report of the problem session organized by A. Katok and R. Zimmer and held on May 25, 1984 dealing with the topics in the title. Another problem session was centred on the rigidity of manifolds of non-positive curvature and related topics concerning their geodesic flows. This is reported on by K. Burns and A. Karok separately [2].