11,857,811 research outputs found
On the wgsc property in some classes of groups
No full textThe 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
On topological filtrations of groups
No full textThe (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
Some considerations on Hydra groups and a new bound for the length of words
No full textAfter 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, writ-
ten 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
On 3-dimensional wgsc inverse-representations of groups
No full textWe study the notion of wgsc inverse-representation of finitely presented groups and use the “-technique” of Poénaru, in order to prove that the universal cover of a closed 3-manifold admitting a wgsc inverse-representation with an extra finiteness condition is simply connected at infinity. Furthermore, we investigate some new relations between wgsc inverse-representations and the qsf property for groups
On some recent investigations of probability in group theory
No full textWe describe some recent contributions on the probability of commuting pairs, introduced by P. Erdos, W. Gustafson and P. Turan around 1968 and 1973. Both combinatorial methods and character theory have significant
application in this context and we illustrate some standard techniques and strategies, once generalizations of the
probability of commuting pairs want to be studied. The importance of the subject is emphasized in some remarks and open questions, which reformulate some classical conjectures in group theory via a probabilistic approach
Factorization number and subgroup commutativity degree via spectral invariants
No full textThe factorization number of a finite group is the number of all possible factorizations of as product of its subgroups and , while the subgroup commutativity degree of is the probability of finding two commuting subgroups in at random. It is known that can be expressed in terms of . Denoting by the subgroups lattice of , the non--permutability graph of subgroups of is the graph with vertices in , where is the smallest sublattice of containing all permutable subgroups of , and edges obtained by joining two vertices such that . The spectral properties of have been recently investigated in connection with and . Here we show a new combinatorial formula, which allows us to express , and so , in terms of adjacency and Laplacian matrices of
On Geometric Simple Connectivity
No full textL'articolo intende dare una visione panoramica su ricerche recenti, molte delle quali sono da attribuire al V.Poenaru, sulla topologia di dimensione basse e sulla teoria geometrica dei gruppi
On the proper homotopy invariance of the Tucker property
No full textA non-compact polyhedron P is Tucker if, for any compact subset K ⊂ P, the fundamental group π 1(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 ma
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