Molecular machines operated by light have been recently

shown to be able to produce oriented motion at the molecular scale 1,2 as well as do macroscopic work when embedded in supramolecular structures [START_REF] Browne | Making molecular machines work[END_REF][START_REF] Chen | Artificial muscle-like function from hierarchical supramolecular assembly of photoresponsive molecular motors[END_REF][START_REF] Van Leeuwen | Dynamic control of function by light-driven molecular motors[END_REF] . However, any supramolecular movement irremediably ceases as soon as the concentration of the interconverting molecular motors or switches reaches a photo-stationary state [START_REF] Eelkema | Molecular machines: Nanomotor rotates microscale objects[END_REF][START_REF] Li | Macroscopic contraction of a gel induced by the integrated motion of light-driven molecular motors[END_REF] . To circumvent this limitation, researchers have typically relied on establishing oscillating illumination conditions-either by modulating the source intensity [START_REF] Foy | Dual-light control of nanomachines that integrate motor and modulator subunits[END_REF][START_REF] Yamada | Photomobile polymer materials: Towards light-driven plastic motors[END_REF] or by using bespoke illumination arrangements [START_REF] Gelebart | Making waves in a photoactive polymer film[END_REF][START_REF] White | A high frequency photodriven polymer oscillator[END_REF][START_REF] Tabe | Photo-induced travelling waves in condensed Langmuir monolayers[END_REF][START_REF] Martinez | Light-driven dynamic Archimedes spirals and periodic oscillatory patterns of topological sol tons in anisotropic soft matter[END_REF] . In contrast, here we report a supramolecular system in which the emergence of oscillating patterns is encoded at the molecular level. Our system comprises chiral liquid crystal structures that revolve continuously when illuminated, under the action of embedded light-driven molecular motors. The rotation at the supramolecular level is sustained by the diffusion of the motors away from a localized illumination area. Above a critical irradiation power, we observe a spontaneous symmetry breaking that dictates the directionality of the supramolecular rotation. The interplay between the twist of the supramolecular structure and the diffusion [START_REF] Epstein | Reaction-diffusion processes at the nano and microscales[END_REF] of the chiral molecular motors creates continuous, regular and unidirectional rotation of the liquid crystal structure under non-equilibrium conditions.

We use overcrowded alkenes Ph-m or Me-m as molecular motors (Fig. 1a). Upon illumination, m transforms into its unstable isomer m*, while the reverse transformation m* → m occurs at room temperature spontaneously. Once these molecular motors are incorporated in an achiral liquid crystal, amplification of chirality yields a photo-responsive cholesteric liquid crystal. This helix-based liquid crystal is further confined between two glass slides, each of them promoting perpendicular orientation of the liquid crystal molecules. When the gap L of the cell is typically smaller than the helical pitch p of the cholesteric liquid crystal (the distance over which the average molecular orientation of the liquid crystal rotates by 2π , and whose sign defines the handedness of the cholesteric helix), the helix is supressed [START_REF] Zeldovich | Fréedericksz transition in cholesteric liquid crystal without external fields[END_REF] . This geometrical frustration can be released when energy kicks in locally, by the use of either electric fields [START_REF] Haas | Electrically variable diffraction in spherulitic liquid crystals[END_REF] or laser beams [START_REF] Smalyukh | Three-dimensional structure and multistable optical switching of triple-twisted particle-like excitations in anisotropic fluids[END_REF] and yields a variety of complex twisted topological structures [START_REF] Ackerman | Two-dimensional skyrmions and other solitonic structures in confinement-frustrated chiral nematics[END_REF] that are all characterized by a constant cholesteric pitch.

Here, the geometrical frustration is released by winding the helix mechanically (by the twisting action of the molecular motor) in a system where m is combined with a shape-persistent chiral co-dopant (Supplementary Fig. 1). Co-doping is necessary because most photo-responsive chiral dopants induce helical unwinding under illumination, and here releasing geometrical frustration requires photo-induced winding. By adjusting the relative proportions of motor and shape-persistent co-dopant, as well as their concentrations, we have prepared cholesteric liquid crystals that display opposite helical handedness at full photo-conversion-for example, provided that the illumination is sufficiently strong (Supplementary Table 1). The initial helical pitch p 0 first increases under illumination, and while the motor continues to photo-convert, the cholesteric helix reverses its handedness and starts winding continuously in the opposite direction (Fig. 1a), until it reaches its saturated photostationary value p ∞ . We use steady irradiation conditions provided by a focused laser beam at wavelength λ = 375 nm (Fig. 1b), and observe the system under crossed linear polarizers (Supplementary Fig. 2). In practice, the emergence of any three-dimensional liquid crystal structure is thus assessed by a two-dimensional pattern. In our earlier work, we have shown that the mechanical winding provided by the motors can generate several types of twisted structures, above a minimal power P min

19

. Here we report on out-of-equilibrium dynamics that occur via a structural transition, when the illumination power is used as a control parameter.

At power values ~. P 1 25 min we can stabilize a twisted liquid crystal structure associated to an axially symmetric and chiral pattern, and that can be either right-handed or left-handed (Fig. 1c,d). Increasing the irradiation power further and up to ~P 2 min , we unveil a breaking in axial symmetry that defines a structural transition: the axial symmetry of the chiral pattern is broken and yields another, much larger chiral pattern that exhibits a kink (Fig. 1e,f), and the handedness of which is pre-determined by the handedness of the axisymmetric pattern (Fig. 1c,d). Strikingly, the emergence of this new pattern is accompanied by its continuous, regular, and unidirectional rotation, in a direction that is pre-determined by its handedness (Supplementary Videos 1,2 and Fig. 2a-c). The fact that the diameter of the beam is three to five times smaller than the diameter of the pattern is a clear manifestation that m* diffuses away from the illuminated zone. When the light is turned off, the pattern fades away.

Constant and stable rotation is sustained under steady illumination conditions-up to forty hours of continuous rotation have been observed for both directions of rotation, without any signs of fatigue. Plotting the correlation coefficient between an image observed at any time t and a reference image, quantifies the similarity between two images over time (Fig. 2d). The corresponding Fourier spectrum provides a quantitative analysis for the rotation rate (Fig. 2e), and highlights that the revolving elastic excitation is

Revolving supramolecular chiral structures powered by light in nanomotor-doped liquid crystals

Tetiana Orlova 1 , Federico Lancia 2 , Charles Loussert 1 , Supitchaya Iamsaard 2 , Nathalie Katsonis 2 * and Etienne Brasselet 1 * robust in its operation. Finally, the emergence of rotation is only marginally influenced by ageing, as demonstrated by two rotating structures observed more than three years apart (Supplementary Videos 1,3). Moreover, this rotation does not depend on the polarization of the laser light, and has been observed with both linearly and circularly polarized light.

We have investigated how the rotation rate depends on the use of a given enantiomer of either Ph-m or Me-m. We conclude that there is a general character to the rotation, as seen in Fig. 3a, where the period of rotation T 0 correlates with the diameter d of any revolving pattern (experimentally, a range of different diameters can be accessed by using different irradiation powers). Specifically, the results evidence an exponential dependence of the rotation period with the diameter of the chiral pattern that is valid for both motors and independently of the direction of rotation. Remarkably, this trend is not compatible with a quadratic law ~∕

T d D 0 2
that can be inferred from a dimensional analysis neglecting the ordered nature of the liquid crystal. This exponential dependence thus suggests that, in addition to diffusion, another parameter plays a key role in establishing the rotation, and this is the long-range orientational order that is characteristic for liquid crystals. Further, the rotation rate is influenced by the bulkiness of the substituents at the stereogenic centre; specifically, it has been shown that an increase of steric hindrance speeds up the rotary motion of the molecular motor 20 . As Me-m* relaxes faster than Ph-m*, the former motor displays a tendency to generate smaller chiral patterns (squares) than the latter (circles), and smaller diameters are associated with faster pattern rotation. Both enantiomers can lead to clockwise and counter-clockwise rotation, which means that the direction of rotation cannot be predicted by choosing one enantiomer over the other, but is instead pre-programmed at the supramolecular level, as illustrated in Fig. 1c-f. Two enantiomers induce the same rotation rates.

The diameter of the chiral pattern d depends on the incident reduced power P (Fig. 3b that are observed experimentally correspond to diffusion coefficients of the molecular motors D 0 (Ph-m) ~ 5.25 ± 3.75 10 -11 m 2 s -1 and D 0 (Me-m) ~ 2.35 ± 1.55 10 -11 m 2 s -1 . This range of values is in good agreement with literature values [START_REF] Pumpa | Slow single-molecule diffusion in liquid crystals[END_REF] and thus confirms that diffusion is key in the emergence of these structures.

We also demonstrate that the occurrence of helix inversion plays a key role in preserving the integrity of the light-fuelled rotor. For a cholesteric liquid crystal with a composition that does not allow helix inversion, the axisymmetric structure is destabilized into the outward growth of fingerprints, and the transition to a rotational regime does not occur (Fig. 4a,b and Supplementary Videos 4,5). Moreover, the diameter of the structure in the case of helix inversion is larger than without, which further indicates that helix inversion promotes radial confinement of the liquid crystal reorientation, as supported by our model (Fig. 4c). Indeed, the latter predicts a much steeper pitch increase along the radial coordinate in helix-inverting cholesteric helices, a feature which is associated with a larger elastic energy barrier, and thus makes it difficult for the chiral pattern to grow. This underpins the role of elastic chiral gradients in the emergence of rotation.

The work produced by the revolving structure can be harnessed into the rotation of a particle-like liquid crystal structure, which behaves like an independent object, of diameter ~30 µ m (Fig. 5 and Supplementar y Video 6). Its trapping results from the complex elastic landscape that emerges as a response to the inhomogeneous liquid crystal orientation. In contrast to what was observed in other systems [START_REF] Eelkema | Molecular machines: Nanomotor rotates microscale objects[END_REF] , the rotation speed of the cargo particle equals that of the revolving chiral pattern, the latter being only marginally influenced by the presence of the cargo. and from a purely kinetic model ('without diffusion'), both neglecting the coupling with the twisted three-dimensional liquid crystal structure (see Methods). The diffusion constant is approximated to be the same for all the motors, and is denoted as D 0 . Sample characteristics and individual analysis of each sample are provided in Supplementary Figs. 3 and4. ; Gaussian beam power, 20 nW (with helix inversion, i.e. p ∞ = -1.6 μ m) and 5 nW (without helix inversion, i.e. p ∞ = + 1.6 μ m); Gaussian beam waist, w= 5 μ m.

Our experimental findings support a physico-chemical mechanistic understanding of the rotation. Localized illumination of nanomotor-doped liquid crystal in confinement generates toroidlike structures. As long as the axisymmetry of the supramolecular structure is preserved (Fig. 1c,d), the diffusion of chiral motors along gradients of concentrations remains invariant per rotation, and thus it never yields any rotation even when m and m* have different diffusion rates. In contrast, once axial symmetry is broken (Fig. 1e,f), the twist of the medium, and thus the propagation rate of the motors, become angle-dependent. This angular dependence impacts the twist gradients, as a result of the long-range orientational order characterizing liquid crystals, in which a local change of twist also affects neighbouring molecules. The new angular dependence also redefines the angular redistribution of the diffusing motors. This leads to the orthoradial propagation of the non-local excitation, which we detect experimentally as a rotating spiral-shaped pattern. In addition, it is the chiral nature of the non-axisymmetric supramolecular structure that provides the directional character to the liquid crystal reorganization.

Given a physico-chemical perturbation on the spatial scale ℓ, we note that the characteristic elastic relaxation time τ

γ = ∕ ℓ K ( ) elast 2
(where γ is the rotational viscosity and K is the typical elastic constant of the liquid crystal, with γ∕K equal to a few 10 10 sm -2 ) and the characteristic diffusion time τ = ∕ ℓ D (1 ) diff 0 2 are of the same order of magnitude, which suggests that a feedback mechanism is established, without time delay.

Overall, we propose that the emergence of rotation is based on a non-local feedback loop that involves the local concentrations of m and m*, and their spatial gradients, as well as twist gradients that occur across the liquid crystal structure. Indeed, the diffusion of a chiral species is influenced by the chirality of the liquid crystal in which they diffuse, and vice versa [START_REF] Galstian | Molecular self-assemblies might discriminate the diffusion of chiral molecules[END_REF][START_REF] Jiang | Chirality differentiation by diffusion in chiral nematic liquid crystals[END_REF] . Therefore, while the spatial distribution of the molecular motors defines the distribution of the pitch, in return, the elastic relaxation and the differential diffusion rates of m and m* define a new chiral landscape (this interplay is illustrated in Supplementary Scheme 1).

