61 research outputs found
Deformations and stability in complex hyperbolic geometry
This paper concerns with deformations of noncompact complex hyperbolic
manifolds (with locally Bergman metric), varieties of discrete representations
of their fundamental groups into and the problem of (quasiconformal)
stability of deformations of such groups and manifolds in the sense of L.Bers
and D.Sullivan.
Despite Goldman-Millson-Yue rigidity results for such complex manifolds of
infinite volume, we present different classes of such manifolds that allow
non-trivial (quasi-Fuchsian) deformations and point out that such flexible
manifolds have a common feature being Stein spaces. While deformations of
complex surfaces from our first class are induced by quasiconformal
homeomorphisms, non-rigid complex surfaces (homotopy equivalent to their
complex analytic submanifolds) from another class are quasiconformally
unstable, but nevertheless their deformations are induced by homeomorphisms
Geometry and topology of complex hyperbolic and CR-manifolds
We study geometry, topology and deformation spaces of noncompact complex
hyperbolic manifolds (geometrically finite, with variable negative curvature),
whose properties make them surprisingly different from real hyperbolic
manifolds with constant negative curvature. This study uses an interaction
between K\"ahler geometry of the complex hyperbolic space and the contact
structure at its infinity (the one-point compactification of the Heisenberg
group), in particular an established structural theorem for discrete group
actions on nilpotent Lie groups
Hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space
We use our new type of bounded locally homeomorphic quasiregular mappings in
the unit 3-ball to address long standing problems for such mappings. The
construction of such mappings comes from our construction of non-trivial
compact 4-dimensional cobordisms with symmetric boundary components and
whose interiors have complete 4-dimensional real hyperbolic structures. Such
bounded locally homeomorphic quasiregular mappings are defined in the unit
3-ball as mappings equivariant with the standard
conformal action of uniform hyperbolic lattices in the unit 3-ball and with its discrete
representation . Here is
the fundamental group of our non-trivial hyperbolic 4-cobordism
and the kernel of the homomorphism is a free group on three generators.Comment: Clarification of the previous submission. 13 pages, 2 figures. arXiv
admin note: text overlap with arXiv:1510.08951, arXiv:1611.0043
Hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space
We use our new type of bounded locally homeomorphic quasiregular mappings in the unit 3-ball to address long standing problems for such mappings, including the Vuorinen injectivity problem. The construction of such mappings comes from our construction of non-trivial compact 4-dimensional cobordisms M with symmetric boundary components and whose interiors have complete 4-dimensional real hyperbolic structures. Such bounded locally homeomorphic quasiregular mappings are defined in the unit 3-ball B³ ⊂ R³ as mappings equivariant with the standard conformal action of uniform hyperbolic lattices Г ⊂ IsomH³ in the unit 3-ball and with its discrete representation G = ρ(Г) ⊂ IsomH⁴. Here, G is the fundamental group of our non-trivial hyperbolic 4-cobordism M = (H⁴∪Ω (G))/G, and the kernel of the homomorphism ρ: Г → G is a free group F₃ on three generators
Complements of tori and Klein bottles in the 4-sphere that have hyperbolic structure
Many noncompact hyperbolic 3-manifolds are topologically complements of links
in the 3-sphere. Generalizing to dimension 4, we construct a dozen examples of
noncompact hyperbolic 4-manifolds, all of which are topologically complements
of varying numbers of tori and Klein bottles in the 4-sphere. Finite covers of
some of those manifolds are then shown to be complements of tori and Klein
bottles in other simply-connected closed 4-manifolds. All the examples are
based on a construction of Ratcliffe and Tschantz, who produced 1171 noncompact
hyperbolic 4-manifolds of minimal volume. Our examples are finite covers of
some of those manifolds.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-41.abs.htm
Infinite index subgroups and finiteness properties of intersections of geometrically finite groups
We explore which types of finiteness properties are possible for intersections of geometrically finite groups of isometries in negatively curved symmetric rank one spaces. Our main tool is a twist construction which takes as input a geometrically finite group containing a normal subgroup of infinite index with given finiteness properties and infinite Abelian quotient, and produces a pair of geometrically finite groups whose intersection is isomorphic to the normal subgroup
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