Y Tian
email: tian@ensta.frb.cottébenjamin.cotte@ensta.fra.chaigneantoine.chaigne@ensta.fr

Wind Turbine Noise Modelling Based on Amiet's Theory

. In this paper, two main broadband noise mechanisms, namely trailing edge noise and turbulent inflow noise, are examined in detail using frequency domain noise prediction models based on Amiet's analytical theory. Improvements are proposed to adapt the original model to wind turbines . First, a wall pressure spectral model proposed recently by Rozenberg, Robert and Moreau that considers an adverse pressure gradient flow (APG) is applied to the trailing edge noise model. This APG model leads to a significant increase in the sound pressure level. Second, an empirical airfoil thickness correction is proposed in the turbulent inflow noise model, which introduces a level reduction that depends on leading-edge thickness, frequency and the ratio of the turbulence integral length scale to the blade chord. The proposed model also includes Doppler effect for rotating blades. Calculation results are validated by comparison with wind tunnel experimental data and with measurements for a full size wind turbine. This model is also used to quantify the amplitude modulation that can be a source of annoyance in the vicinity of a wind farm.

Introduction

Nowadays, wind energy application grows rapidly as a renewable, clean energy. While profiting from wind energy, the noise produced by a modern wind turbine becomes a main problem for the neighbourhood of a wind farm. The annoyance due to amplitude modulation (swish) and to the low frequency noise is especially a concern for the people living close to a wind farm [START_REF] Styles | Monitoring and mitigation of low frequency noise from wind turbines to protect comprehensive test ban seismic monitoring stations[END_REF].

Two main noise sources from modern wind turbines are turbulent inflow noise and trailing edge noise [START_REF] Oerlemans | Location and quantification of noise sources on a wind turbine[END_REF]. The former is induced by the scattering of atmospheric turbulence fluctuation at the leading edge of the blade, sometimes referred to as the leading edge noise; the latter is due to the turbulent boundary layer passing by a sharp edge like the trailing edge. Both noise mechanisms are broadband, and depend on the wind turbine size and on the atmospheric turbulence conditions; the total noise level can be dominated by one or the other.

The goal of this paper is to propose a theoretical noise prediction model, that can be used to study different phenomena related to wind turbine noise (amplitude modulation, wind shear effects, etc). The model is also intended to be coupled to an atmospheric propagation model for the prediction of the far field sound pressure level (SPL). In this paper, a well known analytical model proposed by [START_REF] Amiet | Acoustic radiation from an airfoil in a turbulent stream[END_REF][START_REF] Amiet | Noise due to turbulent flow past a trailing edge[END_REF] is chosen for the noise prediction. The original model is based on a flat plate with semi-infinite chord. As an improvement, an empirical correction on airfoil thickness for turbulent inflow noise is applied. Results show that increasing airfoil reduces the turbulent inflow noise. Furthermore, a wall pressure spectra model proposed by [START_REF] Rozenberg | Wall-pressure spectral model including the adverse pressure gradient effects[END_REF], which considers an adverse pressure gradient (APG) flow is validated and applied to Amiet's trailing edge noise model. This wall pressure model will increase the sound pressure level to a certain level that depends on the flow conditions. The paper is organized as follows: in section 2, a brief introduction to the Amiet's analytical model is given; in section 3, two main improvements, namely, airfoil leading edge thickness and the APG model for the trailing edge noise are explained in detail; section 4 shows the validation for the improved models against the measurements of fixed airfoils; section 5 contains the results from a real size wind turbine calculation validated against the measurements; the conclusions and acknowledgements are in section 6.

Amiet's Analytical model for turbulent inflow noise and trailing edge noise

In this section, Amiet's analytical models for both noise mechanisms are briefly introduced. Section 2.1 is devoted to turbulent inflow noise and section 2.2 to trailing edge noise.

Amiet's model for turbulent inflow noise

An airfoil in a turbulent flow experiences a fluctuation lift loading (unsteady upwash) which will result in the generation of sound. Considering the complex nature of turbulence and its interaction with the airfoil, many previous researchers proposed their models based on some common assumptions [START_REF] Howe | A review of the theory of trailing edge noise[END_REF]: 1. the incoming turbulence fluctuation is considered to be small compared to the mean flow velocity; 2. the interaction between an airfoil and the turbulent flow is inviscid so that the problem is reduced to solving linearized Euler equations. In Amiet's model [START_REF] Amiet | Acoustic radiation from an airfoil in a turbulent stream[END_REF], the turbulent gust properties are assumed unchanged while it is convected by the mean flow, and its velocity fluctuation is represented in terms of chord-wise and span-wise wave number, here, and respectively, (see Figure 1). The airfoil is modelled as a flat plate with no thickness, where the chord is and the span is . The model further assumes that the plate has semi-infinite chord and infinite span extension. If we write the incident gust in the Fourier domain in the form as:

, (1) then the fluctuating velocity field is deduced as:

(2) where is the velocity potential and is the Mach number, . Substituting Equation (2) into the convective wave equation we obtain:

̅ ̅ ̅ where ̅ ̅ ̅ ̅ ̅ ̅
is half chord, is the acoustic wave number, and

̅ ̅ ̅ ̅ ̅
with the free stream velocity. Solving for and using the relationship between acoustic pressure and velocity potential, we can finally get the pressure fluctuating on the airfoil surface as [START_REF] Rozenberg | Modélisation analytique du bruit aérodynamique à large bande des machines tournantes: utilisation de calculs moyennés de mécanique des fluides[END_REF]:

√ ̅ , (3)
with .

Assuming that the source behaves like a dipole source, and applying Lighthill analogy, the power spectrum density of far field acoustic pressure is obtained from Equation (3) as [START_REF] Amiet | Acoustic radiation from an airfoil in a turbulent stream[END_REF][START_REF] Rozenberg | Modélisation analytique du bruit aérodynamique à large bande des machines tournantes: utilisation de calculs moyennés de mécanique des fluides[END_REF]:

⃗ ∫ [ ] (4)
where is the air density, and is the sound velocity. ⃗ is the observer coordinate, is the airfoil response function, , is a modified distance between the source and observer.

is the turbulence energy spectrum. If we assume a infinite span, , the function could be evaluated by a Dirac function, and thus Equation (4) reduces to:

⃗ (5)
Equation ( 5) is referred to as the simplified equation under large aspect ratio assumption.

Amiet's model for trailing edge noise

Trailing edge noise is generated by the scattering of the turbulent boundary layer passing by the trailing edge. Since the origin is the fluctuation of turbulent boundary layer, the derivation is similar to that of turbulent inflow noise, here only the final result, the power spectrum density of far field sound is cited as [START_REF] Amiet | Noise due to turbulent flow past a trailing edge[END_REF][START_REF] Roger | Back-scattering correction and further extensions of Amiet's trailing-edge noise model. Part 1: theory[END_REF]:

⃗ ∫ [ ] (6)
⃗ (7)
which are the exact formula and simplified formula under large aspect ratio assumption respectively. is the power spectral density of wall pressure, and is the spanwise correlation length.

Model improvement: adverse pressure gradient wall pressure model and leading edge thickness correction

3.1 Airfoil thickness correction for turbulent inflow noise [START_REF] Roger | Extensions and limitations of analytical airfoil broadband noise models[END_REF] studied the influence of airfoil leading edge thickness on far field sound pressure level. They show that an increase of leading edge thickness leads to the reduction of turbulent inflow noise. We propose here an empirical correction based on the data shown in Figure 6 of their publication. The reduction level in dB is calculated by linear interpolation based on the these data:

(8)
where is the airfoil maximum thickness, and is the turbulent integral length scale. The subscription stands for the value of reference experimental data, which is: and .