Chirality is thus key to the emergence of rotation, and appears to be a universal design principle [START_REF] Morrow | Transmission of chirality through space and across length scales[END_REF] , as observed in the directional reorientation of passive liquid crystalline media employing unidirectional transfer of optical angular momentum carried by chiral light to achiral liquid crystals [START_REF] Santamato | Collective rotation of molecules driven by the angular momentum of light in a nematic film[END_REF][START_REF] Murazawa | Control of the molecular alignment inside liquid-crystal droplets by use of laser tweezers[END_REF] or unidirectional Marangoni flows driven by absorption of light in chiral liquid crystals [START_REF] Choi | Circular flow formation triggered by Marangoni convection in nematic liquid crystal films w th a free surface[END_REF] . However, the latter systems lack any active molecular component, and therefore they also involve light intensity levels that are at least five orders of magnitude higher than the ~50 mW cm -2 here, which is compatible with sunlight illumination. Another classic example of macroscopic dynamics in chiral liquid crystals is the so-called Lehmann rotation, where chiral liquid crystals are driven by a gradient of temperature along the helical axis [START_REF] Lehmann | Structur, system und magnetisches verhalten flüssiger krystalle und deren mischbarkeit mit festen[END_REF] , and which remains an intriguing phenomenon to date [START_REF] Ignes-Mullol | Continuous rotation of achiral nematic liquid crystal droplets driven by heat flux[END_REF] . The same mechanism yields molecular precession of a chiral liquid-crystalline monolayer spread on a glycerol surface; however, this precession remains short-lived [START_REF] Tabe | Coherent collective precession of molecular rotors with chiral propellers[END_REF] .

By engineering reaction-diffusion processes in a medium characterized by long-range orientational order and taking advantage of large chiral gradients, we demonstrate the emergence of revolving structures, in a phenomenon that appears as a supramolecular and chiral counterpart of the Belousov-Zhabotinsky reaction. These chiral supramolecular structures maintain non-equilibrium conditions at length scales which are typically 10,000 times larger than their individual components. These results contribute to developing autonomous molecular systems that are capable of transforming light into work continuously.

Methods

Methods, including statements of data availability and any associated accession codes and references, are available p.6

Methods

Synthesis of the motors and shape-persistent dopants. The detailed synthesis of Ph-m and Me-m, as well as their purification, are described in the Supplementary Information (for synthetic schemes, see Supplementary Schemes 2 and 3, for NMR characterization, see Supplementary Figures 6789). The shape-persistent dopant 2,2′ -(1,3-propylenedioxy)-1,1′ -binaphthalene was synthesized by using known procedures [START_REF] Van Es | Synthesis and characterization of optically active cyclic 6,6′ -dinitro-1,1′ -binaphthyl-2,2′ -diethers[END_REF] .

Characterization of the motors and shape-persistent dopants. Helical twisting powers (HTP) were measured in the nematic liquid crystal E7 (Merck), by using the Grandjean-Cano wedge method. For each motor enantiomer this yielded the following values: (S,P)-Ph-m HTP wt% = + 92 μ m -1 , (R,M)-Ph-m HTP wt% = -92 μ m -1 , (R,P)-Me-m HTP wt% = + 46.3 μ m -1 , (S,M)-Me-m HTP wt% = -65.1 μ m -1 . The HTPs of the shape-persistent dopants were (R)-BB HTP wt% = -60.2 μ m -1 and (S)-BB HTP wt% = + 60.3 μ m -1 . These values were used to prepare cholesteric liquid crystals with adequate photo-responsive behaviour.

Preparation of the photo-responsive cholesteric liquid crystals. One given enantiomer of molecular motor m and the BB enantiomer w th opposite helical twisting power were dissolved in the commercially available liquid crystal E7 (Merck Japan), in proportions that are determined by using the equation p = Σ i (HTP i c i ee i ) -1 , where c i is the concentration of the ith dopant, and ee i ts enantiomeric purity (this equation is valid only at low concentrations). By using this dependence of the helical pitch to the concentration of dopants, it is possible to pre-program the handedness of the cholesteric helix under irradiation.

For each photo-responsive cholesteric liquid crystal, the value of the pitch was measured before illumination and at the photo-stationary state under uniform illumination with sufficiently high intensity by means of the Cano-Grandjean wedge method, with a wedge angle α defined by α = . tan( ) 0 0269. The handedness of the cholesteric helix was determined using a known procedure [START_REF] Gerber | On the determination of the cholesteric screw sense by the Grandjean-Cano-method[END_REF] .

Further, the liquid crystal was placed into a glass cell of thickness L = 10 µ m that consists of two parallel substrates providing perpendicular orientation for the liquid crystal molecules (E.H.C. Tokyo). The relaxation time of the cholesteric helix from p ∞ to p 0 was determined by monitoring the sample between crossed linear polarizers.

Simplified reaction-diffusion model before spontaneous symmetry breaking.

A simplified analytical model is presented for the light-induced formation of the precursor of the revolving chiral patterns. The model is based on the assumption of a simple reaction scheme: a molecular motor m with an absorption coefficient ε absorbs a photon of light with intensity I at λ = m 375 n , and photo-chemically transforms into m* with a quantum yield φ. At the same time, m* undergoes a spontaneous reverse conversion into m, characterized by a characteristic relaxation time τ. Regarding the diffusion process, we consider the framework of axisymmetric irradiation profile and axisymmetric concentration profiles for the motors m and m*, c and c * , respectively. In a first approximation, we neglect the coupling to the liquid crystalline structure and assume the same diffusion constant D 0 for all motors. Assuming no degradation of the motors, this implies that the sum of c and c * is constant and equals the initial concentration c 0 of the isomer m. Finally, neglecting light intensity depletion along the z coordinate as light passes through the liquid crystal layer, the reaction-diffusion equation for = c c r t ( , ) is expressed as

εφ τ Δ = - + - + c t I c c c D c d d (1) 0 0
After the steady state is reached, namely for τ ≫ t , the dynamical equilibrium is characterized by the inhomogeneous intens ty-dependent concentration

= τ ≫ c r c r t ( ) lim ( , ) (2)
t When introducing the dimensionless radius ρ τ = ∕ r D 0 , the steady-state concentration profile ρ c( ) satisfies the following differential equation: with the boundary conditions

ρ ρ ρ ρ + -+ ~+ = c c c c d d 1 d d (1 I ( )) 0 (3)
ρ = = ρ ρ = →∞ c c c d d 0, lim (4) 0 0
where a reduced light intensity profile has been introduced, namely

ρ ρ = ∕ ∼ I I I ( ) ( ) (5)
s εφτ = ∕ I with 1 ( ) (6)
s From non-uniform molecular motors concentration to spatially varying cholesteric pitch. The pitch p of a cholesteric liquid crystal prepared by dissolving a few percent of chiral dopants in a nematic liquid crystal satisfies, in the absence of intermolecular interactions between the chiral molecules and assuming enantiopure compounds, Extracting parameters from the experimental data using a simple reactiondiffusion model. Although the simplified model presented above cannot predict the broken axial symmetry, the two unknown parameters of the model, I s and D 0 , can be nevertheless determined from experimental observations. Noticeably, I s represents the combination of coefficients describing the photo-chemical isomerization of m to m* and its backward thermal relaxation; its value is fixed for a given composition of the cholesteric liquid crystal. Thus, first the experimental data on P min and d min observed in a given sample are used in the model to estimate the value of I s (Supplementary Fig. 4a). Second, the diameter of twisted pattern as a function of the total beam power is calculated using the obtained I s value and compared with the experimental plots (Supplementary Fig. 4b). Implementing this two-step procedure for all the samples, the range for molecular diffusion coefficient D 0 covering the experimental data is found. For the relaxation time, we use the values τ (Ph-m) = 1,000 s and τ (Me-m) = 330 s [START_REF] Aßhoff | Time-programmed helix inversion in phototunable liquid crystals[END_REF] . The resulting range for D 0 is 10 -11 -10 -10 m 2 s -1 . Changes in the spatial radial distributions of m and m * concentrations depending on the diffusion coefficient demonstrate the effect of molecular diffusion on the system (Supplementary Fig. 5).

β β β = + + - p c c c * * (7)

Data availability.

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

Fig. 1 |

 1 Fig. 1 | Motor-doped liquid crystals winding under irradiation, and associated chiral structures. a, Molecular motor used as photo-active dopant, with a substituent R on the stereogenic centre. Two derivatives were used: Ph-m (R = phenyl group), and Me-m (R = methyl group). A bridged binol BB was used as a passive co-dopant. When mixed with a nematic liquid crystal, the dopant and co-dopant generate a photo-responsive cholesteric liquid crystal by combined chirality transfer. Helix inversion is associated both with the photo-isomerization m → m*, and with its spontaneous reverse transformation m* → m. b, Motor-doped liquid crystal film with thickness L = 10 µ m confined between two glass slides promoting perpendicular orientation of the molecules and illuminated locally using a Gaussian laser beam with a beam waist w= 5 μ m at wavelength λ = 375 nm. c,d, Axisymmetric chiral patterns with opposite handedness observed by polarized optical microscopy between crossed linear polarizers. Such patterns can be sustained for a minimal illumination power of P ~1.25 min . The dashed twisted crosses are a guide for the eyes, to identify the broken chiral symmetry in the pattern. e,f, A secondary structural symmetry breaking takes place spontaneously as the power increases up to P ~2 min , and leads to a non-axisymmetric pattern that features a kink. This pattern rotates continuously, and both its handedness and the direction of rotation are predetermined by the axisymmetric pattern from which it results (c or d). The dashed spirals emphasize the handedness of the pattern and the solid white arrow shows the direction of rotation. Scale bars, 20 μ m.

Fig. 2 | 1 0 0 .

 210 Fig. 2 | Light-driven chiral structures revolving unidirectionally a,b, Snapshots of chiral patterns revolving counter-clockwise and clockwise over one period. c, The polar angle ψ of the structural kink (see white arrow) in the (x,y) plane is plotted against time. The line refers to the best linear fit of the data. d, Correlation coefficient X between any image and the reference image at = t 0. The inset evidences a period T ~2 0 min. e, Fourier spectrum of X(t) plotted around the mean rotation frequency = ∕ f T 1 0 0 . The red curve is a Gaussian fit. Scale bars, 20 μ m.

1 DFig. 3 |

 13 Fig. 3 | universal character of the revolving chiral structures. a, Rotation period T 0 as a function of the structure diameter d for molecular motors Ph-m and Me-m, and for both clockwise and counter-clockwise rotation. The diameter d is defined experimentally as the diameter of the largest bright ring of the pattern. The solid line refers to an exponential fit. Insets: polarizing microscopy snapshots of revolving chiral patterns obtained with Me-m (squares) and Phe-m (circles). Scale bars, 20 µ m. b, Dependence of the structure diameter d as a function of the power of the incident ultraviolet laser beam, where d min and P min refer to the smallest stable and static structure. The experimental data are plotted against data extracted from simulations based on a simplified reaction-diffusion model ('with diffusion')and from a purely kinetic model ('without diffusion'), both neglecting the coupling with the twisted three-dimensional liquid crystal structure (see Methods). The diffusion constant is approximated to be the same for all the motors, and is denoted as D 0 . Sample characteristics and individual analysis of each sample are provided in Supplementary Figs.3 and 4.

Fig. 4 |.

 4 Fig. 4 | helix inversion promotes spatial confinement, and thus preserves the integrity of the revolving patterns. a, Transient dynamics leading to continuous rotation in the presence of helix inversion (147 s). Scale bars, 20 µ m. b, Transient dynamics leading to fingerprint pattern formation without cholesteric helix inversion (~75 s). Scale bar, 20 µ m. c, Simulated dependence of the helix pitch on the distance to the ultraviolet beam axis with d inv and d no-inv the estimated size of the twisted structures with and without helix inversion, which is derived from the frustration condition ∕ = | | p d p ( 2) c . Simulation parameters: p 0 = 16 μ m; p ∞ = 1.6 μ m; diffusion coefficient, = × - -D 3.5 10 m s 0

Fig. 5 |

 5 Fig. 5 | Orbital transport of cargo. Off-axis rotational transport of a cargo, here a particle-like liquid crystal structure, being subjected to an orbiting trajectory under the effect of the revolving pattern. The rotation period is ~ 23 min.

