Date & Time: Thursday, March 10, 2010, 15:00-17:00.
Venue: Ramanujan Hall
Title: "Real and Complex Hyperbolic Geometry
Speaker:Prof. Boris Apanasov, from U of Oklahoma, Norman
Abstract: FIRST COLLOQUIUM TALK:
"Cartan Angular Invariant and Deformations in Rank One Symmetric Spaces"
Abstract:
We develop and study new geometric invariants in the quaternionic hyperbolic space and in the hyperbolic Cayley plane. In these non-commutative and non-associative geometries they are a substitution for Toledo invariant and the Cartan angular invariant well know in the complex hyperbolic geometry. These new invariants are used for the investigation of quasi-Fuchsian deformations of quaternionic and octonionic hyperbolic manifolds. In particular, bendings are defined for such structures, which are the last two classes of locally symmetric structures of rank 1. -----
SECOND COLLOQUIUM TALK;
"Hyperbolic 4-cobordisms and group homomorphisms with infinite kernel"
Abstract:
We will discuss how the global geometry and topology of a non-trivial compact 4-dimensional cobordism $M$ whose interior has a complete hyperbolic structure depend on properties of the variety of discrete representations of the fundamental group of its 3-dimensional boundary $\partial M$. We construct such hyperbolic 4-cobordisms $M$ whose boundary components are covered by the discontinuity set $\Omega (G)\subset S^3$ with two connected components $\Omega_1$ and $\Omega_2$, where the action $\Gamma$ of the fundamental group $\pi_1(\p M)$ is symmetric and has contractible fundamental polyhedra of the same combinatorial (3-hyperbolic) type. Nevertheless we show that a geometric symmetry of boundary components of our hyperbolic 4-cobordism $M(G)$) are not enough to ensure that the group $G=\pi_1(M)$ is quasi-Fuchsian, and in fact our 4-cobordism $M$ is non-trivial. This is related to non-connectedness of the variety of discrete representation of the uniform hyperbolic lattice $\Gamma$ and infinite kernels of the constructed homomorphisms $\Gamma\rightarrow G$.