Adverse pressure gradient wall pressure model for trailing edge noise

In Amiet's trailing edge noise model, one of the most important input parameters is the spectrum of fluctuating wall pressure. An accurate estimation of this spectrum can be done with direct numerical simulation (DNS) or large eddy simulation (LES), but it is really time and cost consuming. [START_REF] Amiet | Acoustic radiation from an airfoil in a turbulent stream[END_REF], as a completion of his model, proposed an experiment based scaling formula, that is, by scaling the measured wall pressure spectrum with certain variables, all the spectra should collapses to a single curve. Later, [START_REF] Goody | Empirical spectral model of surface pressure fluctuations[END_REF] also proposed an improved wall pressure spectrum model that considers Reynolds effect. However, all these scaling models are based on zero gradient flow condition. For an airfoil, at trailing edge, it is often the case that an APG flow exists. [START_REF] Rozenberg | Wall-pressure spectral model including the adverse pressure gradient effects[END_REF] proposed a further improved model based on Goody's model, and takes into account the APG effect. He suggested that the wall pressure spectrum can be presented as:

[ ] [ ] ̃ [ ̃ ] [ ̃] (9)
where the relevant parameters can be calculated as:

̃ [ ] √ [ ],
with the boundary layer displacement thickness, the boundary layer thickness, the momentum thickness, the maximum wall shear stress along the airfoil surface, the ratio of the outer to inner boundary layer time scale, and the strength of the adverse pressure gradient. The model is validated in the original paper of [START_REF] Rozenberg | Wall-pressure spectral model including the adverse pressure gradient effects[END_REF] against 5 different experimental results. In this paper, it is further validated against the data from DTU report [START_REF] Bertagnolio | Low noise airfoil-Final report[END_REF]. The experiments are launched in a wind tunnel with a symmetric airfoil NACA0015. Test velocities are 30 m/s, 40 m/s, 50 m/s, and the angle of attack is 0, 4, 8, and 12 degree. The chord length c is 0.9 m, and the microphone at which wall pressure is recorded is located at x/c = 0.894. Since no detailed boundary layer parameters are provided from the experiment, XFOIL is used to calculate the necessary input parameters for the APG model. For comparison, the test velocity of 30 m/s and 50 m/s at zero angle of attack case are considered. The comparison result is shown in Figure 2.

Model validations against fixed airfoil measurements

Turbulent inflow noise validation

The model is compared to the experimental data from [START_REF] Paterson | Acoustic radiation and surface pressure characteristics of an airfoil due to incident turbulence[END_REF]. Experiments are conducted in a wind tunnel using NACA0012 airfoil. The airfoil dimension is 0.23 m chord and 0.53 m span. The angle of attack is set to zero. The test velocities are 40, 60, 90, 120 and 165 m/s respectively, which corresponds to a free stream Mach number of approximately 0.12, 0.18, 0.27, 0.36, 0.5. The turbulence is generated by a bi-planar grid with a mesh size of 13.3 cm, the resulting turbulence intensity is of the order 4% ~5%, and the span-wise correlation length scale is around 3 cm. The noise is measured at 2.25 m radius from the The calculation results for various test velocities using large aspect ratio formulation are shown in Figure 3, with and without the leading edge thickness correction. The correction is most necessary for low Mach number at higher frequency range.

Trailing edge noise validation

For the validation of trailing edge noise, the experimental data from [START_REF] Brooks | Trailing edge noise prediction from measured surface pressures[END_REF] are considered. The test airfoil is again NACA 0012, with a chord of 0.61 m and a span of 0.46 m. The test velocity chosen for the current validation is 69.5 m/s, and the angle of attack is zero. The reference location is 1.8 cm upstream of the trailing edge and the receiver is located 1.22 m away directly above the trailing edge in the mid-span. The span-wise correlation length that is needed for Equation ( 6) and ( 7) is determined by Corcos's model [START_REF] Moreau | Effect of airfoil aerodynamic loading on trailing-edge noise sources[END_REF]:

with being an experimental constant, and being the convective velocity.

The results for wall pressure spectrum and far field SPL are shown in Figure 4. The results from Goody's wall pressure spectrum model is also plotted for comparison with the APG model. Again we can see that the adverse pressure gradient tends to increase the far field SPL, and that the APG model provides better results than Goody's model. In section 2.1, the power spectral density is simplified under the assumption of large aspect ratio. For the calculation of a wind turbine, since each blade is cut in segments on which the noise models are applied individually, it is not guaranteed that this assumption is valid for all the segments. Thus it is necessary to examine the validity of the large aspect ratio approximation for a wind turbine.

A first test on "segment aspect ratio" is carried out with a fixed NACA0012 airfoil. The chord is set to 0.5 m, and the span is set to 0.25 m, 0.5 m, 1.5 m and 2.5 m respectively, Mach number is 0.2. The observer is located in the mid-span plane 2 m above the airfoil. The results for trailing edge noise are shown in Figure 5, from which we can see that when the aspect ratio is more than 3, the simplified formula provides really close results compared to those from the exact formula. When the aspect ratio is less than 3, the main discrepancy between the two appears for frequency less than 600Hz. The integration over ̅ in Equation ( 6) involves dealing with sub-critical gusts, that exists when ̅ ̅ ̅ . A set of plots of with respect to airfoil response function with respect to ̅ are shown in Figure 6. The parameters are: chord = 0.13 m, span = 0.13 m, Mach number = 0.05 and 0.3 respectively, the test frequency is 1000 Hz. The angles indicated in the figures are the observer angles in the mid-span plane, where 0 o means downstream in the airfoil plane and 90 o means in direction normal to the airfoil. We observe that the airfoil response function decreases rapidly for subcritical cases when the normalized span-wise wave number increased. However, gusts with ̅ close to unity can have a significant effect on the final results.

Exact formular Large Aspect Ratio simplification

To understand the importance of the subcritical gusts for different frequencies, the due to all the super-critical gusts and all the sub-critical gusts respectively, as well as the sum of the two are plotted in Figure 7. We can see that the sub-critical gusts are dominant at low frequency (up to 100Hz), and decrease very fast at higher frequency range. Since for the aspect ratio of 1, the simplified formula could provide satisfying results for most of audible frequency range (200Hz and above), it will be used to calculate wind turbine noise in the next section, even thought the sub-critical gusts are neglected consequently. Doppler effect due to blade rotation is taken into account by relating the PSD of a moving source to a fixed source at the same instant location [START_REF] Schlinker | Helicopter Rotor Trailing Edge Noise[END_REF]:

where is the Doppler factor defined by , is the source frequency and is the observer measured frequency.

Application on a wind turbine and validation against measurements

The comparison is made for a Bonus 300kW wind turbine using data from [START_REF] Zhu | Modelling of aerodynamically generated noise from wind turbines[END_REF]. The main parameters for this type of wind turbine and meteorology conditions are given in Table 1. Wind shear effect is considered in the calculation by taking an experimental log law relation:

( )
where is the velocity at height , is the reference velocity at reference height , taken as the hub height here, and [START_REF] Zhu | Modelling of aerodynamically generated noise from wind turbines[END_REF])

(10)
is the ground roughness. The turbulence intensity and turbulence integral length scale are calculated by the relations given in [START_REF] Zhu | Modelling of aerodynamically generated noise from wind turbines[END_REF], as: and where is the same as defined in Equation 10. The turbulence energy spectrum is calculated using the isotropic Von Kármán spectrum. Plot of the energy spectrum for integral length scale = 100 m with various turbulence intensities and a turbulence intensity = 0.1 with various integral length scales are shown in Figure 8. The calculation procedures are the following: first, each wind turbine blade are artificially cut in several segments, here, 10 segments of 1.5m length for each blade, so that the maximum aspect ratio remains no larger than 1. According to the flow conditions at each segment, the boundary layer parameters required by the APG wall pressure model are calculated by XFOIL (version 6.96) and stored in advance. The noise models for turbulent inflow noise and trailing edge noise are then applied for each segment, with thickness correction and Doppler effect considered. The final total noise perceived by the observer is obtained by logarithm summation over all the segments.