Finally, we introduce the critical pitch = p k L 2 c 2 2 3

 222 β and β * are the HTPs of the m and m*, and β BB and c BB are the HTP and concentration of the shape-persistent chiral dopant. The latter expression can be simplified by introducing the steady-state pitch values at rest, ~= I 0, and in the saturated regime, ~≫ I 1, namely p 0 and p ∞ , respectively, that are given by the expressions following expression for the intensity-dependent pitch profile above which the cholesteric film is frustrated, where L is the cell thickness and = ∕ k K K 2 the ratio between twist and bend Frank elastic constants. In Fig.4c, curve). The radial dependence is obtained by taking into account the Gaussian intensity profile of the light beam

acknowledgements

This research was supported financially by the H2020-MSCA-IF-2014 programme (Grant 661315 to T.O.), the Netherlands Organization for Scientific Research (FOM Grant 13PR3105 to F.L. and N.K.), and the European Research Council (Starting Grant 307784 to N.K.).

Competing interests

The authors declare no competing financial interests.

Correspondence and requests for materials should be addressed to N.K. or E.B.


Daciberg Lima Gonçalves

John Guaschi
email: john.guaschi@unicaen.fr

Oscar Ocampo
email: oscaro@ufba.br

Almost-crystallographic groups as quotients of Artin braid groups

Let n, k ≥ 3. In this paper, we analyse the quotient group B n /Γ k (P n ) of the Artin braid group B n by the subgroup Γ k (P n ) belonging to the lower central series of the Artin pure braid group P n . We prove that it is an almost-crystallographic group. We then focus more specifically on the case k = 3. If n ≥ 5, and i f τ ∈ N is such that gcd(τ, 6) = 1, we show that B n /Γ 3 (P n ) possesses torsion τ if and only if S n does, and we prove that there is a oneto-one correspondence between the conjugacy classes of elements of order τ in B n /Γ 3 (P n ) with those of elements of order τ in the symmetric group S n . We also exhibit a presentation for the almost-crystallographic group B n /Γ 3 (P n ). Finally, we obtain some 4-dimensional almost-Bieberbach subgroups of B 3 /Γ 3 (P 3 ), we explain how to obtain almost-Bieberbach subgroups of B 4 /Γ 3 (P 4 ) and B 3 /Γ 4 (P 3 ), and we exhibit explicit elements of order 5 in B 5 /Γ 3 (P 5 ).

1

Introduction

In this paper, we continue our study of quotients of the Artin braid group B n by elements of the lower central series (Γ k (P n )) k∈N of the Artin pure braid group P n . In the paper [START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF], we analysed the group B n /Γ 2 (P n ) in some detail, and we proved notably that it is a crystallographic group. Using different techniques, I. Marin generalised the results of [START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF] to generalised braid groups associated to arbitrary complex reflection groups [Ma]. In the present paper, we show for all n, k ≥ 3, the quotient B n /Γ k (P n ) of B n by Γ k (P n ) is an almost-crystallographic group, and we investigate more thoroughly the group B n /Γ 3 (P n ). As in [START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF], some natural questions that arise are the existence or not of torsion, the realisation of elements of finite order and that of finite subgroups, their conjugacy classes and the relation with other types of group, such as (almost-) crystallographic groups.

This paper is organised as follows. In Section 2, we recall some definitions and facts about the Artin braid groups, their quotients by the elements of the lower central series (Γ k (P n )) k∈N of P n , and almost-crystallographic groups. In Section 3, we discuss the quotient B n /Γ k (P n ), where n, k ≥ 3. If G is a group, for all q ∈ N, let L q (G) denote the lower central series quotient Γ q (G)/Γ q+1 (G). These quotients have been widely studied, see [Hal, MKS] for example. In the case of P n , it is known that L q (P n ) is a free Abelian group of finite rank, and by [START_REF] Li | Brunnian braids and Lie algebras[END_REF]Theorem 4.6], its rank is given by: rank (L q

(P n )) = 1 q n-1 ∑ j=1 ∑ d|q µ(d)j q/d (1)
where µ is the Möbius function. From this, it follows that the nilpotent group P n /Γ k (P n ) of nilpotency class k -1 is also torsion free (see Lemma 11(a)). Using (1), in Proposi- tion 10, we calculate the Hirsch length of P n /Γ k (P n ), which is equal to ∑ k-1 i=1 rank(L i (P n )). In particular, the Hirsch length of P n /Γ 3 (P n ) and P n /Γ 4 (P n ) is equal to ( n 2 ) + ( n 3 ) and ( n 2 ) + ( n 3 ) + 2( n+1 4 ) respectively (see [CS, Theorem 1.1]). Using a criterion given in [De], we are then able to show that B n /Γ k (P n ) is an almost-crystallographic group. THEOREM 1. Let n, k ≥ 3. The group B n /Γ k (P n ) is an almost-crystallographic group whose holonomy group is S n and whose dimension is equal to ∑ k-1 q=1 1 q ∑ n-1 j=1 ∑ d|q µ(d)j q/d . In particular, the dimension of B n /Γ 3 (P n ) (resp. of B n /Γ 4 (P n )) is equal to ( n 2 ) + ( n 3 ) (resp. to ( n 2 ) + ( n 3 ) + 2( n+1 4 )). Torsion-free almost-crystallographic groups, or almost-Bieberbach groups, are of particular interest because they arise as fundamental groups of infra-nilmanifolds. Infranilmanifolds are manifolds that are finitely covered by a nilmanifold and represent a natural generalisation of flat manifolds. They play an important rôle in dynamical systems, notably in the study of expanding maps and Anosov diffeomorphisms [DD]. The reader may consult [De, Gr, Ru] for more information about these topics.

Another interesting problem is that of the nature of the finite-order elements of B n /Γ k (P n ). Knowledge of the torsion of this group may be used for example to construct almost-Bieberbach subgroups. In this direction, we prove Theorem 2, which generalises [START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF]Theorem 2]. THEOREM 2. Let n, k ≥ 3. Then the quotient group B n /Γ k (P n ) has no elements of order 2 nor of order 3.

From this, we are able to deduce Corollary 15, which proves the existence of almost-Bieberbach groups in B n /Γ k (P n ) for all n, k ≥ 3. In much of the rest of the paper, we focus our attention on the case k = 3. To study the torsion of B n /Γ 3 (P n ), an explicit basis of Γ 2 (P n )/Γ 3 (P n ) is introduced in Section 3.2, and in Section 4.1, we partition this basis into the orbits of the action by conjugation of a certain element δ n of B n /Γ 3 (P n ). In Section 4.2, we exhibit presentations of P n /Γ 3 (P n ) and B n /Γ 3 (P n ). This enables us to study the finite-order elements of B n /Γ 3 (P n ) and their conjugacy classes in Section 4.3. The following result shows that the torsion of B n /Γ 3 (P n ) coincides with that of S n if we remove the elements whose order is divisible by 2 or 3. THEOREM 3. Let n ≥ 5, and let τ ∈ N be such that gcd(τ, 6) = 1. Then the group B n /Γ 3 (P n ) admit finite-order elements of torsion τ if and only if S n does. Further, if x ∈ S n is of order τ, there exists α ∈ S n of order τ such that σ(α) = x, in particular σ(α) and x have the same cycle type.

We end Section 4 with an analysis of the conjugacy classes of the finite-order elements of B n /Γ 3 (P n ), the main result in this direction being the following. THEOREM 4. Let n ≥ 5, and let α and β be two finite-order elements of B n /Γ 3 (P n ) whose associated permutations have the same cycle type. Then α and β are conjugate in B n /Γ 3 (P n ).

In Section 5, we discuss some aspects of the quotients B n /Γ 3 (P n ), where n is small. In Section 5.1, we obtain some almost-Bieberbach subgroups of B 3 /Γ 3 (P 3 ) of dimension 4 that are the fundamental groups of orientable 4-dimensional infra-nilmanifolds, and in Section 5.2, we compute δ 5 5 in terms of the chosen basis of Γ 2 (P 5 )/Γ 3 (P 5 ). Using the constructions of Section 4.3, this allows us to exhibit explicit elements of order 5 in B 5 /Γ 3 (P 5 ).

Another important question in our study is the existence and embedding of finite groups in B n /Γ 3 (P n ). For cyclic groups, the answer is given by Theorem 3, and this may be generalised to Abelian groups in Corollary 24 using Lemma 19(b). We have proved recently in [START_REF] Gonçalves | Embeddings of finite groups in B n /Γ k (P n ) for k = 2[END_REF] that Cayley-type theorems hold for B n /Γ 2 (P n ) and B n /Γ 3 (P n ), namely that if G is any finite group of odd order n (resp. of order n relatively prime with 6) then G embeds in B n /Γ 2 (P n ) (resp. in B n /Γ 3 (P n )). In the case of B n /Γ 2 (P n ), the same result has been proved independently by V. Beck and I. Marin within the more general setting of complex reflection groups [BM]. In the same paper, we also show that with appropriate conditions on n and m, two families of groups of the form

G = Z n ⋊ θ Z m embed in B n /Γ 2 (P n ) and B n /Γ 3 (P n ).
and the CNRS/FAPESP PRC project n o 275209 (France) and n o 2016/50354-1 (Brazil). The third author was initially supported by a project grant n o 151161/2013-5 from CNPq and then partially supported by the FAPESP Projeto Temático Topologia Algébrica, Geométrica e Diferencial n o 2012/24454-8 (Brazil).

Preliminaries

In this section, we recall some definitions and results about Artin braid groups, (almost-) crystallographic groups, and the relations between them that will be used in this paper.

Artin braid groups

We start by recalling some facts about the Artin braid group B n on n strings (see [Han] for more details). It is well known that B n possesses a presentation with generators σ 1 , . . . , σ n-1 that are subject to the following relations:

σ i σ j = σ j σ i for all 1 ≤ i < j ≤ n -1 such that |i -j| ≥ 2 σ i+1 σ i σ i+1 = σ i σ i+1 σ i for all 1 ≤ i ≤ n -2.
(2)

Let σ : B n -→ S n be the homomorphism defined on the given generators of B n by σ(σ i ) = (i, i + 1) for all 1 ≤ i ≤ n -1. Just as for braids, we read permutations from left to right so that if α, β ∈ S n , their product is defined by α

• β(i) = β(α(i)) for i = 1, 2, . . . , n.
The pure braid group P n on n strings is defined to be the kernel of σ, from which we obtain the following short exact sequence:

1 -→ P n -→ B n σ -→ S n -→ 1. ( 3
)
Let G be a group. If g, h ∈ G then [g, h] = ghg -1 h -1 will denote their commutator, and if H, K are subgroups of H then we set [H, K] G). Following P. Hall, for any grouptheoretic property P, G is said to be residually P if for any (non-trivial) element x ∈ G, there exist a group H that possesses property P and a surjective homomorphism

= [h, k] | k ∈ H, k ∈ K . The lower central series {Γ i (G)} i∈N of G is defined inductively by Γ 1 (G) = G, and Γ i+1 (G) = [G, Γ i (G)] for all i ∈ N. If i = 2, Γ 2 (G) is the commutator subgroup of G. For all i, j ∈ N with j > i, Γ j (G) is a normal subgroup of Γ i (
ϕ : G -→ H such that ϕ(x) = 1. It is well known that a group G is residually nilpotent if and only if ∩ i≥1 Γ i (G) = {1}.
The lower central series of groups and their sucessive quotients Γ i (G)/Γ i+1 (G) are isomorphism invariants, and have been widely studied using commutator calculus, in particular for free groups of finite rank [Hal, MKS]. Falk and Randell, and independently Kohno, investigated the lower central series of the pure braid group P n , and proved that P n is residually nilpotent [START_REF] Falk | Pure braid groups and products of free groups[END_REF][START_REF] Kohno | Série de Poincaré-Koszul associée aux groupes de tresses pures[END_REF].

