22,512 research outputs found
Extensions of free groups: algebraic, geometric, and algorithmic aspects
In this work we use geometric techniques in order to study certain natural extensions of free groups, and solve several algorithmic problems on them.
To this end, we consider the family of free-abelian times free groups (Zm x Fn) as a seed towards further generalization in two main directions: semidirect products, and partially commuative groups (PC-groups).
The four principal projects of this thesis are the following:
Direct products of free-abelian and free groups
We begin by studying the structure of the groups Zm x Fn , with special emphasis on their lattice of subgroups, and their endomorphisms (for which an explicit description is given, and both injectivity and surjectiveness are characterized); to then solve on them algorithmic problems involving both subgroups (the membership problem, the finite index problem, and the subgroup and coset intersection problems), and endomorphisms (the fixed points poblem, the Whitehead problems, and the twisted-conjugacy problem).
Algorithmic recognition of infinite-cyclic extensions
In the first part, we prove the algorithmic undecidability of several properties (finite generability, finite presentability, abelianity, finiteness, independence, triviality) of the base group of finitely presented cyclic extensions. In particular, we see that it is not possible to decide algorithmically if a finitely presented Z-extension admits a finitely generated base group. This last result allows us to demonstrate the undecidability of the Bieri-Neumann-Strebel (BNS) invariant.
In the second part, we prove the equivalence between the isomorphism problem within the subclass of unique Z-extensions, and the semi-conjugacy problem for certain type of outer automorphisms, which we characterize algorithmically.
Stallings automata for free-abelian by free groups
After recreating in a purely algorithmic language the classic theory of Stallings associating an automaton to each subgroup of the free group, we extend this theory to semi-direct products of the form Zm ¿ Fn. Specifically, we associate to each subgroup of Zm ¿ Fn , an automaton ("enriched" with vectors in Zm), and we see that in the finitely generated case this construction is algorithmic and allows to solve the membership problem within this family of groups.
The geometric description obtained also shows (even in the case of direct products) not only that the intersection of finitely generated subgroups can be infinitely generated, but that even when it is finitely generated, the rank of the intersection can not be bound in terms of the ranks of the intersected subgroups. This fact is relevant because it denies any possible extension of the celebrated - and recently proven - Hanna-Neumann conjecture in this direction.
Intersection problems for Droms groups
After characterizing those partially commutative groups satisfying the Howson property, we combine the algorithmic version of the theorem of the subgroups of Kurosh given by S.V. Ivanov, with the ideas coming from our work on Zm x Fn, to prove the solvability of the subgroup and coset intersection problems within the subfamily of Droms groups (that is, those PC- groups whose subgroups are always again partially commutative).En aquest treball s'usen tècniques geomètriques per estudiar certes extensions naturals dels grups lliures, i atacar diversos problemes algorÃsmics sobre elles. A aquest efecte, es considera la famÃlia de grups lliure-abelians per lliure (Zm x Fn) com a punt de partida envers generalitzacions en dues direccions principals: productes semidirectes, i grups parcialment commutatius (PC-groups). Els quatre projectes principals d'aquesta tesi es descriuen a continuació. Productes directes de grups lliure-abelians per lliure. Comencem estudiant l'estructura dels grups Zm x Fn, amb especial èmfasi en el seu reticle de subgrups, i el seu monoide d'endomorfismes (per als que es dóna una descripció explÃcita, i es caracteritzen tant la injectivitat com l'exhaustivitat); per després resoldre sobre ells problemes algorÃsmics involucrant tant subgrups (el problema de la pertinença, el problema de l'Ãndex finit, i els problemes de la intersecció de subgrups i classes laterals), com endomorfismes (el problema dels punts fixos, els problemes de Whitehead , i el problema de la "conjugació retorçada" o twisted-conjugacy problem). Reconeixement algorÃtmic d'extensions cÃcliques. A la primera part, es demostra la indecidibilitat algorÃsmica de diverses propietats (generabilitad finita, presentabilitad finita, abelianitat, finitud, llibertat, i trivialitat) del grup base de les extensions cÃcliques finitament presentades. En particular, veiem que no és possible decidir algorÃtmicament si una Z-extensió finitament presentada admet un grup base finitament generat. Aquest últim resultat ens permet demostrar també la indecidibilitat de l'invariant BNS (de Bieri-Neumann-Strebel). A la segona