THE LOWER CENTRAL AND DERIVED SERIES OF THE BRAID GROUPS OF THE SPHERE

In this paper, we determine the lower central and derived series for the braid groups of the sphere. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is important in its own right. The braid groups of the 2-sphere S(2) were studied by Fadell, Van Buskirk and Gillette during the 1960s, and are of particular interest due to the fact that they have torsion elements (which were characterised by Murasugi). We first prove that for all n epsilon N, the lower central series of the n-string braid group B(n)(S(2)) is constant from the commutator subgroup onwards. We obtain a presentation of Gamma(2)(Bn(S(2))), from which we observe that Gamma(2)(B(4)(S(2))) is a semi-direct product of the quaternion group Q(8) of order 8 by a free group F(2) of rank 2. As for the derived series of Bn(S(2)), we show that for all n >= 5, it is constant from the derived subgroup onwards. The group Bn(S(2)) being finite and soluble for n <= 3, the critical case is n = 4 for which the derived subgroup is the above semi-direct product Q(8) (sic) F(2). By proving a general result concerning the structure of the derived subgroup of a semi-direct product, we are able to determine completely the derived series of B(4)(S(2)) which from (B(4)(S(2)))(4) onwards coincides with that of the free group of rank 2, as well as its successive derived series quotients.

THE LOWER CENTRAL AND DERIVED SERIES OF THE BRAID GROUPS OF THE FINITELY-PUNCTURED SPHERE

Motivated in part by the study of Fadell-Neuwirth short exact sequences, we determine the lower central and derived series for the braid groups of the finitely-punctured sphere. For n >= 1, the class of m-string braid groups B(m)(S(2)\{x(1), ... , x(n)}) of the n-punctured sphere includes the usual Artin braid groups B(m) (for n = 1), those of the annulus, which are Artin groups of type B (for n = 2), and affine Artin groups of type (C) over tilde (for n = 3). We first consider the case n = 1. Motivated by the study of almost periodic solutions of algebraic equations with almost periodic coefficients, Gorin and Lin calculated the commutator subgroup of the Artin braid groups. We extend their results, and show that the lower central series (respectively, derived series) of B(m) is completely determined for all m is an element of N (respectively, for all m not equal 4). In the exceptional case m = 4, we obtain some higher elements of the derived series and its quotients. When n >= 2, we prove that the lower central series (respectively, derived series) of B(m)(S(2)\{x(1), ... , x(n)}) is constant from the commutator subgroup onwards for all m >= 3 (respectively, m >= 5). The case m = 1 is that of the free group of rank n - 1. The case n = 2 is of particular interest notably when m = 2 also. In this case, the commutator subgroup is a free group of infinite rank. We then go on to show that B(2)(S(2)\{x(1), x(2)}) admits various interpretations, as the Baumslag-Solitar group BS(2, 2), or as a one-relator group with non-trivial centre for example. We conclude from this latter fact that B(2)(S(2)\{x(1), x(2)}) is residually nilpotent, and that from the commutator subgroup onwards, its lower central series coincides with that of the free product Z(2) * Z. Further, its lower central series quotients Gamma(i)/Gamma(i+1) are direct sums of copies of Z(2), the number of summands being determined explicitly. In the case m >= 3 and n = 2, we obtain a presentation of the derived subgroup, from which we deduce its Abelianization. Finally, in the case n = 3, we obtain partial results for the derived series, and we prove that the lower central series quotients Gamma(i)/Gamma(i+1) are 2-elementary finitely-generated groups.

The classification and the conjugacy classes of the finite subgroups of the sphere braid groups

Let n >= 3. We classify the finite groups which are realised as subgroups of the sphere braid group B(n)(S(2)). Such groups must be of cohomological period 2 or 4. Depending on the value of n, we show that the following are the maximal finite subgroups of B(n)(S(2)): Z(2(n-1)); the dicyclic groups of order 4n and 4(n - 2); the binary tetrahedral group T*; the binary octahedral group O*; and the binary icosahedral group I(*). We give geometric as well as some explicit algebraic constructions of these groups in B(n)(S(2)) and determine the number of conjugacy classes of such finite subgroups. We also reprove Murasugi`s classification of the torsion elements of B(n)(S(2)) and explain how the finite subgroups of B(n)(S(2)) are related to this classification, as well as to the lower central and derived series of B(n)(S(2)).