A presentation of P n is given by the set of generators A i,j 1≤i<j≤n , where:

A i,j = σ j-1 • • • σ i+1 σ 2 i σ -1 i+1 • • • σ -1 j-1 , (4)
subject to the following relations that are expressed in terms of commutators (see [START_REF] Murasugi | A study of braids[END_REF]Remark 3.1,p. 56] or [START_REF] Hansen | Braids and coverings: selected topics[END_REF]Chapter 1,Lemma 4.2]):

           [A r,s , A i,j ] = 1 if 1 ≤ r < s < i < j ≤ n or 1 ≤ r < i < j < s ≤ n [A r,s , A r,j ] = [A -1 s,j , A r,j ] if 1 ≤ r < s < j ≤ n [A r,s , A s,j ] = [A -1 s,j , A -1 r,j ] if 1 ≤ r < s < j ≤ n [A r,i , A s,j ] = [A -1 i,j , A -1 r,j ], A s,j if 1 ≤ r < s < i < j ≤ n.
(5) For notational reasons, if 1 ≤ i < j ≤ n, we set A j,i = A i,j , and if A i,j appears in a word of P n with exponent m i,j ∈ Z, then we let m j,i = m i,j . It follows from the presentation (5) that P n /Γ 2 (P n ) is isomorphic to Z n(n-1)/2 , and that a basis of P n /Γ 2 (P n ) is given by A i,j 1≤i<j≤n , where by abuse of notation, the Γ 2 (P n )-coset of A i,j will also be denoted by A i,j . For all k ≥ 2, equation (3) gives rise to the following short exact sequence:

1 -→ P n /Γ k (P n ) -→ B n /Γ k (P n ) σ -→ S n -→ 1, (6)
where σ : B n /Γ k (P n ) -→ S n is the homomorphism induced by σ. In much of what follows, we shall be interested in the action by conjugation of

B n /Γ 2 (P n ) on Γ 2 (P n )/Γ 3 (P n ).
For all 1 ≤ k ≤ n -1 and 1 ≤ i < j ≤ n, the action of B n on P n described in [START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF]equation (7)] may be rewritten as:

σ k A i,j σ -1 k =        A -1 i,j A i,j-1 A i,j if j = k + 1 and i < k A -1 i,j A i-1,j A i,j if i = k + 1 A σ -1 k (i), σ -1 k (j) otherwise, (7)
where by abuse of notation, we write σ -1 k (i) = σ(σ -1 k )(i), σ being as in equation (3). This action was used in [START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF] to prove the following proposition.

PROPOSITION 5 ([GGO1, Proposition 12]). Let α ∈ B n /Γ 2 (P n ), and let π be the permutation induced by α -1 . Then for all 1 ≤ i < j ≤ n, αA i,j α -1 = A π(i),π(j) in P n /Γ 2 (P n ).

For all α ∈ B n /Γ 3 (P n ), and all 1 ≤ i < j ≤ n and 1 ≤ r < s ≤ n, we claim that:

α[A i,j , A r,s ]α -1 = [αA i,j α -1 , αA r,s α -1 ] = [A π(i),π(j) , A π(r),π(s) ] in P n /Γ 3 (P n ), ( 8
)
where π is the permutation induced by α -1 , and by abuse of notation, A i,j and A r,s are considered as elements of P n /Γ 3 (P n ). To see this, first note that the first equality of (8) clearly holds, and that for all k ∈ N, we have the following short exact sequence:

1 -→ Γ k (P n )/Γ k+1 (P n ) -→ P n /Γ k+1 (P n ) -→ P n /Γ k (P n ) -→ 1. ( 9
)
Taking k = 2 in (9) and using Proposition 5, there exist γ 1 , γ 2 ∈ Γ 2 (P n )/Γ 3 (P n ) such that αA i,j α -1 = γ 1 A π(i),π(j) and αA r,s α -1 = γ 2 A π(r),π(s) . The second equality of (8) then follows using standard commutator relations.

Almost-crystallographic groups

In this section, we recall briefly the definitions of almost-crystallographic and almost-Bieberbach groups, which are natural generalisations of crystallographic and Bieberbach groups, as well as a characterisation of almost-crystallographic groups. For more details about crystallographic groups, see [START_REF] Charlap | Bieberbach groups and flat manifolds[END_REF]Section I.1.1], [START_REF] Dekimpe | Almost-Bieberbach groups: affine and polynomial structures[END_REF]Section 2.1] or [START_REF] Wolf | Spaces of constant curvature[END_REF]Chapter 3].

Given a connected and simply-connected nilpotent Lie group N, the group Aff (N) of affine transformations of N is defined by Aff (N) = N ⋊ Aut (N), and acts on N by: (n, ϕ) • m = nϕ(m) for all m, n ∈ N and ϕ ∈ Aut (N). DEFINITION 6 ([De,Sec. 2.2,p. 15]). Let N be a connected, simply-connected nilpotent Lie group, and consider a maximal compact subgroup C of Aut (N). A uniform dis- crete subgroup E of N ⋊ C is called an almost-crystallographic group, and its dimension is defined to be that of N. A torsion-free, almost-crystallographic group is called an almost-Bieberbach group, and the quotient space E\N is called an infra-nilmanifold. If further E ⊆ N, we say that the space E\N is a nilmanifold.

It is well known that infra-nilmanifolds are classified by their fundamental group that is almost-crystallographic [Au] Every almost-crystallographic subgroup E of the group Aff (N) fits into an extension:

1 -→ Λ -→ E -→ F -→ 1, (10)
where Λ = E ∩ N is a uniform lattice in N, and F is a finite subgroup of C known as the holonomy group of the corresponding infra-nilmanifold E\N [Au]. Let M be an infranilmanifold whose fundamental group E is almost-crystallographic. Following [GPS, Page 788], we recall the construction of a faithful linear representation associated with the extension (10). Suppose that the nilpotent lattice Λ is of class c + 1 i.e. Γ c (Λ) = 1 and

Γ c+1 (Λ) = 1. For i = 1, . . . , c, let Z i = Γ i (Λ) Γ i+1 (Λ)
denote the factor groups of the lower central series {Γ i (Λ)} c+1 i=1 of Λ. We will assume further that these quotients are torsion free, since this will be the case for the groups that we will study in the following sections. Thus Z i ∼ = Z k i for all 1 ≤ i ≤ c and for some k i > 0. The rank or Hirsch number of Λ is equal to ∑ c i=1 k i . The action by conjugation of E on Λ induces an action of E on Z i which factors through an action of the group E/Λ (the holonomy group F), because Λ acts trivially on Z i . This gives rise to a faithful representation θ F : F -→ GL(n, Z) via the composition:

θ F : F -→ GL(k 1 , Z) × • • • × GL(k c , Z) -→ GL(n, Z), (11)
where n, which is the rank of Γ, is also equal to the dimension of N. Using [START_REF] Gasior | Spin structures on almost-flat manifolds[END_REF]Remark 2.5], this representation will be used in Section 5.1 to decide whether M is orientable or not.

In order to prove Theorem 1, we shall use part of the algebraic characterisation of almost-crystallographic groups given in [START_REF] Dekimpe | Almost-Bieberbach groups: affine and polynomial structures[END_REF]Theorem 3.1.3] as follows.

THEOREM 7 ( [START_REF] Dekimpe | Almost-Bieberbach groups: affine and polynomial structures[END_REF]Theorem 3.1.3

]). Let E be a polycyclic-by-finite group. Then E is almostcrystallographic if and only if it has a nilpotent subgroup, and possesses no non-trivial finite normal subgroups.

3 The almost-crystallographic group B n /Γ k (P n )

Let n, k ≥ 3. In this section, we study the group B n /Γ k (P n ). In Section 3.1, we start by recalling some results about the quotient groups Γ k (P n )/Γ k+1 (P n ) that appear in (9). In Section 3.2, we prove Theorems 1 and 2 which state that the groups B n /Γ k (P n ) are almost-crystallographic, and that B n /Γ k (P n ) possesses no element of order 2 or 3 respectively. This allows us to prove in Corollary 15, which shows that if H is a subgroup of S n whose order is not divisible by any prime other than 2 or 3 then the subgroup

σ -1 (H)/Γ k (P n ) of B n /Γ k (P n ) is almost-Bieberbach.

The rank of the free Abelian group

Γ k (P n )/Γ k+1 (P n ) Let n ≥ 2 and k ≥ 1. The group Γ k (P n )/Γ k+1 (P n )
, which we shall denote by L k (P n ), is free Abelian of finite rank, and by [START_REF] Falk | The lower central series of a fiber-type arrangement[END_REF]Theorem 4.2] and [START_REF] Kohno | Série de Poincaré-Koszul associée aux groupes de tresses pures[END_REF]Theorem 4.5], its rank is related to the Poincaré polynomial of certain hyperplane complements. Using Chen groups, Cohen and Suciu gave explicit formulae for rank(L k (P n )) for k ∈ {1, 2, 3} and all n ≥ 2 [CS, Theorem 1.1 and page 46]. More generally, by [LVW, Theorem 4.6], the rank of L k (P n ) is given by equation ( 1). In practice, we may compute these ranks as follows. If k ≥ 2, let k * be the product of the distinct prime divisors of k. Then (1) may be rewritten as:

rank(L k (P n )) = 1 k n-1 ∑ j=1 ∑ d|k * µ(d)j k/d = 1 k ∑ d|k * µ(d)S k/d (n), (12)
where S r (n) = ∑ n-1 j=1 j r . The number of summands in the expression 1

k ∑ d|k * µ(d)j k/d is equal to 2 t . PROPOSITION 8. Let n ≥ 2. Then rank(L k (P n )) is a polynomial of degree k + 1 in the variable n. Proof. By [N, Chapter XVI], the sum S k/d (n) is a polynomial of degree k d + 1 in the variable n, so rank(L k (P n )) is also a polynomial in the variable n. The result follows by noting that d = 1 divides k * . REMARKS 9. (a)
For small values of r, a polynomial expression for S r (n) was computed for example in [AIK, Section 1.2] for 1 ≤ r ≤ 6, and [START_REF] Nielsen | Traité élémentaire des nombres de Bernoulli[END_REF]Chapter XVI,page 296] or [Wi, Tables I andII] for 1 ≤ r ≤ 10. (b) Using the description given in equation ( 12), we have computed rank(L k (P n )) for all 1 ≤ k ≤ 10. If n = 2 or 3, we obtain the equalities rank(L 2 (P n )) = ( n 3 ) and rank (L 3

(P n )) = 2( n+1 4 ) given in [CS, Theorem 1.1]. For 3 ≤ k ≤ 10, rank(L k (P n ))
is the product of ( n+1 4 ) by a polynomial in the variable n of degree k -3. The authors do not know whether this is true in general. (c) The numbers rank(L k (P n )) may be expressed in terms of Bernoulli numbers since the latter are closely related to sums of powers of consecutive integers (see [AIK] for more information about Bernoulli numbers, and especially Formula (1.1) on page 1).

The quotient groups

B n /Γ k (P n )
The proofs of the first two results of this section are straightforward, but will be useful in the analysis of the group B n /Γ k (P n ). First note that P n /Γ k (P n ) is a nilpotent group of nilpotency class k -1, and as we shall see in Lemma 11(a), it is torsion free. PROPOSITION 10. Let n, k ≥ 3. The Hirsch length of the nilpotent group

P n /Γ k (P n ) is equal to ∑ k-1 q=1 1 q ∑ n-1 j=1 ∑ d|q * µ(d)j q/d .
In particular, the Hirsch length of P n /Γ 3 (P n ) (resp. of

P n /Γ 4 (P n )) is equal to ( n 2 ) + ( n 3 ) (resp. to ( n 2 ) + ( n 3 ) + 2( n+1 4 )). Proof.
Let n, k ≥ 3. Since the Hirsch length of a nilpotent group is equal to the sum of the ranks of the consecutive lower central series quotients, the first part of the statement follows from equation ( 12). If q ∈ {3, 4}, the formulae is then a consequence of those given in Remarks 9(b).

LEMMA 11. (a) Let n, k ≥ 2. Then the group P n /Γ k (P n ) is torsion free. (b) Let n ≥ 3, let k ≥ l ≥ 1, and let G be a finite group. If B n /Γ k (P n ) possesses a (normal) subgroup isomorphic to G then B n /Γ l (P n ) possesses a (normal) subgroup isomorphic to G. In particular, if p is prime, and if B n /Γ l (P n ) has no p-torsion then B n /Γ k (P n ) has no p-torsion. Proof. (a) The proof is by induction on k. If k = 2 then P n /Γ 2 (P n ) ∼ = Z n(n-1)/2 ,
which implies the result in this case. Suppose then that the result holds for some k ≥ 2. Then the quotient of the short exact sequence ( 9) is torsion free by induction, and the kernel is torsion free by the results mentioned at the beginning of Section 3.1. It follows that P n /Γ k+1 (P n ) is also torsion free. (b) Assume that B n /Γ k (P n ) possesses a (normal) subgroup isomorphic to G. For all j ≥ 2, we have a central extension of the form:

1 -→ Γ j-1 (P n )/Γ j (P n ) -→ B n /Γ j (P n ) -→ B n /Γ j-1 (P n ) -→ 1. ( 13
)
As we mentioned above, the kernel of the short exact sequence ( 13) is torsion free, so the restriction of the homomorphism In order to prove Theorem 2, we first recall the following lemma from [START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF].

