Two-Dimensional Knots and Representations of Hyperbolic Groups

Abstract

We describe relations between hyperbolic geometry and codimension two knots or, more exactly, between varieties of conjugacy classes of discrete faithful representations of the fundamental groups of hyperbolic n-manifolds M into $\operatorname{SO}^{\circ} (n+2,1)$ and (n-1)-dimensional knots in the (n+1)-sphere. This approach allows us to discover a phenomenon of non-connectedness of these varieties for closed n-manifolds M, $n\geq 3$, with large enough number of disjoint totally geodesic surfaces, to construct quasisymmetric infinitely compounded "Julia" knots $K\subset S^{n+1}$ which are everywhere wild and have recurrent $π_1(M)$-action, and to study circle and 2-plane bundles (with geometric structures) over closed hyperbolic n-manifolds.