Figure 9 shows the sound power level from the calculation compared to the measurement. The results are shown in 1/3-octave bands, A-weighted and averaged over one rotating period. It is needed to say that the measured data includes all the noise mechanisms (stall noise, trailing edge bluntness noise, etc) which are not considered by the models. According to our model, trailing edge noise is dominant over turbulent inflow noise for this type of wind turbine under these meteorological conditions. The agreement between the calculation and the measurement

 = 50m  = 100m  = 150m
is acceptable in the frequency range of 600 Hz to 8000 Hz, but at lower frequency range, the results shows up to 20 dB differences. Turbulent inflow noise level is rather low, which might be due to the large turbulent integral length scale, that ranges between 120 m to 180 m. On the right figure, it is shown that a decrease of turbulent integral length scale will increase the noise level for a fixed turbulence intensity. The comparison for trailing edge noise using APG wall pressure model and Goody's wall pressure model is shown in Figure 10. The results are shown in 1/3-octave bands. It is clear that an adverse pressure flow condition greatly increases the trailing edge noise. By using APG model, the results are much closer to the measurements. Figure 11 shows the thickness correction for turbulent inflow noise. Under this atmospheric turbulence condition, more precisely, when the turbulent integral length scale is much larger than the size of blade chord, this correction is not significant. Figure 12 shows the total SPL variation during one rotating period, the observer is at a 40 m distance from the wind turbine. From the result we can see that when the observer is upwind or downwind direction (corresponds to and ), the perceived SPL is almost constant; but for an observer located in the rotor plane direction (corresponds to and ), even thought the overall SPL is much less, but the variation during one period reaches up to 6 dB. These trends agree with the results from [START_REF] Lee | Numerical modelling of wind turbune aerodynamic noise in the time domain[END_REF].

The directional feature of overall SPL averaged over one rotating period is shown in Figure 13. The wind is coming from the left to the right. The pattern shows a dipole shape, as was shown in many previous work [START_REF] Zhu | Modelling of aerodynamically generated noise from wind turbines[END_REF][START_REF] Lee | Numerical modelling of wind turbune aerodynamic noise in the time domain[END_REF], with a bit lower level in the rotor plane direction, and a higher level in the wind direction.

Conclusions

Amiet's analytical model for turbulent inflow noise and trailing edge noise are extended by considering an APG wall pressure model and a leading edge thickness correction respectively. The extended models are first validated against fixed airfoil measurements, then applied to a full size wind turbine. APG wall pressure model greatly improved the accuracy for trailing edge wind noise prediction, while for the turbulent inflow noise, the thickness correction does not seem significant when the atmospheric turbulent length scale is very large. The results of total SPL as a function of blade position are expected to give an explanation for the 'swish' effect. When the observer is located at rotor plane direction, the amplitude modulation causes a strong 'swish'; while if the observer is located at upwind or downwind direction, the effect is hardly noticeable.

In the future, the model will need to be compared to a more detailed wind turbine experimental campaign, since the comparison with data from the literature is not completely satisfying. Also, the model will be coupled to a propagation model and a more realistic model of the atmospheric boundary layer will be considered. Finally, other noise mechanisms that can give rise to rapid time fluctuations such as stall noise will be investigated.

Fig. 1

 1 Fig.1 Geometry and notations used in the turbulent inflow noise model.

Fig. 2

 2 Fig. 2 Wall pressure spectrum measured by Bertagnolio (2012) (circles), and modeled by the APG model(solid line) and Goody's model(dashed line) respectively. Left, U = 30m/s; right, U = 50m/s.

  the mid-span. Here the data from the microphone located directly above the leading edge are used.

Fig. 3

 3 Fig. 3 SPL of turbulent inflow noise. Symbols: measurement from Paterson and Amiet (1976); solid line: without thickness correction; dashed line: with thickness correction.

Fig. 4

 4 Fig. 4 Comparison between APG model and Goody's model against experiment. Left: wall pressure spectrum comparison; right, far field SPL comparison.

Fig. 5

 5 Fig.5 Trailing edge SPL calculated by the exact and simplified formula for difference aspect ratio respectively.

Figu. 6

 6 Figu.6 Airfoil response function as a function of ̅ and far field trailing edge noise. Blue line: super-critical gust, red line: sub-critical gust.

   Total SPL SPL due to super-critical gusts SPL due to sub-critical gusts

Fig. 8

 8 Fig. 8 Von Kármán turbulence energy spectra for different integral length scales and turbulence intensities.

Fig. 9

 9 Fig. 9 Left: Comparison of turbulent inflow noise and trailing edge noise models against measurement. Right: influence of turbulent integral length scale on the turbulent inflow noise.

Fig. 10

 10 Fig.10 Comparison between APG model and Goody model for trailing edge noise.

Fig

  Fig.11 Turbulent inflow noise with/without leading edge thickness correction.

Figure 13

 13 Figure 13 Directivity of overall SPL in dB(A).

Table 1 Bonus 300kW wind turbine main parameters

 1

	Rotor radius	Tower Height	Rotor angular speed	Airfoil profile	Chord length	Wind speed (10m height)	Observer location	Ground roughness
	15.5 m	31m	3.7 rad/s	NACA 632xx	0.5m-1.5m	8 m/s	40m down wind	30mm

Acknowledgements

The authors would like to thank Michel Roger and Yannick Rozenberg from the Laboratoire de Mécanique des Fluides et d'Acoustique (LMFA) of the Ecole Centrale de Lyon for useful discussions regarding Amiet's theory.


Thu Trang Lê

Abdourrahmane M Atto
email: abdourrahmane.atto@univ-savoie.fr

Emmanuel Trouvé
email: emmanuel.trouve@univ-savoie.fr

SCHATTEN MATRIX NORM BASED POLARIMETRIC SAR DATA REGULARIZATION. APPLICATION OVER CHAMONIX MONT-BLANC

Keywords: PolSAR, filtering, geometric regularity

The paper addresses the filtering of Polarimetry Synthetic Aperture Radar (PolSAR) images. The filtering strategy is based on a regularizing cost function associated with matrix norms called the Schatten p-norms. These norms apply on matrix singular values. The proposed approach is illustrated upon scattering and coherency matrices on RADARSAT-2 PolSAR images over the Chamonix Mont-Blanc site. Several p values of Schatten p-norms are surveyed and their capabilities on filtering PolSAR images is provided in comparison with conventional strategies for filtering PolSAR data.

Introduction

Since the launch of SAR satellites offering full polarization (ALOS, RADASAT-2 and TERRASAR-X, in L band, C band and X band respectively), the use of PolSAR imagery has been studied extensively in many fields, such as biomass and forest height estimation, snow cover mapping, glacier monitoring, damage assessment, etc.

The measurement of polarimetric radar is the complex scattering matrix, which can be expressed as:

S 2 = S hh S hv S vh S vv (1)
where S hh , S vv are the scattering elements of copolarization, and S hv , S vh are the scattering elements of cross-polarization associated with the wave transmitted and received on horizontal/vertical and vertical/horizontal plane respectively.

In monostatic and reciprocal backscattering case, S hv = S vh , the scattering information can be addressed in a complex vector as:

• Lexicographic scattering vector (k 3L ):

k 3L = S hh √ 2S hv S vv T (2)
or

• Pauli scattering vector (k 3P ):

k 3P = 1 √ 2 [S hh + S vv S hh -S vv 2S hv ] T (3)
where superscript "T " refers to the matrix transpose.

Polarimetric information can also be presented by a covariance matrix (C 3 ) based on k 3L or a coherency matrix (T 3 ) based on k 3P as follow:

C 3 = k 3L .k * T 3L =   |S hh | 2 √ 2 S hh S * hv S hh S * vv √ 2 S hv S * hh 2 |S hv | 2 √ 2 S hv S * vv S vv S * hh √ 2 S vv S * hv |S vv | 2   (4)
and

T 3 = k 3P .k * T 3P = 1 2   |S hh + S vv | 2 (S hh -S vv )(S hh + S vv ) * 2 S hv (S hh + S vv ) * • • • • • • (S hh + S vv )(S hh -S vv ) * 2 (S hh + S vv )S * hv |S hh -S vv | 2 2 (S hh -S vv )S * hv 2 S hv (S hh -S vv ) * 4 |S hv | 2   (5)
where superscript " * " indicates the complex conjugate, and • • • denotes mathematical expectation usually estimated by temporal or spatial averaging.