B n /Γ j (P n ) -→ B n /Γ j-1 (P n ) to
LEMMA 13 ([GGO1, Lemma 28]). Let k, n ≥ 3 and r ≥ 0 such that k is odd and r + k ≤ n, and define δ r,k , α r,k ∈ B n /Γ 2 (P n ) by:

δ r,k = σ r+k-1 • • • σ r+ k+1 2 σ -1 r+ k-1 2 • • • σ -1 r+1 and α r,k = σ r+1 • • • σ r+k-1 . ( 14
)
Then δ r,k is of order k in B n /Γ 2 (P n ), and satisfies:

δ r,k = A r+ k+1 2 ,r+k A r+ k+3 2 ,r+k • • • A r+k-1,r+k α -1 r,k . ( 15
)
Let n ≥ 3. We now describe a basis of the free Abelian group Γ 2 (P n )/Γ 3 (P n ) of rank ( n 3 ) by Remark 9(b). This group is generated by the elements of the form [A i,j , A r,s ],

where 1 ≤ i < j ≤ n and 1 ≤ r < s ≤ n, but by equation ( 5),

[A i,j , A r,s ] = 1 in Γ 2 (P n )/Γ 3 (P n ) if Card {i, j, r, s} ∈ {2, 4}, so Γ 2 (P n )/Γ 3 (P n ) is generated by the elements of the form [A i,j , A r,s ],
where Card {i, j, r, s} = 3. Let α i,j,k = [A i,j , A j,k ] for all 1 ≤ i < j < k ≤ n. Using (5) once more, the following equalities hold in Γ 2 (P n )/Γ 3 (P n ):

[A i,k , A j,k ] = α -1 i,j,k , [A i,j , A i,k ] = α -1 i,j,k and [A j,k , A i,j ] = α -1 i,j,k , (16)
and if α i,j,k = α ±1 r,s,t then {i, j, k} = {r, s, t}. So the set

B = α i,j,k 1 ≤ i < j < k ≤ n , ( 17
) generates Γ 2 (P n )/Γ 3 (P n ). Since Card(B) = ( n 3 ) = rank(Γ 2 (P n )/Γ 3 (P n ))
, it follows that B is a basis for Γ 2 (P n )/Γ 3 (P n ), in particular, α i,j,k = 1. Applying equation ( 8) to the Artin generators of B n , and using equation ( 16), the action of B n /Γ 3 (P n ) on the elements of B given in ( 17) is as follows:

σ k α r,s,t σ -1 k =        α σ k (r),σ k (s),σ k (t) if σ k (r) < σ k (s) < σ k (t) α -1 σ k (s),σ k (r),σ k (t) if σ k (r) > σ k (s) α -1 σ k (r),σ k (t),σ k (s) if σ k (s) > σ k (t) (18)
for all 1 ≤ k ≤ n -1 and 1 ≤ r < s < t ≤ n. Using the action of B n on P n given in equation ( 7) and equation ( 16), one may check that in B n /Γ 3 (P n ):

σ k A i,j σ -1 k =      A i,j-1 α -1 i,j-1,j if j = k + 1 and i < k A i-1,j α -1 i-1,i,j if i = k + 1 A σ k (i),σ k (j)
otherwise.

(

) 19
REMARK 14. Consider the action by conjugation of B n /Γ 3 (P n ) on Γ 2 (P n )/Γ 3 (P n ) described by ( 18). The restriction of this action to

P n /Γ 3 (P n ) on Γ 2 (P n )/Γ 3 (P n ) is trivial,
and so the action of B n /Γ 3 (P n ) factors through S n . Further, if n > 3, the action of S n is injective, i.e. if α is a non-trivial permutation, then the induced automorphism is different from the identity by ( 8), ( 16) and the fact that B is a basis.

Proof of Theorem 2. Let n, k ≥ 3. By [GGO1, Theorem 2], B n /Γ 2 (P n ) has no 2-torsion, so applying Lemma 11(b), we conclude that this is also the case for B n /Γ k (P n ). To complete the proof, using Lemma 11(b) once more, it suffices to prove that B n /Γ 3 (P n ) has no 3-torsion. Suppose on the contrary that B n /Γ 3 (P n ) possesses an element ρ of order 3. Let m denote the number of conjugacy classes of elements of B n /Γ 2 (P n ) of order 3. By [GGO1, Theorem 5], m is equal to the number of conjugacy classes of elements of order 3 in S n , so m = ⌊n/3⌋ ≥ 1, and using [GGO1, Proposition 29], representatives of the conjugacy classes of elements of order 3 in B n /Γ 2 (P n ) are given by the elements of the form η t = δ 3t,3 • • • δ 6,3 δ 3,3 δ 0,3 , where 0 ≤ t ≤ m -1. Conjugating ρ if necessary and using the short exact sequence (13), there exists θ ∈ Γ 2 (P n )/Γ 3 (P n ) such that ρ = θη t for some 0 ≤ t ≤ m -1 (here η t is considered as an element of B n /Γ 3 (P n )). Note that η t acts by conjugation on σ -1

(S 3 × 1 × • • • × 1) and σ -1 (1 × 1 × 1 × S n-3
), where σ is as in (6). Further, by ( 8) and ( 16), we see that η t α 1,2,3 η -1 t = α 1,2,3 in B n /Γ 3 (P n ), and that if θ ′ is a word in the elements of B \ {α 1,2,3 } then η t θ ′ η -1 t , written in reduced form as a word in P n /Γ 3 (P n ), does not contain α 1,2,3 . Let θ = α l 1,2,3 θ 1 , where l ∈ Z and θ 1 is a word in the elements of B \ {α 1,2,3 }. For all i ∈ {0, . . . , t}, we have:

δ 3 3i,3 = (σ 3i+2 σ -1 3i+1 ) 3 = σ 2 3i+2 σ -1 3i+2 σ -1 3i+1 σ 3i+2 σ -1 3i+1 σ 3i+2 σ -1 3i+1 = σ 2 3i+2 σ 3i+1 σ -1 3i+2 σ -2 3i+1 σ 3i+2 σ -1 3i+1 = σ 2 3i+2 σ 2 3i+1 σ -2 3i+2 σ -2 3i+1 = [σ 2 3i+1 , σ 2 3i+2 ] -1 = [A 3i+1,3i+2 , A 3i+2,3i+3 ] -1 = α -1 3i+1,3i+2,3i+3 ,
and since the δ 3i,3 commute pairwise, it follows that

η 3 t = α -1 1,2,3 • • • α -1 3t+1,3t+2,3t+3
. Hence:

ρ 3 = (θη t ) 3 = α l 1,2,3 θ 1 η t α l 1,2,3 θ 1 η -1 t η 2 t α l 1,2,3 θ 1 η -2 t η 3 t = α 3l 1,2,3 θ 1 . η t θ 1 η -1 t . η 2 t θ 1 η -2 t . α -1 1,2,3 • • • α -1 3t+1,3t+2,3t+3
in Γ 2 (P n )/Γ 3 (P n ). As we saw above, the words η t θ 1 η -1 t and η 2 t θ 1 η -2 t (written in reduced form in P n /Γ 3 (P n )) do not contain α 1,2,3 , and since ρ 3 = 1, it follows by comparing the coefficients of α 1,2,3 that 3l = 1, which yields a contradiction. So B n /Γ 3 (P n ) has no 3-torsion as required.

The following result is a consequence of Theorems 1 and 2 and the definition of almost-Bieberbach groups.

COROLLARY 15. Let n ≥ 3 and let k ≥ 3. Let H be a subgroup of S n whose order is equal to 2 s 3 t for some s, t ∈ N. Then σ -1 (H)/Γ k (P n ) is an almost-Bieberbach group. In particular, the groups B 3 /Γ 3 (P 3 ) and B 4 /Γ 3 (P 4 ) are almost-Bieberbach groups of dimension 4 and 10 respectively.

Proof. The first part is a consequence of Theorems 1 and 2 and Definition 6. The second part follows by taking H = S n for n ∈ {3, 4}, and by applying the short exact se- quence ( 6) and the results of Table 1.

Torsion elements, conjugacy classes and a presentation

of B n /Γ 3 (P n )

Let n ∈ N. The main aim of this section is to prove Theorems 3 and 4 that describe the torsion elements of B n /Γ 3 (P n ) and their conjugacy classes. We also exhibit a presentation of the quotient group B n /Γ 3 (P n ) in Proposition 18.

4.1 A partition of the basis of Γ 2 (P 3 )/Γ 3 (P 3 )

Let δ n be equal to the element δ 0,n of equation ( 14), considered as an element of the group B n /Γ 3 (P n ). By Lemma 13, δ n n ∈ Γ 2 (P n )/Γ 3 (P n ). We start by partitioning the basis B of Γ 2 (P n )/Γ 3 (P n ) given in (17) into orbits for the action by conjugation by δ n . PROPOSITION 16. Let n ≥ 5. Then the basis B of Γ 2 (P n )/Γ 3 (P n ) given in equation ( 17) is invariant with respect to the action of Z n given by conjugation by δ n . Further, under this action: (a) if gcd(n, 3) = 1, B is the disjoint union of (n-1)(n-2) 6 orbits each of length n.

(b) if gcd(n, 3) = 1, B is the disjoint union of n(n-3) 6 + 1 orbits, one of which is of length n 3 , and the remaining n(n-3) 6

orbits are each of length n.

Proof. Let n ≥ 5, and let 1 ≤ i < j < k ≤ n. Using (8), ( 14) and ( 16), we see that:

δ n α i,j,k δ -1 n = α i-1,j-1,k-1 if i ≥ 2 [A n,j-1 , A j-1,k-1 ] = [A j-1,n , A j-1,k-1 ] = α j-1,k-1,n if i = 1. ( 20
)
So the action of conjugation by δ n permutes the elements of B, which proves the first part of the statement. By (20),

δ n n α i,j,k δ -n n = α i,j,k for all 1 ≤ i < j < k ≤ n. Suppose that 1 ≤ l < n is such that δ l n α i,j,k δ -l n = α i,j,k .
Then by (20), we have:

(i) i -l + n = k, j -l = i and k -l = j, or (ii) i -l + n = j, j -l + n = k and k -l = i.
Summing the three equations in both cases, we see that 3 | n. In particular:

• if 3 ∤ n then all of the orbits are of length n, and there are ( n 3 )/n = (n-1)(n-2) 6 orbits. • if 3 | n then from (i) and (ii), we have either l = n 3 or l = 2n 3 . It follows that j = i + n 3 and k = i + 2n 3 , and for these values of i, j and k, we obtain a single orbit of length n 3 . The remaining orbits are of length n, and so the number of such orbits is given by (( n 3 ) -n 3 )/n, which is equal to n(n-3) 6 .

REMARK 17. Let n = 3q + r, where q ∈ N and r ∈ {0, 1, 2}. Let S be the subset of B defined by:

S = α 1,j,k 2 ≤ j ≤ q + 1 and 2j -1 ≤ k ≤ n -(j -1) if r = 0 α 1,j,k 2 ≤ j ≤ q and 2j -1 ≤ k ≤ n -(j -1) ∪ {α 1,n/3+1,2n/3+1 } if r = 0. ( 21
)
Then S is a transversal for the action of δ n on B described in Proposition 16, and B may be rewritten as:

B =          δ t n α 1,j,k δ -t n 2 ≤ j ≤ q + 1, 2j -1 ≤ k ≤ n -(j -1) and 0 ≤ t ≤ n -1 if r = 0 δ t n α 1,j,k δ -t n 2 ≤ j ≤ q, 2j -1 ≤ k ≤ n -(j -1) and 0 ≤ t ≤ n -1 δ t n α 1,n/3+1,2n/3+1 δ -t n 0 ≤ t ≤ n/3 -1 if r = 0.
With the notation of Proposition 16, let O(n) be equal to the number of orbits of the action on B by conjugation by δ n , and let T = {b i,1 } 1≤i≤O(n) be a transversal for this action (for example, we may take T to be the transversal S defined in ( 21)). We choose T so that b O(n),1 is a representative of the orbit of length n/3 if 3 | n. For 1 ≤ i ≤ O(n), let q(i) be the length of the orbit of b i,1 , so that q(i) = n/3 if 3 | n and i = O(n), and q(i) = n otherwise, and let:

b i,j = δ j-1 n b i,1 δ -(j-1) n for all 1 ≤ j ≤ q(i). ( 22
) Then B = b i,j 1 ≤ i ≤ O(n) and 1 ≤ j ≤ q(i) .