The lower central and derived series of the braid groups of the projective plane

In this paper, we determine the lower central and derived series for the braid groups of the projective plane. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is interesting in its own right. The n-string braid groups B(n)(RP(2)) of the projective plane RP(2) were originally studied by Van Buskirk during the 1960s. and are of particular interest due to the fact that they have torsion. The group B(1)(RP(2)) (resp. B(2)(RP(2))) is isomorphic to the cyclic group Z(2) of order 2 (resp. the generalised quaternion group of order 16) and hence their lower central and derived series are known. If n > 2, we first prove that the lower central series of B(n)(RP(2)) is constant from the commutator subgroup onwards. We observe that Gamma(2)(B(3)(RP(2))) is isomorphic to (F(3) X Q(8)) X Z(3), where F(k) denotes the free group of rank k, and Q(8) denotes the quaternion group of order 8, and that Gamma(2)(B(4)(RP(2))) is an extension of an index 2 subgroup K of P(4)(RP(2)) by Z(2) circle plus Z(2). As for the derived series of B(n)(RP(2)), we show that for all n >= 5, it is constant from the derived subgroup onwards. The group B(n)(RP(2)) being finite and soluble for n <= 2, the critical cases are n = 3, 4. We are able to determine completely the derived series of B(3)(RP(2)). The subgroups (B(3)(RP(2)))((1)), (B(3)(RP(2)))((2)) and (B(3)(RP(2)))((3)) are isomorphic respectively to (F(3) x Q(8)) x Z(3), F(3) X Q(8) and F(9) X Z(2), and we compute the derived series quotients of these groups. From (B(3)(RP(2)))((4)) onwards, the derived series of B(3)(RP(2)), as well as its successive derived series quotients, coincide with those of F(9). We analyse the derived series of B(4)(RP(2)) and its quotients up to (B(4)(RP(2)))((4)), and we show that (B(4)(RP(2)))((4)) is a semi-direct product of F(129) by F(17). Finally, we give a presentation of Gamma(2)(B(n)(RP(2))). (C) 2011 Elsevier Inc. All rights reserved.

Twisted conjugacy classes in symplectic groups, mapping class groups and braid groups (with an appendix written jointly with Francois Dahmani)

We prove that the symplectic group Sp(2n, Z) and the mapping class group Mod(S) of a compact surface S satisfy the R(infinity) property. We also show that B(n)(S), the full braid group on n-strings of a surface S, satisfies the R(infinity) property in the cases where S is either the compact disk D, or the sphere S(2). This means that for any automorphism phi of G, where G is one of the above groups, the number of twisted phi-conjugacy classes is infinite.

Braid groups of non-orientable surfaces and the Fadell-Neuwirth short exact sequence

Let M be a compact, connected non-orientable surface without boundary and of genus g >= 3. We investigate the pure braid groups P,(M) of M, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence 1 -> P(m)(M \ {x(1), ..., x(n)}) hooked right arrow P(n+m)(M) (P*) under right arrow P(n)(M) -> 1, where m, n >= 1, and p* is the homomorphism which corresponds geometrically to forgetting the last m strings. This problem is equivalent to that of the existence of a section for the associated fibration p: F(n+m)(M) -> F(n)(M) of configuration spaces, defined by p((x(1), ..., x(n), x(n+1), ..., x(n+m))) = (x(1), ..., x(n)). We show that p and p* admit a section if and only if n = 1. Together with previous results, this completes the resolution of the splitting problem for surface pure braid groups. (C) 2009 Elsevier B.V. All rights reserved.

Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane

We classify the ( finite and infinite) virtually cyclic subgroups of the pure braid groups P(n)(RP(2)) of the projective plane. The maximal finite subgroups of P(n)(RP(2)) are isomorphic to the quaternion group of order 8 if n = 3, and to Z(4) if n >= 4. Further, for all n >= 3, the following groups are, up to isomorphism, the infinite virtually cyclic subgroups of P(n)(RP(2)): Z, Z(2) x Z and the amalgamated product Z(4)*(Z2)Z(4).

Coincidence Wecken homotopies versus Wecken homotopies relative to a fixed homotopy in one of the maps

We study the 1-parameter Wecken problem versus the restricted Wecken problem, for coincidence free pairs of maps between surfaces. For this we use properties of the function space between two surfaces and of the pure braid group on two strings of a surface. When the target surface is either the 2-sphere or the torus it is known that the two problems are the same. We classify most pairs of homotopy classes of maps according to the answer of the two problems are either the same or different when the target is either projective space or the Klein bottle. Some partial results are given for surfaces of negative Euler characteristic. (C) 2010 Elsevier B.V. All rights reserved.