The intrinsic difficulty on exploiting PolSAR data is the so-called speckle noise involved in coherent acquisition systems. Speckle reduction techniques are often required as a preprocessing step in order to improve the accuracy of image segmentation, classification, edge extraction... There are many approaches in speckle filtering for PolSAR data in literature. Novak and Burl [START_REF] Novak | Optimal speckle reduction in polarimetric SAR imagery[END_REF] proposed the polarimetric whitening filter (PWF), which optimally combines the elements of the scattering matrix and produces a single scalar intensity image in which speckle is reduced but the polarization information is lost. Lee et al. [START_REF] Lee | Speckle reduction in multipolarization and multifrequency SAR imagery[END_REF] used the multiplicative speckle model and minimized the mean square error to filter the diagonal terms of the covariance matrix, which are real and represent the channel powers. Lopes and Sery [START_REF] Lopes | Optimal speckle reduction for the product model in multilook polarimetric SAR imagery and the whishart distribution[END_REF] applied PWF for multilook complex data (MPWF) related to the texture variation using a product model for texture and speckle noise. Vasile et al. [START_REF] Vasile | Intensity-driven adaptive-neighborhood technique for polarimetric and interferometric SAR parameters estimation[END_REF] proposed an adaptive filter for coherency matrix of polarimetric or interferometric SAR data using intensity information to grow the adaptive neighborhood. To preserve the dominant scattering mechanism of each pixel, Lee et al. [START_REF] Lee | Scattering-model-based speckle filtering of polarimetric SAR data[END_REF] proposed an approach that includes in the average only the pixels in the same dominant scattering category.

The two main directions for speckle filtering Pol-SAR data are either the selection of adaptive windows/neighborhoods or the choice of the estimator such as complex multi-looking (ML), locally linear minimum mean square error (LLMMSE), etc. In [START_REF] Atto | Vector and matrix LP norms in polarimetric radar filtering[END_REF], authors introduced an original cost function based on l p norms for multivariate data filtering. In this paper, we address the extension of this cost function when considering the Schatten representation of multi-channel complex PolSAR matrices. We investigate several PolSAR data characterizations issued from standard PolSAR matrix decompositions (ex: S 2 , C 3 , T 3 ,etc.). We consider a PolSAR image I, where each pixel is characterized by a matrix. This considered matrix is still corrupted by speckle noise. Therefore we wish to replace this matrix by a matrix M which minimises a cost function defined as the norm of the difference between matrix M and the observed PolSAR matrices (error norm). There exist many matrix norms. Among these norms, we consider the Schatten p-norms which relates to the sum of the singular values of the difference matrix in our regularization cost function. The filtering strategy then consists in selecting the best matrix M that minimizes the above cost function from neighborhood consideration.

Schatten p-norm based filtering

Let us consider the following quantity:

x p = n i=1 |x i | p 1 p . (6)
This quantity defines the standard ℓ p

• (a) norm when p 1,

• (b) quasi-norm when 0 < p < 1 and

• (c) pseudo-norm when p = 0.
For the sake of simplicity, we will use the terminology "norm" for the general case p 0. Since PolSAR data are matrix based images, we consider matrix norms which makes possible compact representation of the information contained in a given matrix. The following section provides these norms.

Schatten p-norms

Let A be an m × n matrix with the singular value decomposition (SVD) A = U ΣV * , where U and V are, respectively, an m × m and an n × n real or complex unitary matrix, and Σ is an m×n diagonal matrix with non-negative real numbers on the diagonal. The Schatten norm of order p of A is the ℓ p norm of the vector of singular values of A. This norm is:

A p =   min(m,n) k=1 σ p k (A)   1 p (7)
where σ k is the k th singular value of A (that is the (k, k) entry of Σ) [START_REF] Lefkimmiatis | Hessian Schatten-norm regularization for linear inverse problems[END_REF].

In the rest of the paper, we will focus on 3 particular matrix norms among the family of Schatten p-norms:

• The Schatten 1-norm which is also known as the nuclear (or trace) norm.

• The Frobenius norm corresponding to the Schatten 2norm.

• The Schatten 0.5-norm.

It is worth noticing that the singular values of a matrix are known to have a parsimony property: most energy of the matrix is concentrated on the largest singular values. In this respect, the filtering strategy discussed below is expected to be more robust than when using a standard entrywise matrix norm.

Regularization with boxcar neighborhoods and the least Schatten p error norms

Let I h,ℓ be the element located at the (h, l) pixel of image I and

I[h, ℓ, r] = {I h+i,l+j } -r i r,-r j r (8)
be the central pixel of a (2r+1)×(2r+1) analysis window. This window determines the boxcar neighborhood of I h,ℓ concerned by the local optimization. The cost function f p (M, I[h, ℓ, r]) is defined by

f p (M, I[h, ℓ, r])= r i=-r j=-r Q k=1 σ p k (I h+i,l+j -M) 1 p (9)
where Q = min(m, n). This function represents the Schatten p error norms of estimating I[h, ℓ, r] by a constant value M. In practice, we apply the above pixelwise cost function on pixel neighborhoods and seek for the matrix Îh,ℓ that minimizes the sum of pixelwise cost functions, the pixel under consideration pertaining to the pixel neighborhood I[h, ℓ, r]. The matrix Îh,ℓ is denoted by following:

Îh,ℓ = arg M∈I[h,ℓ,r] min f p (M, I[h, ℓ, r]) (10)
The image Î is the filtered image of the optimal values that minimize estimation errors.

Experimental results

The proposed approach is illustrated on a RADARSAT-2 ascending, quad-pol SAR image acquired on 12/24/2009. It covers a high mountain area over Chamonix Mont-Blanc which includes Alpine glaciers as Argentière and Mer de Glace -Leschaux. The test image is the sub-image over Argentière glacier. The study area is a heterogeneous area with mountainous terrain, which are mainly glacier, snow, rock mixture and some villages located sparsely. The considered matrices are scattering matrix (S 2 ) formed from the HH, HV and VV channels and the coherency matrix (T 3 ) formed from the Pauli basis.

The effect of p values to Schatten p-norm based filter

The norms which have been introduced in Section 2, provide vector spaces and their linear operators with measures of size, length and distance. They thus are widely applicable beside customary methods that we often use.

As the case of l p norm, the Schatten p-norm based filter also depends on p values. In this section, we apply the proposed filter to the matrix S 2 and T 3 with p = 0.5; p = 1 and p = 2 using a sliding window of 3x3 pixels. The results obtained over the Argentière glacier image are illustrated in Fig. 6 and Fig. 7 with zoom on textured and homogeneous areas in Fig. 2; 3 for S 2 matrix and Fig. 4; 5 for T 3 matrix.

The results in Fig. 4d; 5d and 7d show that p = 0.5 is not an appropriate value for this filter when applied on T 3 matrix.

The effect of window size to Schatten pnorm based filter

Another parameter which can impact to the efficiency of the filter is the size of analysis window. There are a number of studies about this issue in literature. In this paper, we survey the filtering strategies with window size of 3x3 and 5x5 pixels. On the S 2 matrix, the results of this filter with the window size = 5x5 pixels looks like a kind of segmentation with the patch effect, but the thin structure information is lost. With the window size = 3x3 pixels, the filter operates better. To compare the operation of the filter on S 2 and T 3 , it works better on S 2 .

The fusion of different Schatten p-norms based filtered images with respect to p values

On the complex test-site, the filtering results should preserve structures after speckle removal. The proposed filter concerns a test-site involving both non-smooth structure and smooth areas. A given Schatten p-norm can have desirable properties or limitations depending on the area we focused on. Thus, in order to be more adaptive to the specific site, we consider combining the results of different Schatten p-norms and the original image. The fusion simply consists the sum of the intensities used to obtain the color images. Fig 9, 10, 12, 13, 14 and 15, indicate that the fusion strategy leads a better performance of filtering that the structure features of the original image are preserved and the speckle is also removed significantly. A visual comparison with IDAN filter [START_REF] Vasile | Intensity-driven adaptive-neighborhood technique for polarimetric and interferometric SAR parameters estimation[END_REF] in Fig. 10, 14 and 15 show that the fusion of the Schatten p-norm based filtered images seems to be more speckle removed, clearer with more small preserved details than IDAN filtered image.

