Daniele Ettore Otera

On the wgsc and qsf tameness conditions for finitely presented groups

A finitely presented group is weakly geometrically simply connected (wgsc) if it is the fundamental group of some compact polyhedron whose universal covering is wgsc, i.e., it has an exhaustion by compact connected and simply connected sub-polyhedra. We show that this condition is almost-equivalent to Bricks qsf property, which amounts to finding an exhaustion approximable by finite simply connected complexes, and also to the tame comba- bility introduced and studied by Mihalik and Tschantz. We further observe that a number of standard constructions in group theory yield qsf groups and analyze specific examples. We show that requiring the exhaustion be made of metric balls in some Cayley complex is a strong constraint, not satisfied by general qsf groups. In the second part of this paper we give sufficient conditions under which groups which are extensions of finitely presented groups by finitely generated (but infinitely presented) groups are qsf. We prove, in particular, that the finitely presented HNN extension of the Grigorchuk group is qsf.

On topological filtrations of groups

The (weak) geometric simple connectivity and the quasi-simple filtration are topological notions of manifolds, which may be defined for discrete groups too. It turns out that they are equivalent for finitely presented groups, but the main problem is the absence of examples of groups which do not satisfy them. In this note we study some algebraic classes of groups with respect to these properties.

On Poénaru’s inverse-representations

Abstract The present survey deals with the notion of topological inverse-representation for spaces and groups, a concept introduced by Valentin Poénaru in the 80’s, and largely used in successive works in geometric topology. We recall the origin of this concept as a low-dimensional topology tool for studying compact 3-manifolds and their universal covering spaces. We also illustrate some recent developments in geometric group theory.

Some considerations on Hydra groups and a new bound for the length of words

After a survey on some recent results of Riley and others on Ackermann functions and Hydra groups, we make an analogy between DNA sequences, whose growth is the same of that of Hydra groups, and a musical piece, written with the same algorithmic criterion. This is mainly an aesthetic observation, which emphasizes the importance of the combinatorics of words in two different contexts. A result of specific mathematical interest is placed at the end, where we sharpen some previous bounds on deterministic finite automata in which there are languages with hairpins.