Permeability of different groups of maxillary teeth after 38% hydrogen peroxide internal bleaching

This study evaluated the influence of internal tooth bleaching with 38% hydrogen peroxide (H2O2) on the permeability of the coronal dentin in maxillary anterior teeth and premolars. Seventy teeth (14 per group) were used: central incisors (CI), lateral incisor (LI), canines (C), first premolars (1PM) and second premolars (2PM). Pulp chamber access and transversal sectioning at 2 mm from the cementoenamel junction were performed and the specimens were divided into 2 groups (n= 7): a) no treatment and b) bleaching with 38% H2O2. The bleaching agent was applied to the buccal surface and to the pulp chamber for 10 min. This procedure was repeated 3 times. The specimens were processed histochemically with copper sulfate and rubeanic acid, sectioned longitudinally, and digitalized in a scanner. The area of stained dentin was measured using Image Tool software. Data were analyzed statistically by ANOVA and Tukey's HSD test (?=0.05). There was statistically significant difference (p<0.001) among the untreated groups, CI (0.23 ± 0.26) having the lowest permeability and LI (10.14 ± 1.89) the highest permeability. Among the bleached groups, dentin permeability was increased in all groups of teeth except for 2PM. It may be concluded that bleaching with 38% H2O2 affected dentin permeability near the pulp chamber in maxillary anterior teeth and in first and second premolars.

Dentin permeability of the apical third in different groups of teeth

This ex vivo study evaluated dentin permeability of the root canal in the apical third of different human groups of teeth. Eighty teeth were used, 8 from each dental group: maxillary and mandibular central incisors, lateral incisors and canines, maxillary first premolars (buccal and palatal roots), mandibular first premolars, and maxillary and mandibular second premolars, totalizing 88 roots that were distributed in 11 groups. The root canals were instrumented, irrigated with 1% NaOCl and 15% EDTA. Roots were immersed in 10% copper sulfate for 30 min and then in 1% rubeanic acid alcohol solution for the same period; this chemical reaction reveals dentin permeability by the formation of copper rubeanate, which is a dark-colored compound. Semi-serial 100-µm-thick cross-sections were obtained from the apical third of the roots. Five sections of each apical third were washed, dehydrated, cleared and mounted on glass slides for examination under optical microscopy. The percentage of copper ion infiltration and the amount of tubular dentin were quantified by morphometric analysis. The penetration of copper ions in the apical third ranged from 4.60 to 16.66%. The mandibular central and lateral incisors presented the highest dentin permeability (16.66%), while the maxillary canines and mandibular second and first premolars presented the lowest dentin permeability (4.60%, 4.80% and 5.71%, respectively; p<0.001). The other teeth presented intermediate permeability. In conclusion, dye penetration into dentin tubules at the apical region is strongly dependent on the group of teeth evaluated.

Population dynamics: seasonal variation of phytoplankton functional groups in brazilian reservoirs (Billings and Guarapiranga, São Paulo)

Phytoplankton may function as a "sensor" of changes in aquatic environment and responds rapidly to such changes. In freshwaters, coexistence of species that have similar ecological requirements and show the same environmental requirements frequently occurs; such species groups are named functional groups. The use of phytoplankton functional groups to evaluate these changes has proven to be very useful and effective. Thus, the aim of this study was to evaluate the occurrence of functional groups of phytoplankton in two reservoirs (Billings and Guarapiranga) that supply water to millions of people in São Paulo city Metropolitan Area, southeastern Brazil. Surface water samples were collected monthly and physical, chemical and biological (quantitative and qualitative analyses of the phytoplankton) were performed. The highest biovolume (mm³.L-1) of the descriptor species and functional groups were represented respectively by Anabaena circinalis Rabenh. (H1), Microcystis aeruginosa (Kützing) Kützing (L M/M) and Mougeotia sp. (T) in the Guarapiranga reservoir and Cylindrospermopsis raciborskii (Wolosz.) Seen. and Subba Raju (S N), Microcystis aeruginosa and M. panniformis Komárek et al. (L M/M), Planktothrix agardhii (Gom.) Anagn. and Komárek and P. cf. clathrata (Skuja) Anagn. and Komárek (S1) in the Billings reservoir. The environmental factors that most influenced the phytoplankton dynamics were water temperature, euphotic zone, turbidity, conductivity, pH, dissolved oxygen, nitrate and total phosphorous.

Catalytic enantioselective arylations: boron to zinc exchange as a powerful tool for the generation of transferable aryl groups

The transmetalation between boron and zinc is of great importance for application in organic synthesis, since it allows the formation of new carbon-carbon bonds between organometallic units and electrophiles. The direct arylation of aldehydes or more scarcely ketones, in a catalytic, enantioselective manner using chiral catalysts has been described recently. The enantiomerically enriched diarylmethanols obtained in these reactions are valuable precursors for important bioactive molecules. This review provides a synopsis of this ever-growing field and highlights some of the challenges that still remain.