Conclusions and perspectives

The original method presented in this paper provided several early results which have demonstrated its potential in filtering PolSAR data.

• This filter can operate with many kind of data format of PolSAR data. • If we choose an appropriate p value, we can take the advantage of Schatten p-norms (mentioned in Section 2.1) to increase sparsity.

The further works to develop this method may concern:

• implementing performance assessment for filtered images

• looking for initial transform to exploit sparsity

• studying fusion strategies to reach a better performance

• applying the proposed method in different decomposition/coding to exploit polarization information of Pol-SAR data.

Figure 1 :

 1 Figure 1: Chamonix Mont-Blanc test-site. (a) Optical image warped to a DEM (b) RADARSAT2 descending image on 06/22/2009 (Pauli basis, R:HH-VV, G:2HV, B:HH+VV)

5 Figure 2 :

 52 Figure 2: Zoom of filtering scattering matrix S 2 using different values of p on structure area.

5 Figure 3 :

 53 Figure 3: Zoom of filtering scattering matrix S 2 using different values of p on homogeneous area.

Figure 4 :

 4 Figure 4: Zoom of filtering coherency matrix T 3 using different values of p on structure area.

Figure 5 :

 5 Figure 5: Zoom of filtering coherency matrix T 3 using different values of p on homogeneous area.

5 Figure 6 :

 56 Figure 6: Filtering scattering matrix S 2 using different values of p.

Figure 7 :

 7 Figure 7: Filtering coherency matrix T 3 using different values of p.

Figure 8 :Figure 9 :Figure 10 :

 8910 Figure 8: Filtering scattering matrix S 2 using different window size.

Figure 11 :

 11 Figure 11: Zoom of filtering scattering matrix S 2 using different window size.

Figure 12 :Figure 13 :Figure 14 :Figure 15 :

 12131415 Figure 12: Zoom of filtering scattering matrix S 2 with the fusion of several "p" results on structure area. (a) Original S 2 image; (b) and (c) are the filtered images with the fusion between original image, filtered images p = 0.5; p = 1; p = 2 with window size = 3x3 and 5x5 pixels, respectively


Jacques Oberto

Marie Gaudin

Matteo Cossu

Aurore Gorlas

Alexeï Slesarev

Evelyne Marguet

Patrick Forterre

Genome Sequence of a Hyperthermophilic Archaeon, Thermococcus nautili 30-1, That Produces Viral Vesicles

Thermococcus nautili 30-1 (formerly Thermococcus nautilus), an anaerobic hyperthermophilic marine archaeon, was isolated in 1999 from a deep-sea hydrothermal vent during the Amistad campaign. Here, we present the complete sequence of T. nautili, which is able to produce membrane vesicles containing plasmid DNA. This property makes T. nautili a model organism to study horizontal gene transfer.

T hermococcus nautili 30-1 (formerly Thermococcus nautilus) was isolated on the East Pacific ridge near a black smoker, 2,600 m below the ocean surface [START_REF] Lepage | Molecular diversity of new Thermococcales isolates from a single area of hydrothermal deep-sea vents as revealed by randomly amplified polymorphic DNA fingerprinting and 16S rRNA gene sequence analysis[END_REF]. T. nautili qualifies as a new species and the characterization of its physiology and metabolism has been reported [START_REF] Gorlas | Thermococcus nautili sp. nov., a hyperthermophilic ar-chaeon isolated from a hydrothermal deep sea vent (East Pacific Ridge)[END_REF].

T. nautili was grown anaerobically in ZB medium at 85°C (1), and genomic DNA was purified and sequenced using paired-end Illumina libraries to generate 16 ϫ 10 6 reads assembled with Velvet (3). Genome assembly was performed with Consed (4). The remaining gaps were filled using bacterial artificial chromosome (BAC) sequencing with Thermofidelase and Fimers [START_REF] Malykh | Finishing "working draft" BAC projects by directed sequencing with ThermoFidelase and Fimers[END_REF]. Genes were annotated using FGENESB (Softberry). Additional functional annotations were obtained with Integrated Microbial Genomes [START_REF] Markowitz | IMG ER: a system for microbial genome annotation expert review and curation[END_REF]. The chromosome of T. nautili is 1,976,356 bp long and harbors 2,288 genes. The noncoding genes comprise 4 rRNA genes (23S, BD01_1261, and 16S, BD01_1259), with a duplication of the 5S rRNA gene (BD01_1590 and BD01_2049), and 46 tRNA genes. The chromosomal origin of replication was identified with FIT-BAR (7) using published origin recognition complex (ORC) sequences [START_REF] Matsunaga | Genomewide and biochemical analyses of DNA-binding activity of Cdc6/Orc1 and Mcm proteins in Pyrococcus sp[END_REF]. It is positioned upstream of the CDC6 gene (BD01_1824). A similar approach located the replication termination region at coordinate 772,784 using archaeal dif sites [START_REF] Cortez | Evidence for a Xer/dif system for chromosome resolution in archaea[END_REF].

Several strains of Thermococcus contain either one or two plasmids (10-13). Our group previously sequenced and characterized two plasmids of 3.6 and 13 kb from T. nautili that were named pTN1 and pTN2, respectively [START_REF] Soler | The rolling-circle plasmid pTN1 from the hyperthermophilic archaeon Thermococcus nautilus[END_REF][START_REF] Soler | Two novel families of plasmids from hyperthermophilic archaea encoding new families of replication proteins[END_REF][START_REF] Gaudin | Extracellular membrane vesicles harbouring viral genomes[END_REF]. The rolling-circle plasmid pTN1 was used to construct the first shuttle vector able to replicate in both Escherichia coli and the model organism Thermococcus kodakaraensis [START_REF] Santangelo | Shuttle vector expression in Thermococcus kodakaraensis: contributions of cis elements to protein synthesis in a hyperthermophilic archaeon[END_REF] and contributed to the development of T. kodakaraensis as a host for genetics studies [START_REF] Farkas | Genetic techniques for the archaea[END_REF]. Plasmid pTN2 encodes a new family of DNA primase formed by the fusion of protein domains homologous to the archaeo-eukaryotic primases PriS and PriL [START_REF] Gill | A highly divergent archaeo-eukaryotic primase from the Thermococcus nautilus plasmid, pTN2[END_REF]. DNA sequencing revealed the presence of an additional plasmid of 18 kb, pTN3 [START_REF] Gaudin | Extracellular membrane vesicles harbouring viral genomes[END_REF]. This extrachromosomal element, related to the virus-like element TKV4 from T. kodakaraensis [START_REF] Krupovic | Archaeal proviruses TKV4 and MVV extend the PRD1-adenovirus lineage to the phylum Euryarchaeota[END_REF], is found as a freely replicating plasmid and also integrated at a specific position on the chromosome into a tRNA Leu gene (BD01_0018). The plasmid attachment site (attP)is located within the integrase-encoding gene TnaP3-13 [START_REF] Gaudin | Extracellular membrane vesicles harbouring viral genomes[END_REF]. The attP and attB sites share an identical stretch of 41 nucleotides (nt), as observed previously in the case of the TKV4 integron of T. kodakaraensis [START_REF] Krupovic | Archaeal proviruses TKV4 and MVV extend the PRD1-adenovirus lineage to the phylum Euryarchaeota[END_REF]. In the chromosome of T. nautili, 6 clustered regularly interspaced short palindromic repeat (CRISPR) loci were found using CRISPRFinder [START_REF] Grissa | CRISPRFinder: a web tool to identify clustered regularly interspaced short palindromic repeats[END_REF]. Interestingly, two CRISPR spacers are directed against resident plasmid pTN3.

