Equivariant map

About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.

On the lifting of structure groups

Abstract: § i. General principal bundles i. F-structures Let P=(P,~,B,G) be a principal bundle where P and B are topological spaces and @ is a topological group. Let 0: F+G be a continuous homomorphism from a topological group F onto G with kernel K • 0 will be called central, if (i) K is discrete (ii) K is contained in the center of F Thus, in particular, K is abelian. It is easy to check that (ii) follows from (i) if F is connected. Thus every covering projection P: F÷G (F connected) satisfies the conditions above. Throughout this paper we shall assume that p is central. A r-structure on P is a F-principal bundle P=(P,~,B,F) together with a strong bundle map ~: P÷P which is equivariant under the right actions of the structure groups; that is

Hyperconvex representations and exponential growth

Abstract: Let $G$ be a real algebraic semi-simple Lie group and $\Gamma$ be the fundamental group of a compact negatively curved manifold. In this article we study the limit cone, introduced by Benoist, and the growth indicator function, introduced by Quint, for a class of representations $\rho:\Gamma\to G$ admitting a equivariant map from $\partial\Gamma$ to the Furstenberg boundary of $G$'s symmetric space together with a transversality condition. We then study how these objects vary with the representation.

Median structures on asymptotic cones and homomorphisms into mapping class groups

Abstract: The main goal of this paper is a detailed study of asymptotic cones of the mapping class groups. In particular, we prove that every asymptotic cone of a mapping class group has a bi-Lipschitz equivariant embedding into a product of real trees, sending limits of hierarchy paths onto geodesics, and with image a median subspace. One of the applications is that a group with Kazhdan’s property (T) can have only finitely many pairwise non-conjugate homomorphisms into a mapping class group. We also give a new proof of the rank conjecture of Brock and Farb (previously proved by Behrstock and Minsky, and independently by Hamenstaedt).

Lattice polarized toric K3 surfaces

Abstract: When studying mirror symmetry in the context of K3 surfaces, the hyperkaehler structure of K3 makes the notion of exchanging Kaehler and complex moduli ambiguous. On the other hand, the metric is not renormalized due to the higher amount of supersymmetry of the underlying superconformal field theory. Thus one can define a natural mapping from the classical K3 moduli space to the moduli space of conformal field theories. Apart from the generalization of mirror constructions for Calabi-Yau threefolds, there is a formulation of mirror symmetry in terms of orthogonal lattices and global moduli space arguments. In many cases both approaches agree perfectly - with a long outstanding exception: Batyrev's mirror construction for K3 hypersurfaces in toric varieties does not fit into the lattice picture whenever the Picard group of the K3 surface is not generated by the pullbacks of the equivariant divisors of the ambient toric variety. In this case, not even the ranks of the corresponding Picard lattices add up as expected. In this paper the connection is clarified by refining the lattice picture. We show (by explicit calculation with a computer) mirror symmetry for all families of toric K3 hypersurfaces corresponding to dual reflexive polyhedra, including the formerly problematic cases.