Frequency of LCT-13910C > T single nucleotide polymorphism associated with adult-type hypolactasia/lactase persistence among Brazilians of different ethnic groups

Background: Adult-type hypolactasia, the physiological decline of lactase some time after weaning, was previously associated with the LCT -13910C>T polymorphism worldwide except in Africa. Lactase non-persistence is the most common phenotype in humans, except in northwestern Europe with its long history of pastoralism and milking. We had previously shown association of LCT -13910C>T polymorphism with adult-type hypolactasia in Brazilians; thus, we assessed its frequency among different Brazilian ethnic groups. Methods: We investigated the ethnicity-related frequency of this polymorphism in 567 Brazilians [mean age, 42.1 +/- 16.8 years; 157 (27.7%) men]; 399 (70.4%) White, 50 (8.8%) Black, 65 (11.5%) Brown, and 53 (9.3%) Japanese-Brazilian. DNA was extracted from leukocytes; LCT -13910C>T polymorphism was analyzed by PCR-restriction fragment length polymorphism. Results: Prevalence of the CC genotype associated with hypolactasia was similar (57%) among White and Brown groups; however, prevalence was higher among Blacks (80%) and those of Japanese descent (100%). Only 2 (4%) Blacks had TT genotype, and 8 (16%) had the CT genotype. Assuming an association between CC genotype and hypolactasia, and CT and TT genotypes with lactase persistence, 356 (62.8%) individuals had hypolactasia and 211 (37.2%) had lactase persistence. The White and Brown groups had the same hypolactasia prevalence (similar to 57%); nevertheless, was 80% among Black individuals and 100% among Japanese-Brazilians (P < 0.01). Conclusion: The lactase persistence allele, LCT -13910T, was found in about 43% of both White and Brown and 20% of the Black Brazilians, but was absent among all Japanese Brazilians studied.

A simplified minimal residual disease polymerase chain reaction method at early treatment points can stratify children with acute lymphoblastic leukemia into good and poor outcome groups

Background Minimal residual disease is an important independent prognostic factor in childhood acute lymphoblastic leukemia. The classical detection methods such as multiparameter flow cytometry and real-time quantitative polymerase chain reaction analysis are expensive, time-consuming and complex, and require considerable technical expertise. Design and Methods We analyzed 229 consecutive children with acute lymphoblastic leukemia treated according to the GBTLI-99 protocol at three different Brazilian centers. Minimal residual disease was analyzed in bone marrow samples at diagnosis and on days 14 and 28 by conventional homo/heteroduplex polymerase chain reaction using a simplified approach with consensus primers for IG and TCR gene rearrangements. Results At least one marker was detected by polymerase chain reaction in 96.4%, of the patients. By combining the minimal residual disease results obtained on days 14 and 28, three different prognostic groups were identified: minimal residual disease negative on days 14 and 28, positive on day 14/negative on day 28, and positive on both. Five-year event-free survival rates were 85%, 75.6%,, and 27.8%, respectively (p<0.0001). The same pattern of stratification held true for the group of intensively treated children. When analyzed in other subgroups of patients such as those at standard and high risk at diagnosis, those with positive B-derived CD10, patients positive for the TEL/AML1 transcript, and patients in morphological remission on a day 28 marrow, the event-free survival rate was found to be significantly lower in patients with positive minimal residual disease on day 28. Multivariate analysis demonstrated that the detection of minimal residual disease on day 28 is the most significant prognostic factor. Conclusions This simplified strategy for detection of minimal residual disease was feasible, reproducible, cheaper and simpler when compared with other methods, and allowed powerful discrimination between children with acute lymphoblastic leukemia with a good and poor outcome.

Fossil groups in the Millennium Simulation Evolution of the brightest galaxies

Aims. We create a catalogue of simulated fossil groups and study their properties, in particular the merging histories of their first-ranked galaxies. We compare the simulated fossil group properties with those of both simulated non-fossil and observed fossil groups. Methods. Using simulations and a mock galaxy catalogue, we searched for massive (>5 x 10(13) h(-1) M-circle dot) fossil groups in the Millennium Simulation Galaxy Catalogue. In addition, we attempted to identify observed fossil groups in the Sloan Digital Sky Survey Data Release 6 using identical selection criteria. Results. Our predictions on the basis of the simulation data are: (a) fossil groups comprise about 5.5% of the total population of groups/clusters with masses larger than 5 x 10(13) h(-1) M-circle dot. This fraction is consistent with the fraction of fossil groups identified in the SDSS, after all observational biases have been taken into account; (b) about 88% of the dominant central objects in fossil groups are elliptical galaxies that have a median R-band absolute magnitude of similar to-23.5-5 log h, which is typical of the observed fossil groups known in the literature; (c) first-ranked galaxies of systems with M > 5 x 10(13) h(-1) M-circle dot, regardless of whether they are either fossil or non-fossil, are mainly formed by gas-poor mergers; (d) although fossil groups, in general, assembled most of their virial masses at higher redshifts in comparison with non-fossil groups, first-ranked galaxies in fossil groups merged later, i.e. at lower redshifts, compared with their non-fossil-group counterparts. Conclusions. We therefore expect to observe a number of luminous galaxies in the centres of fossil groups that show signs of a recent major merger.