Discrete symmetry with compact fundamental domain, and geometric simple connectivity - A provisional Outline of work in Progress -

Abstract

We show that a certain geometric property, the QSF introduced by S. Brick and M. Mihalik, is universally true for {\ibf all} finitely presented groups $Γ$. One way of defining this property is the existence of a smooth compact manifold $M$ with $π_1 M = Γ$, such that $\tilde M$ is geometrically simply-connected ({\it i.e.} without handles of index $λ= 1$). There are also alternative, more group-theoretical definitions, which are presentation independent. But $Γ\in {\rm QSF}$ is not only a universal property, it is quite highly non-trivial too; its very special case for $Γ= π_1 M^3$ (where it means $π_1^{\infty} \tilde M^3 = 0$) is actually already known, as a corollary of G. Perelman's big breakthrough on the Geometrization of 3-Manifolds.