A presentation of B n /Γ 3 (P n )

In this section, we exhibit a presentation of B n /Γ 3 (P n ) by applying the techniques of [START_REF] Johnson | Presentation of groups, LMS Lecture Notes 22[END_REF]Proposition 1,p. 139] to obtain a presentation of a group extension to the short exact sequence (9).

PROPOSITION 18. Let n ≥ 3. (a) The group P n /Γ 3 (P n ) has a presentation given by the generating set X n = A i,j 1≤i<j≤n ∪ {α r,s,t } 1≤r<s<t≤n , subject to the following relations:

The following results allow us to decide whether certain elements of a group are of finite order or not. They will be used to obtain finite-order elements in B n /Γ 3 (P n ), as well as in the proof of Theorem 3. LEMMA 20. Let d ≥ 2, let G be a group that has no d-torsion, and let α ∈ G. Suppose that σ : G -→ S n is a surjective homomorphism whose kernel K is torsion free, and such that d divides the order of σ(α). Then for all θ ∈ K, θα is of infinite order in G.

Proof. Suppose on the contrary that there exists θ ∈ K for which θα is of finite order, m say, in G. Let q denote the order of σ(α). Since θ ∈ K, Id = σ((θα) m ) = (σ(α)) m , and thus q | m. In a similar manner, we see that σ((θα) q ) = Id, so (θα) q ∈ K. Now 1 = (θα) m = ((θα) q ) m/q , and the fact that K is torsion free implies that (θα) q = 1, hence q = m. By hypothesis, d | q, and so ((θα) q/d ) d = 1. This implies that the order of (θα) q/d divides d. On the other hand, the fact that σ(θα) is of order q implies that σ((θα) q/d ) is of order d, and so the order of (θα) q/d cannot be strictly less than d. Hence (θα) q/d is of order d, but this contradicts the fact that G has no d-torsion.

PROPOSITION 21. Let n ≥ 3. (a) If n is even then for all k ∈ N and θ ∈ P n /Γ 3 (P n ), (θδ n ) k / ∈ Γ 2 (P n )/Γ 3 (P n ). (b) Suppose that n is odd. Then the element δ n n belongs to Γ 2 (P n )/Γ 3 (P n )
, and with the notation of ( 22), we have:

δ n n = O(n) ∏ i=1 q(i) ∏ j=1 b m i,j i,j , ( 23
)
where m i,1 = m i,2 = • • • = m i,q(i) for all 1 ≤ i ≤ O(n). Further, if 3 | n then for all k ∈ N and θ ∈ P n /Γ 3 (P n ), (θδ n ) k is non trivial in B n /Γ 3 (P n ).
Proof.

(a) Let n be even, and assume on the contrary that (θδ n

) k ∈ Γ 2 (P n )/Γ 3 (P n ) for some k ∈ N and θ ∈ P n /Γ 3 (P n ). Let f : B n /Γ 3 (P n ) -→ B n /Γ 2 (P n ) be the homomorphism given in (13) with j = 3. Then (θδ n ) k = 1 in B n /Γ 2 (P n ), where θ = f (θ) and δ n = f (δ n ).
We now apply Lemma 20 to the short exact sequence (6), taking k = 2, G = B n /Γ 2 (P n ), α = δ n , and the homomorphism σ of that lemma to be σ. Note that σ(δ n ) is of order n, so 2 | n, and B n /Γ 2 (P n ) has no 2-torsion by [START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF]Theorem 2]. Since P n /Γ 2 (P n ) is torsion free, it follows from Lemma 20 that θδ n is of infinite order in B n /Γ 2 (P n ), and we obtain a contradiction. We conclude that (θδ

n ) k / ∈ Γ 2 (P n )/Γ 3 (P n ). (b) Since δ n n = 1 in B n /Γ 2 (P n ) by Lemma 13, it follows from the short exact sequence (13) with j = 3 that δ n n ∈ Γ 2 (P n )/Γ 3 (P n ).
Hence in terms of the basis B of Γ 2 (P n )/Γ 3 (P n ) given by ( 17), and with the notation of ( 22), for all 1 ≤ i ≤ O(n) and 1 ≤ j ≤ q(i), there exist m i,j ∈ Z that are unique and for which (23) holds. The equality m i,1 = m i,2 = • • • = m i,q(i) follows by conjugating (23) by δ n and using ( 22). The last part of the state- ment follows by applying Lemma 20 to the short exact sequence (6), where we take G = B n /Γ 3 (P n ), d = 3 and α = δ n , and using Theorem 2. THEOREM 22. Let n ∈ N, let m ≤ n, and let s be the largest divisor of m for which gcd(s, 6) = 1. If s > 1, the group B n /Γ 3 (P n ) possesses infinitely-many elements of order s.

Proof. Let 1 ≤ m ≤ n, and let s be the largest divisor of m for which gcd(s, 6) = 1. Since we assume that s > 1, we must have n ≥ 5. Further, the fact that s ≤ n implies using Lemma 19(a) that the homomorphism ι : B s /Γ 3 (P s ) -→ B n /Γ 3 (P n ) is injective. So it suffices to prove that if n ∈ N is relatively prime with 6 then B n /Γ 3 (P n ) possesses elements of order n. In this case, δ n n is as given in ( 23), where O(n) = (n -1)(n -2)/6 by Proposition 16(a), and q(i) = n for all 1 ≤ i ≤ O(n). Let:

δ n = θδ n , (24)
where we take θ to be an element of

Γ 2 (P n )/Γ 3 (P n ), so θ = ∏ O(n) i=1 ∏ n j=1 b r i,j
i,j , where r i,j ∈ Z for all 1 ≤ i ≤ O(n) and 1 ≤ j ≤ n. Then by ( 22) and Proposition 21(b), we have:

δ n n = (θδ n ) n = θ. δ n θδ -1 n . δ 2 n θδ -2 n • • • δ n-1 n θδ -(n-1) n . δ n n = O(n) ∏ i=1 n ∏ j=1 b i,j ∑ n j=1 r i,j . O(n) ∏ i=1 n ∏ j=1 b m i,1 i,j = O(n) ∏ i=1 n ∏ j=1 b i,j ∑ n j=1 r i,j +m i,1 . ( 25
)
Taking k = 3 in (6), σ( δ n ) is of order n in S n , and it follows from this and ( 25) that δ n is of order n if and only if:

n ∑ j=1 r i,j = -m i,1 for all i = 1, . . . , O(n). (26)
This system of equations has infinitely-many solutions in the r i,j , which yields infinitelymany elements in B n /Γ 3 (P n ) of order n.

REMARK 23. If n ∈ N and gcd(n, 6) = 1, the proof of Theorem 22 shows how to obtain explicit finite-order elements of B n /Γ 3 (P n ). However, as part of the process, we need to determine δ n n in terms of the elements of the basis B. This seems to be an arduous computation in general, as the calculation given in Section 5.2 in the case n = 5 indicates.

We are now able to prove Theorem 3.

Proof of Theorem 3. Let n ≥ 5, and let τ ∈ N be such that gcd(τ, 6) = 1. If β is an element of B n /Γ 3 (P n ) of order τ then using (6) and the fact that P n /Γ 3 (P n ) is torsion free by Lemma 11(a), it follows that σ(β) is an element of S n of order τ. Conversely, suppose that x ∈ S n is an element of order τ, and let η 1 • • • η t be the cycle decomposition of x. Then τ = lcm(n 1 , . . . , n t ), where for i = 1, . . . , t, η i is of length n i . In particular, gcd(n i , 6) = 1 for all i = 1, . . . , t, and ∑ t i=1 n i ≤ n by [Ho]. Let y ∈ S n be such that

yxy -1 = (1, . . . , n 1 )(n 1 + 1, . . . , n 1 + n 2 ) • • • (n 1 + • • • + n t-1 + 1, . . . , n 1 + • • • + n t ). ( 27
)
Since σ is surjective, there exists ρ ∈ B n /Γ 3 (P n ) such that σ(ρ) = y. For i = 1, . . . , t, let δ n i ∈ B n i /Γ 3 (P n i ) be as defined in ( 24), where the coefficients of θ satisfy (26). From the proof of Theorem 22,

δ n i is of order n i . Let β = ζ( δ n 1 , . . . , δ n t ) in B n /Γ 3 (P n ).
By taking k = 3 in Lemma 19(b), we see that β is of order τ, and since the permutation associated to δ n i is the n i -cycle (1, . . . , n i ), it follows that σ(β) = yxy -1 , and hence σ(α) = x, where α = ρ -1 βρ. In particular, B n /Γ 3 (P n ) has elements of order τ if and only if S n does.

One consequence of Theorem 3 is the classification of the isomorphism classes of the finite cyclic subgroups of B n /Γ 3 (P n ). Using Lemma 19(b), we may also caracterise the isomorphism classes of the finite Abelian subgroups of B n /Γ 3 (P n ) in a manner similar to that of B n /Γ 2 (P n ) given in [START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF]Theorem 6].

COROLLARY 24. Let n ≥ 3. Then there is a one-to-one correspondence between the isomorphism classes of the finite Abelian subgroups of B n /Γ 3 (P n ) and those of the finite Abelian subgroups of S n whose order is relatively prime with 6.

Proof. The proof is similar to that of [START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF]Theorem 6], using Lemma 19(b) and the finite-order elements of the form (24) constructed in the proof of Theorem 22.

To end this section, we prove Theorem 4, which says that if n ≥ 5, two finite-order elements of B n /Γ 3 (P n ) are conjugate if and only if their associated permutations are conjugate.

Proof of Theorem 4. Let n ≥ 5, and let α and β be two finite-order elements of B n /Γ 3 (P n ) whose associated permutations have the same cycle type. Conjugating β in B n /Γ 3 (P n ) if necessary, we may suppose that β is as in the proof of Theorem 3, in particular, it is of order τ = lcm(n 1 , . . . , n t ), where gcd(n i , 6) = 1 for all i = 1, . . . , t, and σ(β) is equal to the permutation given in (27). To complete the proof of the theorem, it suffices to show that if α ∈ B n /Γ 3 (P n ) is any finite-order element such that σ(α) has the same cycle type as σ(β), then α and β are conjugate in B n /Γ 3 (P n ). Conjugating α if necessary, we may suppose further that σ(α) = σ(β). Since Ker (σ) = P n /Γ 3 (P n ) is torsion free by Lemma 11(a), it follows that α is also of order τ.

Let B ′ be the union of the elements of the basis B defined in (17) and their inverses. From ( 8) and ( 16), β acts on B ′ by conjugation, and if βα i,j,k β -1 = α -1 r,s,t for some 1 ≤ i < j < k ≤ n and 1 ≤ r < j < s ≤ t, then either

i = n 1 + • • • + n m + 1 < j ≤ n 1 + • • • + n m+1 < k for some 0 ≤ m < t, or i < j = n 1 + • • • + n m + 1 < k ≤ n 1 + • • • + n m+1
for some 1 ≤ m < t. From this and the fact that n l is odd for all l = 1, . . . , t, we see that under this action, the orbits of α i,j,k and α -1 i,j,k are disjoint for all 1 ≤ i < j < k ≤ n, and that the orbit of α -1 i,j,k is obtained by taking inverses of the elements of the orbit of α i,j,k . We then choose a transversal b i,1 , b -1 i,1 1 ≤ i ≤ q for this action on B ′ , where for 1 ≤ i ≤ q, b i,1 ∈ B, and we let s i denote the length of the orbit of b i,1 . Observe that s i divides τ. Then B ′′ = b i,j 1 ≤ i ≤ q and 1 ≤ j ≤ s i is a basis of Γ 2 (P n )/Γ 3 (P n ), where:

b i,j = β j-1 b i,1 β -(j-1) for all 1 ≤ i ≤ q and 1 ≤ j ≤ s i .