part, es demostra l'equivalència entre el problema de l'isomorfisme dins de la subclasse de Z-extensions úniques, i el problema de la semi-conjugació per a cert tipus d'automorfismes exteriors, que caracteritzem algorÃsmicament. Autòmats d'Stallings per a grups lliure-abelians by lliure. Després de recrear en un llenguatge purament algorÃsmic la teoria clà ssica d'Stallings associant un autòmat a cada subgrup del grup lliure, estenem aquesta teoria a productes semidirectes de la forma Zm x Fn . Concretament associem un autòmat "enriquit" amb vectors de Zm a cada subgrup de Zm x Fn , i veiem que en el cas de subgrups finitament generats aquesta construcció és algorÃsmica i permet resoldre el problema de la pertinença dins d'aquesta famÃlia de grups. La descripció geomètrica obtinguda mostra a més (fins i tot en el cas de productes directes), no només que la intersecció de subgrups finitament generats pot ser infinitament generada, sinó que, fins i tot quan és finitament generada, no es pot afitar el rang de la intersecció en termes dels rangs dels subgrups intersecats. Aquest fet és rellevant perquè denega qualsevol possible extensió de la celebrada - i recentment provada - conjectura de Hanna Neumann en aquesta direcció. Problemes de la intersecció per a grups de Droms. Després de caracteritzar els grups parcialment commutatius que satisfan la propietat de Howson, combinem la versió algorÃsmica del teorema dels subgrups de Kurosh donada per S.V. Ivanov, amb les idees provinents del nostre treball sobre Zm x Fn, per demostrar la resolubilitat dels problemes de la intersecció de subgrups, de classes laterals (i afins) dins la subfamÃlia de PC-grups de Droms (i.e., aquells PC-grups en que tots els subgrups son de nou parcialment commutatius)
Automaton Semigroups and Groups: On the Undecidability of Problems Related to Freeness and Finiteness
In this paper, we study algorithmic problems for automaton semigroups and
automaton groups related to freeness and finiteness. In the course of this
study, we also exhibit some connections between the algebraic structure of
automaton (semi)groups and their dynamics on the boundary. First, we show that
it is undecidable to check whether the group generated by a given invertible
automaton has a positive relation, i.e. a relation p = 1 such that p only
contains positive generators. Besides its obvious relation to the freeness of
the group, the absence of positive relations has previously been studied and is
connected to the triviality of some stabilizers of the boundary. We show that
the emptiness of the set of positive relations is equivalent to the dynamical
property that all (directed positive) orbital graphs centered at non-singular
points are acyclic.
Gillibert showed that the finiteness problem for automaton semigroups is
undecidable. In the second part of the paper, we show that this undecidability
result also holds if the input is restricted to be bi-reversible and invertible
(but, in general, not complete). As an immediate consequence, we obtain that
the finiteness problem for automaton subsemigroups of semigroups generated by
invertible, yet partial automata, so called automaton-inverse semigroups, is
also undecidable.
Erratum: Contrary to a statement in a previous version of the paper, our
approach does not show that that the freeness problem for automaton semigroups
is undecidable. We discuss this in an erratum at the end of the paper
Minsky machines and algorithmic problems
This is a survey of using Minsky machines to study algorithmic problems in
semigroups, groups and other algebraic systems.Comment: 19 page
Correlation of Automorphism Group Size and Topological Properties with Program-size Complexity Evaluations of Graphs and Complex Networks
We show that numerical approximations of Kolmogorov complexity (K) applied to
graph adjacency matrices capture some group-theoretic and topological
properties of graphs and empirical networks ranging from metabolic to social
networks. That K and the size of the group of automorphisms of a graph are
correlated opens up interesting connections to problems in computational
geometry, and thus connects several measures and concepts from complexity
science. We show that approximations of K characterise synthetic and natural
networks by their generating mechanisms, assigning lower algorithmic randomness
to complex network models (Watts-Strogatz and Barabasi-Albert networks) and
high Kolmogorov complexity to (random) Erdos-Renyi graphs. We derive these
results via two different Kolmogorov complexity approximation methods applied
to the adjacency matrices of the graphs and networks. The methods used are the
traditional lossless compression approach to Kolmogorov complexity, and a
normalised version of a Block Decomposition Method (BDM) measure, based on
algorithmic probability theory.Comment: 15 2-column pages, 20 figures. Forthcoming in Physica A: Statistical
Mechanics and its Application
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