A major interest in the study of T. nautili is its ability to generate membrane vesicles containing plasmid DNA [START_REF] Soler | Plasmids, viruses and virus-like membrane vesicles from Thermococcales[END_REF], including the virus-like genome of pTN3 (viral vesicles) [START_REF] Gaudin | Extracellular membrane vesicles harbouring viral genomes[END_REF]. The discovery that these vesicles can transfer genetic information between related Thermococcales [START_REF] Gaudin | Hyperthermophilic archaea produce membrane vesicles that can transfer DNA[END_REF] provides an important tool for the study of horizontal gene transfer across species or even domains, from which genomic evolution models can be inferred. The genetic determinism of the biosynthesis of these viral vesicles is presently being investigated and relies primarily on the T. nautili genome sequence presented here.

Nucleotide sequence accession number. The T. nautili genome sequence has been deposited in the NCBI repository under the accession number CP007264.

Genome AnnouncementsMarch/April 2014 Volume 2 Issue 2 e00243-14 genomea.asm.org 1

Genome Announcements 2 genomea.asm.org March/April 2014 Volume 2 Issue 2 e00243-14

ACKNOWLEDGMENTS

This work was supported by the Centre National de la Recherche Scientifique (Program Marine Genome Biology) and the Agence Nationale de la Recherche (ANR 12-BSV3-0023-01).


Rob De

Alexander D Rahm
email: alexander.rahm@nuigalway.ie

THE IMAGE OF THE BOREL-SERRE BORDIFICATION IN ALGEBRAIC K -THEORY

We give a method for constructing explicit non-trivial elements in the third K-group (modulo torsion) of an imaginary quadratic number field. These arise from the relative homology of the map attaching the Borel-Serre boundary to the orbit space of the SL2 group over the ring of imaginary quadratic integers on its symmetric space -hyperbolic three-space. We provide an algorithm which produces a chain of matrix quadruples specifying our element of K3 of the field, modulo torsion. We carry out the algorithm for the Eisenteinian integers as well as for the imaginary quadratic integers of discriminant -7.

Introduction

We use the SL 2 -group of the ring of integers O := O Q( √ -m ) of an imaginary quadratic number field Q( √ -m ), where m is a square-free positive integer, in order to obtain explicit elements of infinite order in the algebraic K-group K 3 (O)/torsion = K 3 (k)/torsion.

Let Γ be a subgroup of finite index in SL 2 (O), with torsion-free image of Γ in P SL 2 (O); and let Γ 1 , . . . , Γ s denote the stabilisers inside Γ of the cusps of the action on hyperbolic 3-space. Let P • be a projective resolution for Γ, and denote the homology of the cone of the natural map

⊕ i P • (Γ i ) ⊗ Z[Γ i ] Z → P • (Γ) ⊗ Z[Γ] Z ,
by H * (Γ; Γ 1 , . . . , Γ s ). Then we have a long exact sequence

• • • → ⊕ i H n (Γ i ) → H n (Γ) → H n (Γ; Γ 1 , . . . , Γ s ) → ⊕ i H n-1 (Γ i ) → H n-1 (Γ) → • • • .
Let ∆ i be the stabiliser of the ith cusp inside ∆ = GL 2 (Q( √ -m )), so that Γ is a subgroup of ∆ and each Γ i is a subgroup of ∆ i . We can replace Γ with ∆ and each Γ i with ∆ i in the above, obtaining a long exact sequence of relative homology involving ∆ and the ∆ i , and the natural homomorphisms give rise to a commutative diagram . . .

/ / ⊕ i H n (Γ i ) αn / / H n (Γ) / / H n (Γ; Γ 1 , . . . , Γ s ) / / ⊕ i H n-1 (Γ i ) α n-1 / / . . . . . . / / ⊕ i H n (∆ i ) / / H n (∆) / / H n (∆; ∆ 1 , . . . , ∆ s ) / / ⊕ i H n-1 (∆ i ) / / . . . . (1.1)
Using a result of Suslin, one can see that the map Lemma 2.10). Therefore the image of a class in H 3 (Γ; Γ 1 , . . . , Γ s ) lifts to a class in H 3 (∆).

⊕ i H n-1 (Γ i ) → ⊕ i H n-1 (∆ i ) is trivial for n ≥ 2 (cf.
From this lift, one can obtain an element in K 3 (k)/torsion ≃ Z by standard methods. However, in order to show that the resulting element is non-trivial we need some more control. In Section 2 we therefore carry out a variation of this process, and apply it to suitable homology classes obtained from triangulations in Section 4. For the Eisenstein integers, our triangulation actually provides an element of H 3 (Γ; Γ 1 , . . . , Γ s ), but in the case of O a non-principal ideal domain, the fact that there are several SL 2 (O)-orbits of cusps does force us to bypass H 3 (Γ; Γ 1 , . . . , Γ s ) completely, which we achieve in Section 4, yielding an algorithm that works for all rings of imaginary quadratic integers.

The relation with K 3 (k)

In this section we modify some by now classical constructions in [START_REF] Suslin | K3 of a field, and the Bloch group[END_REF] and [3, p. 73] to suit our current needs. We shall use this in Theorem 4.6 below to show that our algorithm in Section 4 can be used to obtain a non-trivial element in K 3 (k)/torsion if k is an imaginary quadratic number field.

In order to be able to take non-trivial torsion in stabilisers into account, and be able to work in P SL 2 (O) instead of SL 2 (O) if desired, we work in somewhat greater generality.

Let L be a field, and fix two subgroups Λ ⊆ Λ of L * . (Typically, we have in mind Λ = {1} or {±1}, and Λ = O * torsion with O the ring of algebraic integers in a number field.) Let ∆ = GL 2 (L)/Λ. Let L be the set of orbits for the action of Λ on L 2 \ {(0, 0)} given by scalar multiplication, which has a natural map to P 1 L . For n ≥ 0 we let C n (L 2 ) be the free Abelian group with as generators (n + 1)tuples (l 0 , . . . , l n ) of elements in L such that if l i 1 and l i 2 have the same image in P 1

L then l i 1 = l i 2 . We shall call such a tuple (l 0 , . . . , l n ) with all l i distinct in L (or equivalently, in P 1 L ) non-degenerate, and we shall call it degenerate otherwise. Then ∆ acts on C n (L 2 ) as Λ ⊆ Λ, and with the usual boundary map d : We then obtain a group homomorphism f 2,L : C 2 (L 2 ) → ∧2 L * / Λ ∧L * by letting this be trivial on a degenerate generator (l 0 , l 1 , l 2 ), and by mapping a non-degenerate generator (l 0 , l 1 , l 2 ) to cr 2 (p 0 , p 1 , p 2 ) with p i a point in l i .

C n (L 2 ) → C n-1 (L 2 ) for n ≥ 1 given by d(l 0 , . . . , l n ) = n i=0 (-1) i (l 0 , . . . , l i , . . . , l n ) we get a complex • • • d / / C 4 (L 2 ) d / / C 3 (L 2 ) d / / C 2 (L 2 ) d / / C 1 (L 2 ) d / / C 0 (L 2 ) of Z[∆]-modules. Let ∧2 L * = L * ⊗ Z L * (x) ⊗ (-x) with x in L * .
We also define a group homomorphism f 3,L :

C 3 (L 2 ) → Z[L ♭ ], with L ♭ = L\{0, 1}.
We define it to be trivial on a degenerate generator (l 0 , l 1 , l 2 , l 3 ), and we let it map a non-degenerate generator (l 0 , l 1 , l 2 , l 3 ) to [cr 3 (l 0 , l 1 , l 2 , l 3 )], the cross-ratio of the images of the points in P 1 L . Recall that cr 3 is defined by similar rules as those for cr 2 :

• cr 3 (gl 0 , gl 1 , gl 2 , gl 3 ) = cr 3 (l 0 , l 1 , l 2 , l 3 ) for every g in GL 2 (L);

• cr 3 ([1, 0], [0, 1], [1, 1], [x, 1]) = x for x = 0, 1.
We define δ 2,L : Z[L ♭ ] → ∧2 L * / Λ ∧L * by mapping a generator [x] to the class of (1 -x) ∧(x). Then the diagram

