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.

Surface Properties of Calcinated Titanium Dioxide Probed by Solvatochromic Indicators: Relevance to Catalytic Applications

Titanium dioxide was obtained by hydrolysis of the corresponding ethoxide, followed by washing, drying, and calcination at 80, 160, 240, 320, 400, and 700 C, respectively. The following surface properties of the solids obtained were determined as a function of the calcinations temperature: T(Calcn); area by the BET method; BrOnsted acidity by titration with sodium hydroxide; empirical polarity, ET(30); Lewis acidity, alpha(Surf); Lewis basicity, beta(Surf); and dipolarity/polarizability pi*(Sturf), by use of solvatochromic indicators. Except for le surf whose value increased slightly, heating the samples resulted in a decrease of all of the above-mentioned surface properties, due to the decrease of surface hydroxyl groups. This conclusion has been corroborated by FTIR. Values of E(T)(30), alpha(Surf), and pi*(Surf) are higher than those of water and alcohols; the BrOnsted and Lewis acidities of the samples correlate linearly. The advantages of using solvatochromic indicators to probe the surface properties and relevance of the results to the applications of TiO(2) are discussed.

Magnetic coupled electrochemistry: Exploring the use of superparamagnetic nanoparticles for capturing, transporting and concentrating trace amounts of analytes

We propose the use of functionalized superparamagnetic nanoparticles for capturing, and transporting analytes, in association with an external miniature magnet to deposit such nanocarrier species at the electrode surface. This approach can be employed for the electroanalytical determination of chemical species capable of interacting with the nanoparticles, or in the opposite case, to block their response at the electrode surface. The concept was successfully demonstrated by using aminofunctionalized nanoparticles to block the discharge of hexacyanoferrate(II) ions, and to enhance the signals of aquapentacyanoferrate(II) ions via coordination to the surface amino groups. Selective analysis was also performed for silver ions, surpassing the stripping methods in terms of versatility and usefulness. (C) 2010 Elsevier B.V. All rights reserved.

Effect of biofunctionalized implant surface on osseointegration: a histomorphometric study in dogs

Among the different properties that influence bone apposition around implants, the chemical or biochemical composition of implant surface may interfere on its acceptance by the surrounding bone. The aim of this study was to investigate if a biofunctionalization of implant surface influences the bone apposition in a dog model and to compare it with other surfaces, such as a microstructured created by the grit-blasting/acid-etching process. Eight young adult male mongrel dogs had the bilateral mandibular premolars extracted and each one received 6 implants after 12 weeks, totaling 48 implants in the experiment. Four groups of implants were formed with the same microrough topography with or without some kind of biofunctionalization treatment. After histomorphometric analysis, it was observed that the modified microstructured surface with a "low concentration of the bioactive peptide" provided a higher adjacent bone density (54.6%) when compared to the other groups (microstructured + HA coating = 46.0%, microstructured only = 45.3% and microstructured + "high concentration of the bioactive peptide" = 40.7%), but this difference was not statistically significant. In conclusion, biofunctionalization of the implant surface might interfere in the bone apposition around implants, especially in terms of bone density. Different concentrations of bioactive peptide lead to different results.

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.

Effect of fluoride-containing solutions on the surface of cast commercially pure titanium

This study evaluated the effects of fluoride-containing solutions on the surface of commercially pure titanium (CP Ti) obtained by casting. CP Ti specimens were fabricated and randomly assigned to 5 groups (n=10): group 1: stored in distilled water at 37 ± 1ºC; group 2: stored in distilled water at 37 ± 1ºC and daily immersed in 0.05% NaF for 3 min; group 3: stored in distilled water at 37 ± 1ºC and daily immersed in 0.2% NaF for 3 min; group 4: stored in distilled water at 37 ± 1ºC; and immersed in 0.05% NaF every 15 days for 3 min; and group 5: stored in distilled water at 37 ± 1ºC and immersed in 0.2% NaF every 15 days for 3 min. Surface roughness was measured with a profilometer immediately after metallographic polishing of the specimens (T0) and at 15-day intervals until completing 60 days of experiment (T15, T30, T45, T60). Data were analyzed statistically by ANOVA and Tukey's test (α=0.05). There was no statistically significant difference (p>0.05) in surface roughness among the solutions. In conclusion, fluoride-containing solutions (pH 7.0) used as mouthwashes do not damage the surface of cast CP Ti and can be used by patients with titanium-based restorations.

Surface and subsurface erosion of primary enamel by acid beverages over time

This study evaluated the influence of a cola-type soft drink and a soy-based orange juice on the surface and subsurface erosion of primary enamel, as a function of the exposure time. Seventy-five primary incisors were divided for microhardness test (n=45) or scanning electron microscopy (SEM) analysis (n=30). The specimens were randomly assigned to 3 groups: 1 - artificial saliva (control); 2 - cola-type soft drink; and 3 - soy-based orange juice. Immersion cycles in the beverages were undertaken under agitation for 5 min, 3 times a day, during 60 days. Surface microhardness was measured at 7, 15, 30, 45 and 60 days. After 60 days, specimens were bisected and subsurface microhardness was measured at 30, 60, 90, 120, 150 and 200 µm from the surface exposed. Data were analyzed by ANOVA and Tukey’s test (a=0.05). Groups 2 and 3 presented similar decrease of surface microhardness. Regarding subsurface microhardness, group 2 presented the lowest values. SEM images revealed that after 60 days the surfaces clearly exhibited structural loss, unlike those immersed in artificial saliva. It may be concluded that erosion of the surfaces exposed to the cola-type soft drink was more accentuated and directly proportional to the exposure time.

Microleakage in conservative cavities varying the preparation method and surface treatment

OBJECTIVE: To assess microleakage in conservative class V cavities prepared with aluminum-oxide air abrasion or turbine and restored with self-etching or etch-and-rinse adhesive systems. Materials and Methods: Forty premolars were randomly assigned to 4 groups (I and II: air abrasion; III and IV: turbine) and class V cavities were prepared on the buccal surfaces. Conditioning approaches were: groups I/III - 37% phosphoric acid; groups II/IV - self-priming etchant (Tyrian-SPE). Cavities were restored with One Step Plus/Filtek Z250. After finishing, specimens were thermocycled, immersed in 50% silver nitrate, and serially sectioned. Microleakage at the occlusal and cervical interfaces was measured in mm and calculated by a software. Data were subjected to ANOVA and Tukey's test (α=0.05). RESULTS: Marginal seal provided by air abrasion was similar to high-speed handpiece, except for group I. There was SIGNIFICANT difference between enamel and dentin/cementum margins for to group I and II: air abrasion. The etch-and-rinse adhesive system promoted a better marginal seal. At enamel and dentin/cementum margins, the highest microleakage values were found in cavities treated with the self-etching adhesive system. At dentin/cementum margins, high-speed handpiece preparations associated with etch-and-rinse system provided the least dye penetration. CONCLUSION: Marginal seal of cavities prepared with aluminum-oxide air abrasion was different from that of conventionally prepared cavities, and the etch-and-rinse system promoted higher marginal seal at both enamel and dentin margins.