Let F be a hyperbolic group and G be the isometry group of a Gromov-hyperbolic and geodesic metric space. We study the action of the outer automorphism group Out(F) on the set X(F, G) of conjugacy classes of representations of F into G. For some Aut(F)-invariant subset A C F, we study the properties of A-stable representations which are generalizations of primitive stable representations defined by Minsky. Their conjugacy classes form Out(F)-invariant subsets of X(F, G) which contains the set of conjugacy classes of quasi-isometrically embedded representations. We establish a sufficient condition for the induced action to be properly discontinuous and use it when A is the set of commutators of F. Working with the derived subgroup [F, F], we also prove that [F, F]-stability is equivalent to the [F, F]- well-displacing property. Focusing on representations in PSL2(C), we find new characterizations of convex cocompact subgroups of PSL2(C).
A stratified space is a topological space equipped with a stratification, which is a decomposition or partition of the topological space satisfying certain extra conditions. More recently, the notion of poset-stratified space, i.e., topological space endowed with a continuous map to a poset with its Alexandrov topology, has been popularized. Both notions of stratified spaces are ubiquitous in mathematics, ranging from investigations of singular structures in algebraic geometry to extensions of the homotopy hypothesis in higher category theory. In this article we study the precise mathematical relation between these different approaches to stratified spaces.
We prove that knots and links that have a3-highly twisted irreducible diagram with more than two twist regions are hyperbolic. Furthermore, this result is sharp. The result is obtained using combinatorial techniques, using a new approach involving the Euler characteristic. By using geometric techniques, Futer and Purcell proved hyperbolicity under the assumption that the diagram is6-highly twisted.
