Discrete symmetry with compact fundamental domain, and geometric simple connectivity - A provisional Outline of work in Progress -
Résumé
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 $\Gamma$. One way of defining this property is the existence of a smooth compact manifold $M$ with $\pi_1 \, M = \Gamma$, 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 $\Gamma \in {\rm QSF}$ is not only a universal property, it is quite highly non-trivial too; its very special case for $\Gamma = \pi_1 \, M^3$ (where it means $\pi_1^{\infty} \, \tilde M^3 = 0$) is actually already known, as a corollary of G.~Perelman's big breakthrough on the Geometrization of 3-Manifolds.
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