9,480 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)
Finiteness results for subgroups of finite extensions
We discuss in the context of finite extensions two classical theorems of
Takahasi and Howson on subgroups of free groups. We provide bounds for the rank
of the intersection of subgroups within classes of groups such as virtually
free groups, virtually nilpotent groups or fundamental groups of finite graphs
of groups with virtually polycyclic vertex groups and finite edge groups. As an
application of our generalization of Takahasi's Theorem, we provide an uniform
bound for the rank of the periodic subgroup of any endomorphism of the
fundamental group of a given finite graph of groups with finitely generated
virtually nilpotent vertex groups and finite edge groups.Comment: 20 pages; no figures. Keywords: finite extensions, Howson's Theorem,
Hanna Neumann Conjecture, Takahasi's Theorem, periodic subgroup
Intersection problem for Droms RAAGs
We solve the subgroup intersection problem (SIP) for any RAAG G of Droms type
(i.e., with defining graph not containing induced squares or paths of length
3): there is an algorithm which, given finite sets of generators for two
subgroups H,K of G, decides whether is finitely generated or not,
and, in the affirmative case, it computes a set of generators for .
Taking advantage of the recursive characterization of Droms groups, the proof
consists in separately showing that the solvability of SIP passes through free
products, and through direct products with free-abelian groups. We note that
most of RAAGs are not Howson, and many (e.g. F_2 x F_2) even have unsolvable
SIP.Comment: 33 pages, 12 figures (revised following the referee's suggestions
Intersection problem for Droms RAAGs
We solve the subgroup intersection problem (SIP) for any RAAG G of Droms type
(i.e., with defining graph not containing induced squares or paths of length
3): there is an algorithm which, given finite sets of generators for two
subgroups H,K of G, decides whether is finitely generated or not,
and, in the affirmative case, it computes a set of generators for .
Taking advantage of the recursive characterization of Droms groups, the proof
consists in separately showing that the solvability of SIP passes through free
products, and through direct products with free-abelian groups. We note that
most of RAAGs are not Howson, and many (e.g. F_2 x F_2) even have unsolvable
SIP.Comment: 33 pages, 12 figures (revised following the referee's suggestions
On the intersection of free subgroups in free products of groups
Let (G_i | i in I) be a family of groups, let F be a free group, and let G =
F *(*I G_i), the free product of F and all the G_i. Let FF denote the set of
all finitely generated subgroups H of G which have the property that, for each
g in G and each i in I, H \cap G_i^{g} = {1}. By the Kurosh Subgroup Theorem,
every element of FF is a free group. For each free group H, the reduced rank of
H is defined as r(H) = max{rank(H) -1, 0} in \naturals \cup {\infty} \subseteq
[0,\infty]. To avoid the vacuous case, we make the additional assumption that
FF contains a non-cyclic group, and we define sigma := sup{r(H\cap
K)/(r(H)r(K)) : H, K in FF and r(H)r(K) \ne 0}, sigma in [1,\infty]. We are
interested in precise bounds for sigma. In the special case where I is empty,
Hanna Neumann proved that sigma in [1,2], and conjectured that sigma = 1;
almost fifty years later, this interval has not been reduced. With the
understanding that \infty/(\infty -2) = 1, we define theta := max{|L|/(|L|-2) :
L is a subgroup of G and |L| > 2}, theta in [1,3]. Generalizing Hanna Neumann's
theorem, we prove that sigma in [theta, 2 theta], and, moreover, sigma = 2
theta if G has 2-torsion. Since sigma is finite, FF is closed under finite
intersections. Generalizing Hanna Neumann's conjecture, we conjecture that
sigma = theta whenever G does not have 2-torsion.Comment: 28 pages, no figure
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