In this paper we prove that a wild knot K which is the limit set of a Kleinian group acting conformally on the unit 3-sphere, with its standard metric, is homogeneous: given two points p, q ∈ K, there exists a homeomorphism f of the sphere such that f(K) = K and f(p) = q. We also show that if the wild knot is a fibered knot then we can choose an f which preserves the fibers.
Given an origami (square-tiled surface) with automorphism group , we compute the decomposition of the first homology group of into isotypic -submodules. Through the action of the affine group of on the homology group, we deduce some consequences for the multiplicities of the Lyapunov exponents of the Kontsevich-Zorich cocycle. We also construct and study several families of interesting origamis illustrating our results.
We prove, for any , that there is a closed connected orientable surface so that the hyperbolic space almost-isometrically embeds into the Teichmüller space of , with quasi-convex image lying in the thick part. As a consequence, quasi-isometrically embeds in the curve complex of .
Applying the “exact WKB method” (cf. Delabaere-Dillinger-Pham) to the stationary one-dimensional Schrödinger equation with polynomial potential, one is led to a multivalued complex action-integral function. This function is a (hyper)elliptic integral; the sheet structure of its Riemann surface above the plane of its values has interesting properties: the projection of its branch-points is in general a dense subset of the plane, and there is a group of symmetries acting on the surface. The distribution...
This paper is mainly intended as a survey of the recent work of a number of authors concerning certain infinite group actions on spheres and to raise some as yet unanswered questions. The main thrust of the current research in this area has been to decide what topological and geometrical properties characterise the infinite conformal or Möbius groups. One should then obtain reasonable topological or geometrical restrictions on a subgroup G of the homeomorphism group of a sphere which will imply...
We definite invariants of translation surfaces which refine Veech groups. These aid in exact determination of Veech groups. We give examples where two surfaces of isomorphic Veech group cannot even share a common tree of balanced affine coverings. We also show that there exist translation surfaces of isomorphic Veech groups which cannot affinely cover any common surface. We also extend a result of Gutkin and Judge and thereby give the first examples of noncompact Fuchsian...