The research topic we propose lies in the intersection of Group Theory, Geometry and (low-dimensional) Topology. In this project we wish to explore the geometry and the topology at infinity of discrete groups. The geometrical viewpoint for groups has sparked the interest of geometers, topologists and group theorists since the seminal work of M.Gromov on the asymptotic invariants of groups. We would like to look at groups from a topological viewpoint, and to study some topological properties (at infinity) of groups. In particular we will mainly focus on the geometric simple connectivity (g.s.c.) and the simple connectivity at infinity. The simple connectivity at infinity is an important tameness condition on the ends of the space, and it has been used to characterize Euclidean spaces among contractible open topological manifolds. Whereas the geometric simple connectivity is a related notion developed by V.Poenaru (mostly in dimensions 3 and 4), in his work concerning the Poincaré Conjecture. It is worthy to note that it can be shown that all reasonable examples of groups (e.g. word hyperbolic, semi-hyperbolic, CAT(0), group extensions, one relator groups) are g.s.c. Hence it would be very interesting to find an example of a finitely presented group which fails to be g.s.c. Discrete groups which are not g.s.c. (if they exist) would lay at the opposite extreme to hyperbolic (or CAT(0)) groups and thus they should be non generic, in a probabilistic sense. The first step will be to find some combinatorial property equivalent to the g.s.c. On the other hand, if one can show that ANY group is geometrically simply connected, then the g.s.c. would be the first non-trivial property which holds true for all groups (contradicting the underlying philosophy of the Geometric Group Theory). Both cases will have a deep impact in the understanding of the space of groups and for their (geometrical) classification.
Field of science
- /natural sciences/mathematics/pure mathematics/topology
- /natural sciences/mathematics/pure mathematics/geometry
- /humanities/philosophy, ethics and religion/philosophy
Call for proposal
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