37,958 research outputs found
Structural stability of meandering-hyperbolic group actions
In his 1985 paper Sullivan sketched a proof of his structural stability
theorem for group actions satisfying certain expansion-hyperbolicity axioms. In
this paper we relax Sullivan's axioms and introduce a notion of meandering
hyperbolicity for group actions on general metric spaces. This generalization
is substantial enough to encompass actions of certain non-hyperbolic groups,
such as actions of uniform lattices in semisimple Lie groups on flag manifolds.
At the same time, our notion is sufficiently robust and we prove that
meandering-hyperbolic actions are still structurally stable. We also prove some
basic results on meandering-hyperbolic actions and give other examples of such
actions.Comment: 58 pages, 5 figures; [v2] Corollary 3.19 is wrong and thus removed;
[v3] Introduced a new notion of meandering hyperbolicity, generalized the
main structural stability theorem even further, and added a new Section 5 on
uniform lattices and their structural stabilit
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
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