C 3 (L 2 ) d / / f 3,L C 2 (L 2 ) f 2,L Z[L ♭ ] δ 2,L / / ∧2 L * / Λ ∧L *
commutes. For the non-degenerate case this is a direct calculation: using the GL 2 (L)-invariance of both f 3,L and f 2,L and the GL 2 (L)-equivariance of d we can start with a generator (l 0 , l 1 , l 2 , l 3 ) containing the classes of (a, 0), (0, b), (1, 1) and (xc, c) in L for some and a, b and c in L * and x in L ♭ , which results in [x] in Z[L ♭ ] under f 3,L and the class of (1 -x) ∧(x) under f 2,L • d. For a degenerate tuple (l 0 , l 1 , l 2 , l 3 ) the commutativity is obvious if {l 0 , l 1 , l 2 , l 3 } has at most two elements as then f 2,L is trivial on every term in d(l 0 , l 1 , l 2 , l 3 ). If it consists of three classes A, B and C with A occurring twice among l 0 , l 1 , l 2 and l 3 , then up to permuting B and C the six possibilities for (l 0 , l 1 , l 2 , l 3 ) are (A, A, B, C), (A, B, A, C), (A, B, C, A), (B, A, A, C), (B, A, C, A) and (B, C, A, A). After cancellation of identical terms with opposite signs in d(l 0 , l 1 , l 2 , l 3 ), we see that commutativity follows because f 2,L is alternating.

From the above diagram we now obtain the commutative diagram

• • • d / / C 4 (L 2 ) d / / C 3 (L 2 ) f 3,L d / / C 2 (L 2 ) f 2,L d / / C 1 (L 2 ) d / / C 0 (L 2 ) 0 / / B 2 (L) δ 2,L / / ∧2 L * / Λ ∧L * / / 0 (2.2) with (2.3) B 2 (L) = Z[L ♭ ] f 3,L • d(C 4 (L 2 ))
.

We observe that we could take GL 2 (L)-coinvariants in the top row because of the properties of f 3,L and f 2,L . In particular,

f 3,L induces a homomorphism H 3 (C • (L 2 ) ∆ ) → ker(δ 2,L ).
In order to describe B 2 (L) more explicitly, we note that f 3,L • d maps a non-degenerate generator (l 0 , . . . , l 4 ) to 5 i=1 (-1) i cr 3 (l 0 , . . . , l i , . . . , l 4 ) with l 0 , . . . , l 4 distinct points in P 1 L . Using the invariance of cr 3 under the action of GL 2 (L), we may use the points [1, 0], [0, 1], [1,1], [x, 1] and [y, 1] in P 1

L for x and y in L ♭ with x = y. This then under f 3,L • d yields the 5-term relation

(2.4) [x] -[y] + [y/x] -[(1 -y)/(1 -x)] + [(1 -y -1 )/(1 -x -1 )] .
For a degenerate generator (l 0 , . . . , l 4 ) we note that f 3,L • d is trivial if {l 0 , . . . , l 4 } has at most three elements because all the terms in d(l 0 , . . . , l 4 ) are degenerate. If this set contains four classes, then after cancelling possible identical terms in d(l 0 , . . . , l 4 ) and applying f 3,L to the result we see that it is of the form [cr 3 (m 1 , . . . , m 4 )] -sgn(σ)[cr 3 (m σ(1) , . . . , m σ( 4) )] for a permutation σ in S 4 with sign sgn(σ), and four distinct points m i in P 1 L . Those images generate the subgroup (2.5)

[x] + [x -1 ] with x in L ♭ + [y] + [1 -y] with y in L ♭ of Z[L ♭ ].
Remark 2.6. According to (the proofs of) Lemmas 1.2, 1.3 and 1.4 of [START_REF] Suslin | K3 of a field, and the Bloch group[END_REF], if L has at least four elements, then the group B 2 (L) is a quotient of the group

p(L) = Z[L ♭ ] [x] -[y] + [y/x] + [(1 -x)/(1 -y)] -[(1 -x -1 )/(1 -y -1 )] with x, y in L ♭ , x = y of loc. cit.
It is obtained from p(L) by dividing out certain elements of exponent dividing 6 (and their exponent divides 2 if x 2 -x + 1 has a solution in L).

Recall from loc. cit. that if L is infinite, then there is a short exact sequence

0 → Tor(L * , L * ) ∼ → K 3 (L) ind → B(L) → 0 ,
where Tor(L * , L * ) ∼ denotes the unique non-trivial extension of Tor(L * , L * ) by Z/2Z if L has characteristic different from 2, and Tor(L * , L * ) otherwise,

K 3 (L) ind = coker(K M 3 (L) → K 3 (L)), and B(F ) is the kernel of p(F ) → F * ⊗ Z F * / x ⊗ y + y ⊗ x under the map [x] → x ⊗ (1 -x).
By mapping x⊗y to (y) ∧(x), we get a map from the complex 0 → p(L) → L * ⊗ Z L * / x⊗y+y⊗x → 0 to the complex in the bottom row of (2.2). Since 2(-x) ⊗ x = 2(x ⊗ x) it is clear from the definition of ∧2 L * / Λ ∧L * in (2.1) that the map F * ⊗ F * / x ⊗ y + y ⊗ x → ∧2 L * / Λ ∧L * is surjective with kernel of exponent dividing a = lcm(2, | Λ|) whenever Λ is finite. Multiplying by a then gives a map ker(δ 2,L )/torsion → B(L)/torsion, with the latter isomorphic with K 3 (L) ind /torsion if L is infinite.

Fix a non-zero point p in L 2 , and for n ≥ 0 let P n (L) = P n (L, p, Λ) be the free Abelian group with as generators the (n + 1)-tuples (g 0 , . . . , g n ) of elements in ∆ such that if g i 1 p and g i 2 p give the same point in P 1 L then g i 1 p = g i 2 p already in L. Clearly, P n (L) is a projective Z[∆]-module, and P • (L) with the boundary d : P n (L) → P n-1 (L) for n ≥ 1 given by d(g 0 , . . . , g n ) = n i=0 (-1) i (g 0 , . . . , g i , . . . , g n ) is a complex of projective Z[∆]-modules. The evaluation homomorphism P n (L) → C n (L 2 ) mapping (g 0 , . . . , g n ) to (g 0 p, . . . , g n p) gives the commutative diagram

• • • d / / P 4 (L) d / / P 3 (L) d / / P 2 (L) d / / P 1 (L) d / / P 0 (L) • • • d / / C 4 (L 2 ) d / / C 3 (L 2 ) d / / C 2 (L 2 ) d / / C 1 (L 2 ) d / / C 0 (L 2 ) (2.7)
of Z[∆]-modules, and we obtain a homomorphism

H 3 (P • (L) ∆ ) → H 3 (C • (L 2 ) ∆ ).
Combining it with the map in (2.2), we obtain a homomorphism (2.8)

H 3 (P • (L) ∆ ) → ker(δ 2,L ) .
Remark 2.9. There are augmentations P

• (L) → Z and C • (L 2 ) → Z, and if L is infinite those give Z[∆]-resolutions of Z,
with the first being a projective resolution. Clearly P • (L) is a subcomplex of the standard resolution for ∆, so we obtain a canonical isomorphism

H 3 (P • (L) ∆ ) ≃ → H 3 (∆, Z). In this case the morphism P • (L) → C • (L 2 ) is the (up to chain homotopy unique) morphism of Z[∆]-resolutions of Z.

