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

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 $\mathrm{\Gamma }$. One way of defining this property is the existence of a smooth compact manifold $M$ with ${\pi }_{1}M=\mathrm{\Gamma }$, such that $\tilde{M}$ is geometrically simply-connected ({\it i.e.} without handles of index $\lambda = 1$). There are also alternative, more group-theoretical definitions, which are presentation independent. But $\mathrm{\Gamma }\in \mathrm{Q}\mathrm{S}\mathrm{F}$ is not only a universal property, it is quite highly non-trivial too; its very special case for $\Gamma = \pi_1 M^3$(whereitmeans$\pi_1^{\infty} \tilde M^3 = 0$) is actually already known, as a corollary of G. Perelman’s big breakthrough on the Geometrization of 3-Manifolds.