As in the proof of Proposition 21(a), let f : B n /Γ 3 (P n ) -→ B n /Γ 2 (P n ) be the projection given in (13) with j = 3, and let σ ′ : B n /Γ 2 (P n ) -→ S n be the homomorphism given in (6

) with k = 2. Since Ker ( f ) = Γ 2 (P n )/Γ 3 (P n ) is torsion free, f (α) and f (β) are elements of B n /Γ 2 (P n ) of order τ. Further, σ = σ ′ • f , so σ ′ ( f (α)) = σ ′ ( f (β)), and applying [GGO1, Theorem 5], there exists ξ ∈ B n /Γ 2 (P n ) such that ξ f (α)ξ -1 = f (β). Since f is surjective, there exists ξ ′ ∈ B n /Γ 3 (P n ) such that f (ξ ′ αξ ′-1 ) = f (β). So con- jugating α if necessary, there exists θ ∈ Γ 2 (P n )/Γ 3 (P n ) such that θβ = α.
It suffices to show that there exists Ω ∈ Γ 2 (P n )/Γ 3 (P n ) such that ΩαΩ -1 = β, or equivalently that: θ = (βΩβ -1 )Ω -1 , (29) using the fact that θ commutes with Ω. Let θ = ∏ 1≤i≤q 1≤j≤s i b r i,j i,j , and Ω = ∏ 1≤i≤q 1≤j≤s i b

x i,j i,j , where r i,j , x i,j ∈ Z for all 1 ≤ i ≤ q and 1 ≤ j ≤ s i . Since θβ = α, the elements β and θβ are of order τ, and so:

1 = (θβ) τ = θ(βθβ -1 ) • • • (β τ-1 θβ -(τ-1) )β τ = θ(βθβ -1 ) • • • (β τ-1 θβ -(τ-1) ). ( 30
)
From the construction of the basis B ′′ , it follows from (30) that:

1 = θ(βθβ -1 ) • • • (β τ-1 θβ -(τ-1) ) = ∏ 1≤i≤q ∏ 1≤j≤s i b i,j
τ(∑ 1≤j≤s i r i,j )/s i , from which we conclude that ∑ 1≤j≤s i r i,j = 0 for all 1 ≤ i ≤ q. In a similar manner, (29) may be written as:

∏ 1≤i≤q 1≤j≤s i b r i,j i,j = ∏ 1≤i≤q 1≤j≤s i b x i,j-1 -x i,j i,j
, where the index j -1 of x i,j-1 is taken modulo s i . So for all i = 1, . . . , q, we obtain a system of equations x i,j-1x i,j = r i,j for all 1 ≤ j ≤ s i that is subject to the compatibility condition ∑ 1≤j≤s i r i,j = 0, and it may be seen easily that each such system admits a solution. Using (29), we conclude that α and β are conjugate as required.

5 Some examples with a small numbers of strings

In this section, we study a couple of examples where the number of strings is small. In Section 5.1, we determine, up to conjugacy, the almost-Bierberbach subgroups of B 3 /Γ 3 (P 3 ) that contain P 3 /Γ 3 (P 3 ), and we identify them using the classification of [De]. In Section 5.2, we calculate explicitly δ 5 5 in Γ 2 (P 5 )/Γ 3 (P 5 ) in terms of the basis B. This example illustrates the computational difficulties that we encounter with respect to the constructions of Proposition 21 and Theorem 22. We finish the paper with a remark concerning the study of the quotients B n /Γ k (P n ) for k > 3.

5.1 Some almost-Bieberbach subgroups of B 3 /Γ 3 (P 3 )

In this section, we describe the almost-Bieberbach groups σ -1 (H)/Γ 3 (P 3 ), where H is a subgroup of S 3 . Recall that representatives of the conjugacy classes of subgroups of S 3 are given by {Id}, (1, 2) , (1, 2, 3) and S 3 . As we shall see now in Theorem 25, these groups are of dimension 4 and their holonomy group is H, and that any subgroup of B 3 /Γ 3 (P 3 ) containing P 3 /Γ 3 (P 3 ) is in fact of the form σ -1 (H)/Γ 3 (P 3 ), where H is a subgroup of S 3 . THEOREM 25. (a) Let K be a subgroup of B 3 /Γ 3 (P 3 ) that contains P 3 /Γ 3 (P 3 ). Then K is conjugate to σ -1 (H)/Γ 3 (P 3 ), where H is one of the subgroups {Id}, (1, 2) , (1, 2, 3) or S 3 of S 3 . (b) Consider the four subgroups of S 3 given in part (a). (i) If H = {Id}, the group σ -1 (H)/Γ 3 (P 3 ) = P 3 /Γ 3 (P 3 ) has a presentation whose generators are a = A 1,3 , b = A 2,3 , c = A 1,2 and d = [A 1,2 , A 2,3 ], and that are subject to the following relations:

By Remarks 12(a), the groups of the form σ -1 (H)/Γ 3 (P 3 ) described in Theorem 25 are almost-crystallographic. Further, since Ker (σ) is torsion free by Lemma 11(a) and the torsion of S 3 divides 6, it follows from Theorem 2 that these groups are also almost-Bieberbach. Using Theorem 25, we now identify these groups with those given in the classification of 4-dimensional almost-Bieberbach groups with 2-step nilpotent subgroup given in [START_REF] Dekimpe | Almost-Bieberbach groups: affine and polynomial structures[END_REF]Section 7.2]. Note that by [START_REF] Gasior | Spin structures on almost-flat manifolds[END_REF]Remark 2.5], if M is an infranilmanifold whose fundamental group is E, then it is orientable if and only if the image of the representation θ F : F -→ GL(n, Z) given by equation ( 11) is contained in SL(n, Z).

COROLLARY 26. Let H be a subgroup of S 3 . Then σ -1 (H)/Γ 3 (P 3 ) is a 4-dimensional almost-Bieberbach group with 2-step nilpotent subgroup, is the fundamental group of an orientable 4-infra-nilmanifold X H , and is isomorphic to: (a) group number 1, Q = P1, given in [START_REF] Dekimpe | Almost-Bieberbach groups: affine and polynomial structures[END_REF]p. 169] [De, pp. 173-174] [De, p. 207] [START_REF] Dekimpe | Almost-Bieberbach groups: affine and polynomial structures[END_REF]p. 209]

with k 1 = k 3 = 1 and k 2 = -1 if H = {1}. (b) group number 9, Q = Cc, given in
with k 1 = 1, k 2 = -1, k 3 = k 4 = 0 and non-trivial action if H = (1, 2) . (c) group number 146, Q = R3, given in
with k 1 = k 2 = 1 and k 3 = k 4 = -1 if H = (1, 2, 3) . (d) group number 161, Q = R3c, given in
with k 1 = 1 and k 2 = k 3 = k 4 = k 5 = 0 if H = S 3 . REMARK 27. If G is a group and a, b ∈ G, then the notation used in [De] for the com- mutator [a, b], namely [a, b] = a -1 b -1 ab, is different to that used in this paper. However [a, b] ≡ [a -1 , b -1 ] modulo Γ 3 (G)
, so the difference in notation does not cause any problems in the identification of our subgroups with those of [De].

Proof of Corollary 26. By Remarks 9(b) and Proposition 10, the groups P 3 /Γ 2 (P 3 ) and Γ 2 (P 3 )/Γ 3 (P 3 ) are torsion free and their respective ranks are 3 and 1, and the Hirsch length of P 3 /Γ 3 (P 3 ) is equal to 4. So by Corollary 15, σ -1 (H)/Γ 3 (P 3 ) is a 4-dimensional almost-Bieberbach group with 2-step nilpotent subgroup. In order to identify the group σ -1 (H)/Γ 3 (P 3 ) with the corresponding group of [De] for each subgroup H of S 3 , it suffices to use Theorem 25 and to apply the classification of [START_REF] Dekimpe | Almost-Bieberbach groups: affine and polynomial structures[END_REF]Section 7.2], where the k i are as given in the statement of parts (a)-(d).

It remains to show that for each subgroup H of S 3 , the manifold X H are orientable. To see this, by the paragraph preceding the statement of Corollary 26, it suffices to show that the image of each representation θ H : H -→ GL(4, Z) lies in SL(4, Z). We exhibit the matrices θ H (α) and θ H (β), where α and β are the generators given in Theorem 25 that act on the (ordered) elements a, b, c and d. If H = {Id}, the representation θ H is clearly trivial, and the result follows. For the remaining cases, consider the elements M 1 = 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 -1 and M 2 = 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1 of SL(4, Z). If H = (1, 2) , θ H (α) = M 1 , if H = (1, 2, 3) , θ H (α) = M 2 , and if H = S 3 , θ H (α) = M 2 and θ H (β) = M 1 . REMARK 28. Since S 4 has 11 non-conjugate subgroups (including the trivial group and the whole group), then in a similar manner, we may show that there are eleven nonisomorphic almost-Bieberbach subgroups of B 4 /Γ 3 (P 4 ) of the form σ -1 (H)/Γ 3 (P 4 ) with holonomy group H, each of dimension 10 using Proposition 10, where H runs through the subgroups of S 4 .

5.2 Some explicit finite-order elements in B 5 /Γ 3 (P 5 ) By [GGO1, Corollary 4], B 5 /Γ 2 (P 5 ) possesses elements of order 3 and 5. From Theorem 2, B 5 /Γ 3 (P 5 ) does not have elements of order 3, but Theorem 22 implies that there exist elements of order 5. Further, if gcd(n, 6) = 1, elements of order n in B n /Γ 3 (P n ) may be determined explicitly using the construction given in the proof of Theorem 22 provided we are able to compute δ n n in terms of the elements of the basis B of the group Γ 2 (P n )/Γ 3 (P n ) described in (17). We now carry out this calculation in the case n = 5.

Using equation ( 7) and with the notation of Lemma 13, we start by describing the action by conjugation of α -1 5 = α -1 0,5 , where α 0,5 = σ 1 σ 2 σ 3 σ 4 , on the elements of the generating set A i,j 1≤i<j≤5 of P 5 :

α -1 5 = σ -1 4 σ -1 3 σ -1 2 σ -1 1 :                A 1,2 -→ [A 1,2 A 1,3 A 1,4 , A 1,5 ]A 1,5 A 1,3 -→ [A 1,2 A 2,3 A 2,4 , A 2,5 ]A 2,5
A 1,4 -→ [A 1,3 A 2,3 A 3,4 , A 3,5 ]A 3,5 A 1,5 -→ [A 1,4 A 2,4 A 3,4 , A 4,5 ]A 4,5 A i,j -→ A i-1,j-1 if 2 ≤ i < j ≤ 5.

(31)

From equation ( 17), the following 10 elements form a basis of Γ 2 (P 5 )/Γ 3 (P 5 ):

a 1 = [A 1,2 , A 2,3 ] , a 2 = [A 1,2 , A 2,5 ] , a 3 = [A 1,4 , A 4,5 ] , a 4 = [A 3,4 , A 4,5 ] a 5 = [A 2,3 , A 3,4 ] , b 1 = [A 1,2 , A 2,4 ] , b 2 = [A 1,3 , A 3,5 ] , b 3 = [A 2,4 , A 4,5 ] b 4 = [A 1,3 , A 3,4 ] , b 5 = [A 2,3 , A 3,5 ] , (32)
and that using (8) and ( 16), under the action by conjugation by α -1 5 (considered as an element of B 5 /Γ 3 (P 5 )), this basis splits into two orbits of length 5 of the form: In order to obtain an element α of finite order in B 5 /Γ 3 (B 5 ), by the construction of the proof of Theorem 22, it suffices to compute δ 5 5 in terms of the basis of Γ 2 (P 5 )/Γ 3 (P 5 ) given in (32). Let:

c 1 = [A 1,2 A 2,3 A 2,4 , A 2,5 ], c 2 = [A 1,5 A 1,2 A 1,3 , A 1,4 ] and c 3 = [A 1,2 A 1,3 A 1,4 , A 1,5 ].
In Γ 2 (P 5 )/Γ 3 (P 5 ), we have 16), in Γ 2 (P 5 )/Γ 3 (P 5 ) we obtain:

c 1 c 2 c 3 = b -1 1 b -1 2 b -1 3 b -1 4 b -
c 1 c 2 c 3 = [A 1,2 A 2,3 A 2,4 , A 2,5 ][A 1,5 A 1,2 A 1,3 , A 1,4 ][A 1,2 A 1,3 A 1,4 , A 1,5 ] = [A 1,2 , A 2,5 ][A 2,3 , A 2,5 ][A 2,4 , A 2,5 ][A 1,5 , A 1,4 ][A 1,2 , A 1,4 ][A 1,3 , A 1,4 ]• • [A 1,2 , A 1,5 ][A 1,3 , A 1,5 ][A 1,4 , A 1,5 ] = [A 1,2 , A 2,5 ][A 2,3 , A 3,5 ] -1 [A 2,4 , A 4,5 ] -1 [A 1,4 , A 1,5 ] -1 [A 1,2 , A 2,4 ] -1 [A 1,3 , A 3,4 ] -1 • • [A 1,2 , A 2,5 ] -1 [A 1,3 , A 3,5 ] -1 [A 1,4 , A 1,5 ] = [A 2,3 , A 3,5 ] -1 [A 2,4 , A 4,5 ] -1 [A 1,2 , A 2,4 ] -1 [A 1,3 , A 3,4 ] -1 [A 1,3 , A 3,5 ] -1 = b -1 1 b -1 2 b -1 3 b -1 4 b -1 5 .

  a 1 -→ a 2 -→ a 3 -→ a 4 -→ a 5 and b 1 -→ b 2 -→ b 3 -→ b 4 -→ b 5 .

