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Valentin Poenaru

Publications and source records attributed to Valentin Poenaru.

6 recordsLinked to original sources
arXiv2016

Finitely presented groups and the Whitehead nightmare

We define a `nice representation' of a finitely presented group G as being a non-degenerate essentially surjective simplicial map f from a `nice' space X into a 3-complex associated to a presentation of G, with a strong control over the singularities of f, and such that X is WGSC (weakly geometrically simply connected), meaning that it admits a filtration by simply connected and compact subcomplexes. In this paper we study such representations for a very large class of groups, namely QSF (quasi-simply filtered) groups, where QSF is a topological tameness condition of groups that is similar, but weaker, than WGSC. In particular, we prove that any QSF group admits a WGSC representation which is locally finite, equivariant and whose double point set is closed.

math.GT↗
arXiv2007

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 $Γ$. 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.

math.GT↗