Dr. Boris Apanasov

Abstract:

We discuss several aspects of geometry and topology of knotted surfaces where the unifying theme is the discrete holonomy groups of corresponding geometric structures, which also involves algebra of varieties of discrete group representations and dynamics of their action in different senses - from dynamics of group orbits in considered spaces to ergodicity of group action, dynamical systems and dynamics of equivariant mappings with bounded distortion (quasiconformal, quasisymmetric and quasiregular). An interesting and unusual aspect is given by the wild properties of obtained knotted surfaces (in particular almost everywhere wildly knotted spheres - cf. A. T. Fomenko's art).

Our approach has a combinatorial flavor based on our method of "hyperbolic block-building", Siamese twins construction resulting in dis-crete representations with arbitrary large kernels (applications of our recently introduced conformal interbreeding generalizing the Gromov-Piatetskii hyperbolic interbreeding). These methods let us construct everywhere wild nontrivial 2-knots and surfaces in 4-sphere and solve well known problems in geometric analysis. Created wild surfaces have ergodic dynamics of uniform hyperbolic lattices and are obtained by constructed wild quasisymmetric maps equivariant with respect to the action of uniform hyperbolic lattices. This is connected to theory of conformal deformations of hyperbolic structures, their Teichmuller spaces (varieties of discrete reprs of hyperbolic lattices) and nontrivial homology 4-cobordisms. For related material (negatively curved locally symmetric rank one spaces, their Teichmuller spaces, reprs-n of uniform hyperbolic lattices, hyperbolic 4-cobordisms and several their appls to algebra, geometry, topology and geom analysis we refer to our new book "Dynamics of Discrete Group Action" published in the series De Gruyter Advances in Analysis and Geometry, 10.