Final Lift Actions Associated with Topological Functors

For a topological functor U:EB, the fiber U ^–1(b), bB, is a cocomplete poset and the left action, induced by final lift, of the endomorphism monoid B(b,b) on U ^–1(b) is cocontinuous. It is shown that every cocontinuous left action of B(b,b) on any cocomplete poset can be realized as the final lift action associated to a canonically defined topological functor over B. If B is a Grothendieck topos and b=, the subobject classifier, then B(,) inherits both a monoidal and a cocomplete poset structure. In the case B= Sets, all cocontinuous left actions of B(,) on itself are explicitly described and each is shown to arise as the final lift action associated to a specific subcategory of a certain fixed category, referred to as the category of LR-spaces. Relationships between these LR-spaces and several other well known topological categories are also considered.     

A note on the crossed product $$ \mathcal{R}(\mathcal{A},\alpha ) $$ associated with PSL2(ℝ)

Given the hyperbolic measure dxdy/y ² on the upper half plane ℍ, the rational actions of PSL₂(ℝ) on ℍ induces a continuous unitary representation α of this group on the Hilbert space L ²(ℍ, dxdy/y ²). Supposing that = {M _f : f ∈ L ^∞ (ℍ, dxdy/y ²)}, we show that the crossed product is of type I. In fact, the crossed product is *-isomorphic to the von Neumann algebra , where is the abelian group von Neumann algebra generated by the left regular representation of K. This work was supported by the Youth Foundation of Sichuan Education Department of China (Grant No. 2003B017)     

Group Actions, Double Cosets, and Homomorphisms: Unifying Concepts for the Constructive Theory of Discrete Structures

In the present paper we describe the use of group actions, double cosets and homomorphisms in the constructive theory of discrete structures, as we found it useful from both a theoretical and a practical point of view. By means of examples we should like to demonstrate that these methods are useful both as unifying principles and as efficient methods for applications.     

On the absence of invariant measures with locally maximal entropy for a class of shifts of finite type

We prove that for a class of shifts of finite type, , any invariant measure which is not a measure of maximal entropy can be perturbed a small amount in the weak* topology to an invariant measure of higher entropy. Namely, there are no invariant measures which are strictly local maxima for the entropy function.

     

Polar actions on compact Euclidean hypersurfaces

Given an isometric immersion of a compact Riemannian manifold of dimension n ≥ 3 into Euclidean space of dimension n + 1, we prove that the identity component Iso ⁰(M ⁿ ) of the isometry group Iso(M ⁿ ) of M ⁿ admits an orthogonal representation such that for every . If G is a closed connected subgroup of Iso(M ⁿ ) acting polarly on M ⁿ , we prove that Φ(G) acts polarly on , and we obtain that f(M ⁿ ) is given as Φ(G)(L), where L is a hypersurface of a section which is invariant under the Weyl group of the Φ(G)-action. We also find several sufficient conditions for such an f to be a rotation hypersurface. Finally, we show that compact Euclidean rotation hypersurfaces of dimension n ≥ 3 are characterized by their underlying warped product structure.      

Amenable actions and almost invariant sets

In this paper, we study the connections between properties of the action of a countable group on a countable set and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of on , where is a measure space. In particular, we show that the action of on is amenable iff the shift has almost invariant sets.

     

Grassmannian Fixed Point Ratios

We provide estimates for the fixed point ratios in the permutation representations of a finite classical group over a field of order q on k-subspaces of its natural n-dimensional module. For sufficiently large n, each element must either have a negligible ratio or act linearly with a large eigenspace. We obtain an asymptotic estimate in the latter case, which in most cases is q^–dk where d is the codimension of the large eigenspace. The results here are tailored for our forthcoming proof of a conjecture of Guralnick and Thompson on composition factors of monodromy groups.     

Polar and coisotropic actions on Kähler manifolds

The main result of the paper is that a polar action on a compact irreducible homogeneous Kähler manifold is coisotropic. This is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics.