Examples of retracts in a free group that are not the fixed subgroup of any group of automorphisms

Let F be a free group of rank at least three. We show that some retracts of F previously studied by Martino-Ventura are not equal to the fixed subgroup of any group of automorphisms of F. This shows that, in F, there exist subgroups that are equal to the fixed subgroup of some set of endomorphisms but are not equal to the fixed subgroup of any set of automorphisms. Moreover, we determine the Galois monoids of these retracts, where, by the Galois monoid of a subgroup H of F, we mean the monoid consisting of all endomorphisms of F that fix H.

The density of injective endomorphisms of a free group

"Vegeu el resum a l'inici del document del fitxer adjunt."

BASS CYCLIC UNITS AS FACTORS IN A FREE GROUP IN INTEGRAL GROUP RING UNITS

Marciniak and Sehgal showed that if u is a non-trivial bicyclic unit of an integral group ring then there is a bicyclic unit v such that u and v generate a non-abelian free group. A similar result does not hold for Bass cyclic units of infinite order based on non-central elements as some of them have finite order modulo the center. We prove a theorem that suggests that this is the only limitation to obtain a non-abelian free group from a given Bass cyclic unit. More precisely, we prove that if u is a Bass cyclic unit of an integral group ring ZG of a solvable and finite group G, such that u has infinite order modulo the center of U(ZG) and it is based on an element of prime order, then there is a non-abelian free group generated by a power of u and a power of a unit in ZG which is either a Bass cyclic unit or a bicyclic unit.

FREE GROUP ALGEBRAS IN DIVISION RINGS

Let k be an algebraically closed field of characteristic zero and let L be an algebraic function field over k. Let sigma : L -> L be a k-automorphism of infinite order, and let D be the skew field of fractions of the skew polynomial ring L[t; sigma]. We show that D contains the group algebra kF of the free group F of rank 2.

Invariants of Unipotent Transformations Acting on Noetherian Relatively Free Algebras

2000 Mathematics Subject Classification: 16R10, 16R30.

Experimenting with infinite groups, I

A group is termed parafree if it is residually nilpotent and has the same nilpotent quotients as a given free group. Since free groups are residually nilpotent, they are parafree. Nonfree parafree groups abound and they all have many properties in common with free groups. Finitely presented parafree groups have solvable word problems, but little is known about the conjugacy and isomorphism problems. The conjugacy problem plays an important part in determining whether an automorphism is inner, which we term the inner automorphism problem. We will attack these and other problems about parafree groups experimentally, in a series of papers, of which this is the first and which is concerned with the isomorphism problem. The approach that we take here is to distinguish some parafree groups by computing the number of epimorphisms onto selected finite groups. It turns out, rather unexpectedly, that an understanding of the quotients of certain groups leads to some new results about equations in free and relatively free groups. We touch on this only lightly here but will discuss this in more depth in a future paper.

Factoriality and the pin-reutenauer procedure

We consider implicit signatures over finite semigroups determined by sets of pseudonatural numbers. We prove that, under relatively simple hypotheses on a pseudovariety V of semigroups, the finitely generated free algebra for the largest such signature is closed under taking factors within the free pro-V semigroup on the same set of generators. Furthermore, we show that the natural analogue of the Pin-Reutenauer descriptive procedure for the closure of a rational language in the free group with respect to the profinite topology holds for the pseudovariety of all finite semigroups. As an application, we establish that a pseudovariety enjoys this property if and only if it is full.

Polynomial-time complexity for instances of the endomorphism problem in free groups

We say the endomorphism problem is solvable for an element W in a free group F if it can be decided effectively whether, given U in F, there is an endomorphism Φ of F sending W to U. This work analyzes an approach due to C. Edmunds and improved by C. Sims. Here we prove that the approach provides an efficient algorithm for solving the endomorphism problem when W is a two- generator word. We show that when W is a two-generator word this algorithm solves the problem in time polynomial in the length of U. This result gives a polynomial-time algorithm for solving, in free groups, two-variable equations in which all the variables occur on one side of the equality and all the constants on the other side.

Statistical properties of subgroups of free groups

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution: subgroups are generated (finite presentations are determined) by randomly chosen k-tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so-called graph-based distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the raph-based distribution, while they are exponentially generic in the word-based distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group.

Acute seizures in acute ischemic stroke: does thrombolysis have a role to play?

Seizures appear at stroke presentation, during the acute phase or as a late complication of stroke. Thrombolysis has not been investigated as a risk factor despite its potential neurotoxic effect. We try to identify risk factors for seizures during the acute phase of ischemic stroke in a cohort including thrombolysed patients. We undertook a case-control study at a single stroke center using data from Acute Stroke Registry and Analyse of Lausanne (ASTRAL). Patients with seizure occurring during the first 7 days following stroke were retrospectively identified. Bi-variable and multivariable statistical analyses were applied to compare cases and randomly selected controls. We identified 28 patients experiencing from seizures in 2,327 acute ischemic strokes (1.2 %). All seizures occurred during the first 72 h. Cortical involvement, thrombolysis with rt-PA, arterial recanalization, and higher initial NIHSS were statistically associated with seizures in univariated analysis. Backward linear regression identified cortical involvement (OR 7.53, 95 % CI 1.6-35.2, p < 0.01) and thrombolysis (OR 4.6, 95 % CI 1.6-13.4, p = 0.01) as being independently associated with seizure occurrence. Overall, 3-month outcome measured by the modified Rankin scale (mRS) was comparable in both groups. In the subgroup of thrombolysed patients, outcome was significantly worse at 3 months in the seizure group with 9/12 (75 %) patients with mRS ≥3, compared to 6/18 (33.3 %) in the seizure-free group (p = 0.03). Acute seizures in acute ischemic stroke were relatively infrequent. Cortical involvement and thrombolysis with rt-PA are the principal risk factors. Seizures have a potential negative influence on clinical outcome in thrombolysed patients.

Free groups in central simple algebras without Tits` theorem

Let R be a noncommutative central simple algebra, the center k of which is not absolutely algebraic, and consider units a,b of R such that {a,a(b)} freely generate a free group. It is shown that such b can be chosen from suitable Zariski dense open subsets of R, while the a can be chosen from a set of cardinality \k\ (which need not be open).

TMD and GOHAI indices of Brazilian institutionalised and community-dwelling elderly

Considering the controversy in the literature regarding several aspects of temporomandibular dysfunction (TMD) in elderly populations and the absence of reliable data on elderly Brazilians in this field, this study consisted of an evaluation of TMD prevalence and the self-perception of oral health among institutionalised and community-dwelling elderly in Sao JosE dos Campos, Brazil.Two hundred and fifteen community-dwelling and 185 institutionalised elderly people were evaluated by the Helkimo anamnestic (Ai) and clinical dysfunction (Di) indices and answered a questionnaire using the Geriatric Oral Health Assessment Index (GOHAI).The major prevalence of TMD symptoms was for the Ai0 (symptom-free) group (69.5%), while the major prevalence of clinical signs was for the DiI (mild) group (56%). Women presented a higher AiII classification than men (chi(2) test, p = 0.049). Community-dwelling elderly presented a significantly lower Ai0 classification than the institutionalised ones (Two ratios equality test, p < 0.001). There was no relationship between the institutionalised status and the clinical dysfunction index for Di0 and DiIII classification (Two ratios equality test, p = 0.194 and 0.535 respectively). The institutionalised elderly presented greater (One-way anova = 0.005) self-perception of oral health (33.45) than did the community-dwelling group (32.66). There were only weak Pearson's correlations among the anamnestic (-33.0%) or clinical (-14.7%) findings by the TMD and GOHAI indices. Symptom-free (Ai0) institutionalised elderly presented better scores in all GOHAI dimensions and elderly representing an absence of clinical TMD signs (Di0) presented higher GOHAI physical dimension scores in both groups.The prevalence of TMD symptoms among this sample of elderly individuals was relatively low, self-perception of oral health was reasonable and a weak, inverse correlation was found between TMD signs and symptoms and elderly self-perception of oral health measured by the GOHAI index.

THE STRUCTURE OF FREE AUTOMORPHIC MOUFANG LOOPS

We describe the structure of a free loop of rank n in the variety of automorphic Moufang loops as a subdirect product of a free group and a free commutative Moufang loop, both of rank n. In particular, the variety of automorphic Moufang loops is the join of the variety of groups and the variety of commutative Moufang loops.

A SURVEY ON FREE OBJECTS IN DIVISION RINGS AND IN DIVISION RINGS WITH AN INVOLUTION

Let D be a division ring with center k, and let D-dagger be its multiplicative group. We investigate the existence of free groups in D-dagger, and free algebras and free group algebras in D. We also go through the case when D has an involution * and consider the existence of free symmetric and unitary pairs in D-dagger.

Walter Lorenz – Wegbegleiter und Orientierungspunkt in der interkulturellen Sozialen Arbeit

First meeting, common interests and ways together The European Centre for Social Welfare, Training and Research situated in Vienna, was our first meeting place. W. Lorenz was interested in the international comparison of the different concepts and perspectives of social welfare problems in the European countries and the different developments in the training of social professions in Europe. The challenge of intercultural, antiracist social work in the context of Erasmus-Intensive Seminars To organize an intensive seminar with the aim to train students and colleagues for intercultural and antiracist competence in social professions, we formed an European network of European universities and schools of s.w. in Vienna (VIENNET), with the support of ECCE (European Centre of Community Education) in Koblenz. “The group discovered that working on these issues in an international context raises issues of ‘difference’ with renewed acuteness”(cit. W. Lorenz). We learned to cope with a variety of differences: biographical, language, theoretical and institutional backgrounds and discourse traditions. A Venue for an Intensive Seminar In choosing a venue for an Intensive Seminar we were relatively free. We locked for a place, “one dream about”, to support in the best way our seminar aims, to promote a base built on knowledge, skills and values particularly in the area of inner/outer borders, disadvantage, ignorance, minorities, majorities, vulnerable groups, racism and xenophobia. In a small village in Burgenland (Austria), very close to the Hungarian border, we thought to have found it. Future Prospect Are we only representatives of our background institutions or did we act and exposed ourselves as persons with a very specific biography and training experience. Can we sustain this created network, as a network of experts and friends in the field of intercultural, antiracist social work? This question is still open.