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

TL;DR

The paper demonstrates that the QSF property, related to geometric simple connectivity, is universally valid for all finitely presented groups, linking group theory with geometric topology and 3-manifold geometrization.

Contribution

It proves the universal applicability of the QSF property to all finitely presented groups, extending previous results in 3-manifold topology.

Findings

QSF property holds for all finitely presented groups

Connection between geometric simple connectivity and group properties

Extension of 3-manifold geometrization results to broader group classes

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 $\Gamma$. 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.