1 5 .

 5 To see this, recall that if a, b and c are elements of a group G, we have a Witt-Hall identity [ab, c] = [a, [b, c]][b, c][a, c] [MKS, Theorem 5.1]. So using equation (

Table 1 :

 1 this subgroup is injective, and B n /Γ j-1 (P n ) also has a (normal) subgroup isomorphic to G. The first part of the statement then follows by reverse induction on j. In particular, if B n /Γ k (P n ) has p-torsion, the second part of the statement follows by taking G = Z p .Proof of Theorem 1. Let n, k ≥ 3. In order to prove that the group B n /Γ k (P n ) is almostcrystallographic, by Theorem 7, it suffices to show that it does not have finite (nontrivial) normal subgroups. Suppose on the contrary that it possesses such a subgroup H. Taking l = 2 (resp. l = 1) in Lemma 11(b), it follows that B n /Γ 2 (P n ) (resp. S ′ has 3-torsion, and since the elements of B 3 /Γ 2 (P 3 ) of order 3 are pairwise conjugate[START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF] Theorem 5], and H ′ is normal in B 3 /Γ 2 (P 3 ), the only elements of B 3 /Γ 2 (P 3 ) of order 3 are those belonging to H ′ , but this contradicts the fact that B 3 /Γ 2 (P 3 ) possesses infinitely many elements of order 3[START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF] Proposition 21]. This proves the first part of the statement of Theorem 1. For the second part, the dimension of the almost-crystallographic group B n /Γ k (P n ) is equal to the Hirsch length of P n /Γ k (P n ) given in Proposition 10, and the associated holonomy group is the quotient group B n /Γ k (P n )/P n /Γ k (P n ), which is isomorphic to S n . As in[START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF] Corollary 13], the first part of Theorem 1 may be generalised as follows: let n, k ≥ 3, and let H be a subgroup of S n . Then the group σ -1 (H)/Γ k (P n ) is an almost almost-crystallographic group whose holonomy group is H and whose dimension is equal to that of B n /Γ k (P n ). (b) Using Theorem 1, we may compute the dimension of the almost-crystallographic group B n /Γ k (P n ). In Table1, we exhibit these dimensions for small values of n and k. This gives us an idea about their growth in terms of k and n. The dimension of the almost-crystallographic group B n /Γ k (P n ) for n = 3, 4, 5, 6 and k = 2, 3, 4, 5.

	We now prove Theorem 1.

n ) has a normal subgroup H ′ (resp. H) isomorphic to H. Hence the groups H, H ′ and H are either isomorphic to S n , or to the alternating subgroup A n , or if n = 4, to Z 2 ⊕ Z 2 . If n ≥ 4, or if n = 3 and H = S 3 , it follows that H ′ possesses 2-torsion, but this contradicts [GGO1,

Theorem 2]

. It remains to analyse the case n = 3 and H = A 3 ∼ = Z 3 . Then H

Acknowledgements

Work on this paper started in 2014, and took place during several visits to the Departamento de Matemática, Universidade Federal de Bahia, the Departamento de Matemática do IME -Universidade de São Paulo and to the Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, Université de Caen Normandie. The first two authors were partially supported by the FAPESP Projeto Temático Topologia Algébrica, Geométrica e Diferencial 2012/24454-8 (Brazil), by the Réseau Franco-Brésilien en Mathématiques, by the CNRS/FAPESP project n o 226555 (France) and n o 2014/50131-7 (Brazil),

(i) the α r,s,t commute pairwise and with the A i,j .

(ii) [A i,j , A l,m ] =      α i,j,m if Card {i, j, l, m} = 3 and j = l α -1 i,k,m if Card {i, j, l, m} = 3, where k = j if i = l, and k = l if j = m 1 if Card {i, j, l, m} ∈ {2, 4}. (b) The group B n /Γ 3 (P n ) has a presentation given by the generating set {σ k } 1≤k≤n-1 ∪ X n , subject to the following relations: (i) relations (a)(i) and (a)(ii) emanating from those of P n /Γ 3 (P n ).

(ii) the Artin braid relations (2), viewed in B n /Γ 3 (P n ).

(iii) the conjugacy relations described in (18) and(19). Proof. To prove parts (a) and (b), it suffices to apply [START_REF] Johnson | Presentation of groups, LMS Lecture Notes 22[END_REF]Proposition 1,p. 139] first to (9) taking k = 2, and then to (6) taking k = 3 using also part (a).

Torsion elements and conjugacy classes in B n /Γ 3 (P n )

In this section, we prove Theorems 3 and 4. We start by proving the following lemma whose result generalises [START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF]Theorem 3(a)].

. . , n t be integers greater than or equal to 2 for which ∑ t i=1 n i ≤ m, and let ζ :

)) be the homomorphism ι (resp. ι) given in the statement. By equation (4), ι n,m restricts to an injective homomorphism ι n,m | P n : P n -→ P m given by ι n,m | P n (A i,j ) = A i,j for all 1 ≤ i < j ≤ n. For n ≤ q ≤ m -1, let ι q : P q -→ P q+1 denote the homomorphism ι q,q+1 P q . We claim that the homomorphism

, where the ho- momorphism ι q Γ k (P q )/Γ k+1 (P q ) : Γ k (P q )/Γ k+1 (P q ) -→ Γ k (P q+1 )/Γ k+1 (P q+1 ) is induced by ι q for all n ≤ q ≤ m -1, it suffices to prove that ι q Γ k (P q )/Γ k+1 (P q ) is in- jective. To do so, consider the Fadell-Neuwirth short exact sequence 1 -→ Ker (p) -→ P q+1 p -→ P q -→ 1, where p is the homomorphism given geometrically by forgetting 13 the last string. Using the presentation given in equation ( 5), we obtain two well-known facts, first that ι q is a section for p, and secondly that the resulting semi-direct product is almost direct, i.e. the action induced by P q on the Abelianisation Ker (p) /Γ 2 (Ker (p)) of Ker (p) is trivial. It follows from [FR1, Theorem 3.1] that the induced sequence

Γ k (P q )/Γ k+1 (P q ) -→ 1 is split short exact, and that the homomorphism ι q Γ k (P q )/Γ k+1 (P q ) is a section for p. In particular, ι q Γ k (P q )/Γ k+1 (P q ) is injective, and the claim follows. Now consider the following commutative diagram of short exact sequences:

where the rows are the short exact sequences given by the central extension ( 13). The statement of part (a) is then a consequence of applying the 5-Lemma to this diagram, the above claim, induction on k, and the fact that ι :

Let n 1 , . . . , n t be integers greater than or equal to 2 such that

i=1 n i denote the usual homomorphism that to a braid associates its permutation, and let B n 1 ,...,n t denote the corresponding mixed braid group, namely the preimage under σ of the subgroup

..,n t denote the embedding of B n i into the i th factor of B n 1 ,...,n t . Note that ζ is the equal to the following composition:

..,n t -→ B n i /Γ k (P n i ) be the composition of the projection onto the i th factor of B n 1 ,...,n t , followed by the canonical projection

From the constructions of ϕ i and ψ i , we see that

for all 1 ≤ i ≤ t, and so the composition

injective, and the composition

, which is the homomorphism ζ of the statement, may be seen to be injective using part (a) and the injectivity of the homomorphism B n 1 ,...,n t ֒-→ B ∑ t i=1 n i .

(

For each of the remaining groups, a presentation is obtained by adding extra generators and relations to those of P 3 /Γ 3 (P 3 ) given in (i). In each case, we will just indicate these extra generators and relations.

(ii) If H = (1, 2) , the group σ -1 (H)/Γ 3 (P 3 ) has a presentation with one extra generator α = σ 1 and five extra relations:

(1)

1 and five extra relations:

(1)

(iv) If H = S 3 , the group σ -1 (H)/Γ 3 (P 3 ) = B 3 /Γ 3 (P 3 ) has two extra generators α = σ 2 σ 1 and β = σ 1 and eleven extra relations:

(1)

Proof. (a) First suppose that K 1 and K 2 are subgroups of B 3 /Γ 3 (P 3 ) that contain P 3 /Γ 3 (P 3 ), and for which σ(K 1 ) = σ(K 2 ). We claim that K 1 = K 2 . To see this, let x ∈ K 1 . Since σ(K 1 ) = σ(K 2 ), there exists y ∈ K 2 such that σ(x) = σ(y), and so there exists z ∈ P 3 /Γ 3 (P 3 ) such that y

, where H = σ(K) is a subgroup of S 3 . Since all such subgroups are normal in S 3 , with the exception of those of order 2, to complete the proof of part (a), it suffices to show that if K 1 and K 2 are subgroups of B 3 /Γ 3 (P 3 ) that contain P 3 /Γ 3 (P 3 ), and for which σ(K 1 ) = (1, 2) and σ(K 2 ) = τ, where τ ∈ {(1, 3), (2, 3)}, then K 1 and K 2 are conjugate in B 3 /Γ 3 (P 3 ). To see this, let τ ′ ∈ S 3 be such that τ ′ (1, 2)τ ′-1 = τ, and let τ ′ ∈ B 3 /Γ 3 (P 3 ) be such that σ( τ ′ ) = τ ′ . Then τ ′ K 1 τ ′-1 contains Ker (σ) = P 3 /Γ 3 (P 3 ), and σ( τ ′ K 1 τ ′-1 ) = σ(K 2 ) = τ . The result then follows from the first part of the proof. (b) The case (i) follows from Proposition 18(a). For cases (ii)-(iv) we apply the techniques of [START_REF] Johnson | Presentation of groups, LMS Lecture Notes 22[END_REF]Proposition 1,p. 139] to the extension

The extra relations involving the action by conjugacy of H on the kernel follow from equations ( 18) and ( 19). 5 α -5

The idea of the above computation is first to eliminate all of the terms involving α 5 using equation ( 31), and then to move each of the underlined terms to the left oneby-one in order to create the word A 1,2 A 1,3 A 1,4 A 1,5 A 2,3 A 2,4 A 2,5 A 3,4 A 3,5 A 4,5 , which we know to be the full twist braid α 5 5 in P 5 , and so in P 5 /Γ 3 (P 5 ). In doing so, we introduce basis elements of Γ 2 (P 5 )/Γ 3 (P 5 ) given by (32), perhaps written in one of the forms of equation ( 16). Note that the result is coherent with that of Proposition 21, and with the notation of that proposition, we have m. Setting θ = ∏ 2 i=1 ∏ 5 j=1 b r i,j i,j , where b 1,j = a j , b 2,j = b j and r i,j ∈ Z for all i = 1, 2 and j = 1, . . . , 5, using the notation of Proposition 21(b), and applying the construction of Theorem 22, by ( 33) and ( 26), we have m 1,j = 0, m 2,j = -1, and θδ 5 is of order 5 if and only if ∑ 5 j=1 r 1,j = 0 and ∑ 5 j=1 r 2,j = 1. So to obtain an explicit element θ, it suffices to pick any integers satisfying these two relations. For example, if r 2,1 = 1 and r 2,2 = • • • = r 2,5 = r 1,1 = • • • = r 1,5 = 0 then the element b 1 δ 5 = [A 1,2 , A 2,4 ](σ 4 σ 3 σ -1 2 σ -1 1 ) is of order 5 in B 5 /Γ 3 (P 5 ). REMARK 29. The study of the quotients B n /Γ k (P n ) for k > 3 does not appear to be an easy problem. Our approach requires a description of a basis of Γ k (P n )/Γ k (P n ). For example, if n = 3 and k = 4, a long and arduous calculation show that a basis of the group Γ 3 (P 3 )/Γ 4 (P 3 ), which is free Abelian of rank 2 by Remark 9(b), is given by {[[A 1,2 , A 2,3 ], A 1,3 ], [[A 2,3 , A 1,3 ], A 1,2 ]}. Since S 3 has 4 subgroups (up to isomorphism), arguing as in Remark 28, we may exhibit 4 non-isomorphic almost-Bieberbach subgroups of B 3 /Γ 4 (P 3 ) of dimension 6 of the form σ -1 (H)/Γ 4 (P 3 ) with holonomy group H, where H is a subgroup of S 3 and σ is as in (3).