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

Abstract 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.

The end-depth

Some refinements of the (simple) connectivity at infinity

Topological tameness conditions for infinite groups

On the WGSC Property in Some Classes of Groups

The property of quasi-simple filtration (or qsf) for groups has been introduced in literature more than 10 years ago by S. Brick. This is equivalent, for groups, to the weak geometric simple connectivity (or wgsc). The main interest of these notions is that there is still not known whether all finitely presented groups are wgsc (qsf) or not. The present note deals with the wgsc property for solvable groups and generalized FC-groups. Moreover, a relation between the almost-convexity condition and the Tucker property, which is related to the wgsc property, has been considered for 3-manifold groups.

Alcune condizioni topologiche di gruppi discreti

Asymptotic Topology of Groups and Connectivity at Infinity

A Survey on Just-Non-X Groups

Let be a class of groups. A group which does not belong to but all of whose proper quotient groups belong to is called just-non- group. The present note is a survey of recent results on the topic with a special attention to topological groups.

On the proper homotopy invariance of the Tucker property

A non-compact polyhedron P is Tucker if, for any compact subset K ⊂ P, the fundamental group π1(P − K) is finitely generated. The main result of this note is that a manifold which is proper homotopy equivalent to a Tucker polyhedron is Tucker. We use Poenaru’s theory of the equivalence relations forced by the singularities of a non-degenerate simplicial map.

A topological property for discrete groups

Some algebraic and topological properties of the nonabelian tensor product

Several authors investigated the properties which are invariant under the passage from a group to its nonabelian tensor square. In the present note we study this problem from the viewpoint of the classes of groups and the methods allow us to prove a result of invariance for some geometric properties of discrete groups.

Some remarks on the ends of groups

The geometric simple connectivity for groups