Now let p be the point [1, 0] in P 1

L and consider its stabiliser

∆ ∞ = a b 0 c with a, c in L * , b in L /Λ • id 2 in ∆. Let N 1 = {( 1 0 b c ) with b in L, c in L * } ⊂ GL 2 (L) and N 1,1 the subgroup of N 1 where c = 1. Let ∆ ∞ 1 be the image of Λ • N 1 in ∆ ∞ and let ∆ ∞ 1,1 be a subgroup of the image of Λ • N 1,1 . Lemma 2.10. Let m ≥ 1 and assume [L : Λ] is finite but L is infinite. (1) The image of the map H m (∆ ∞ 1,1 , Z) → H m (∆ ∞ 1 , Z) is killed by multiplication by [ Λ : Λ]. (2) If Λ/Λ is cyclic and m is even then the image of the map H m (∆ ∞ 1,1 , Z) → H m (∆ ∞ 1 , Z) is trivial. Proof. It will suffice to prove those results for ∆ ∞ 1,1 equal to the image of Λ • N 1,1 in ∆ ∞ 1 . Note that we have compatible isomorphisms ∆ ∞ 1 ≃ Λ/Λ × N 1 and ∆ ∞ 1,1 ≃ Λ/Λ × N 1,1 . If N ′ 1 ⊂ N 1 consists of the matrices ( 1 0 0 c ) with c in L * then by [10, Theorem 1.9] the natural map N 1 → N ′ 1 induces an isomorphism on homology. The composition ∆ ∞ 1 ≃ Λ/Λ × N 1 → Λ/L × N ′
1 then induces an isomorphism on homology. Since this composition maps ∆ ∞ 1,1 to Λ/L × {( 1 0 0 1 )}, the result is clear. Let p 1 , . . . , p s with s ≥ 1 be distinct points in

P 1 L . Fix elements A 1 , . . . , A s in ∆ with p j = A j [1, 0]. Then A -1 j 1 A j 2 is in ∆ ∞ 1 if
and only if j 1 = j 2 . We shall consider elements in P 3 (L) satisfying the following. (In Proposition 4.5 we shall prove that it is fulfilled by the elements obtained from our algorithm in Section 4.) Assumption 2.11. Suppose α in P 3 (L) is such that the following hold.

(1) It is a sum of (g 0 , g 1 , g 2 , g 3 ) with each g i in a coset A j i ∆ ∞ 1 .

(2) In P 2 (L) ∆ , we can write dα = s j=1 A j β j for β j in the kernel of

d : Z[∆ ∞ 1,1 × ∆ ∞ 1,1 × ∆ ∞ 1,1 ] ∆ ∞ 1,1 → Z[∆ ∞ 1,1 × ∆ ∞ 1,1 ] ∆ ∞ 1,1 .
Note that each (g 0 , g 1 , g 2 , g 3 ) in the first condition is in P 3 (L) for p = (1, 0):

A j 1 γ 1 [1, 0] = A j 2 γ 2 [1, 0] with the γ i in ∆ ∞
1 implies j 1 = j 2 , and γ 1 (1, 0) = γ 2 (1, 0) already in L because of the definition of ∆ ∞ 1 . We now prove that under mild hypotheses we can associate an element in ker(δ 2,L ) to any α satisfying Assumption 2.11.

Proposition 2.12. Assume [ Λ : Λ] is finite, and let t ≥ 1 be an integer such that the image of

H 2 (∆ ∞ 1,1 , Z) in H 2 (∆ ∞ 1 , Z
) is annihilated by multiplication by t. To α satisfying Assumption 2.11 we can then associate a well-defined element in ker(δ 2,L ) by computing the map in (2.8) with p = [1, 0] on those (g 0 , g 1 , g 2 , g 3 ) in tα for which the g 0 , . . . , g 3 lie in four distinct cosets A j ∆ ∞ 1 . Proof. Note that a t exists by Lemma 2.10 if L is infinite, and it obviously exists if L is finite. We can write

tβ j = d βj in Z[∆ ∞ 1 × ∆ ∞ 1 × ∆ ∞ 1 ] ∆ ∞ 1 for some βj in Z[∆ ∞ 1 × ∆ ∞ 1 × ∆ ∞ 1 × ∆ ∞ 1 ]
, which is unique up to adding an element in the kernel

Z[∆ ∞ 1 × ∆ ∞ 1 × ∆ ∞ 1 × ∆ ∞ 1 ] → Z[∆ ∞ 1 × ∆ ∞ 1 × ∆ ∞ 1 ] ∆ ∞ 1 .
From the definitions we clearly have that the standard resolution of ∆ ∞ 1 is a subcomplex of P • (L), so that α = tα -

s j=1 A j βj gives a well-defined class in H 3 (P • (L) ∆ ) modulo the image of H 3 (∆ ∞ 1 , Z).
Note that under the composition (2.8) of the homomorphisms obtained from (2.7) and (2.2), the βj do not contribute to the image (and hence this image is independent of the choice of the βj ) because γ ′ [1, 0] = [1, 0] for all γ ′ in ∆ ∞ 1 , so they only result in combinations of degenerate generators in C 3 (L 2 ), which are killed by f 3,L . Similarly, the only terms (A j 1 γ 1 , A j 2 γ 2 , A j 3 γ 3 , A j 4 γ 4 ) (with all γ i in ∆ ∞ 1 ) in α that contribute are those where all A j i are distinct. Remark 2.13. If Λ is finite, and a = lcm(2, | Λ|), then according to Remark 2.6, multiplication by a gives us a map ker(δ 2,L )/torsion → B(L)/torsion. If L is infinite then the latter is isomorphic with K 3 (L) ind /torsion. So for finite Λ and infinite L, Proposition 2.12 gives a way of constructing an element in K 3 (L) ind /torsion out of α. We shall use this in Section 4 in order to show that for an imaginary quadratic number field k, we can obtain elements of infinite order in K 3 (k)/torsion ≃ Z from suitable triangulations of hyperbolic 3-space.

The Borel-Serre boundary

Let k be an imaginary quadratic number field, and let O be its ring of algebraic integers. We work with the Borel-Serre bordification H of hyperbolic 3-space H, which yields a compact orbit space Γ \ H [7, appendix]. Recall from [7, page 512] the identification, for a torsion-free subgroup Γ of finite index in SL 2 (O), between the integral homology of the cusp stabiliser Γ i and the integral homology of the torus T i joined at the ith cusp by the Borel-Serre compactification. Namely,

⊕ i H n (Γ i ; Z) ∼ = ⊕ i H n (T i ; Z) ∼ = H n (∂( Γ \H); Z),
where the direct sum runs over the orbits of cusps modulo Γ, and the identification passes over to the boundary of the quotient of hyperbolic 3-space H by Γ. Consider the map α induced on homology when attaching the boundary in the Borel-Serre compactification of Γ \H.

It turns out [START_REF] Rahm | On a question of Serre[END_REF] that the above relative homology group H 3 (Γ; Γ 1 , . . . , Γ s ) is precisely the kernel of the map α 2 induced on second degree homology by the attaching map α. So the answer given in [START_REF] Rahm | On a question of Serre[END_REF] to a question of Serre provides an isomorphism from H 3 (Γ; Γ 1 , . . . , Γ s ) to the one-generator group

∪ i T i ∼ = Z.
This is the case because we can locate the induced maps α n in the first of the long exact sequences (1.1). Taking into account that the cohomological dimension of Γ is 2, this long exact sequence concentrates in . . .

H 3 (Γ) = 0 ∪ i T i i T i i T i / ∪ i T i ⊕ H cusp 2 H cusp 2 i x i i x i , y i i y i ⊕ H cusp 1 H cusp 1 ⊕{ vanishing ideal } i Z Z → 0 α 3 = 0 α 2 α 1 α 0 augmentation
where x i and y i denote the cycles generating H 1 (T i ), and the maps without labels are the obvious restriction maps making the sequence exact. Note that the cuspidal homology satisfies equality of ranks between H cusp 2 and H cusp 1 , because the naive Euler-Poincaré characteristic vanishes. The isomorphism from H 3 (Γ; Γ 1 , . . . , Γ s ) to the one-generator group ∪ i T i ∼ = Z tells us that a representative for the generator of H 3 (Γ; Γ 1 , . . . , Γ s ) is given by the Borel-Serre bordification F of an ideal fundamental polyhedron F for the action of Γ on H, because the boundary of Γ \ F is ∪ i T i . In H 3 (Γ; Γ 1 , . . . , Γ s ), the polyhedron F is given as a chain of matrix quadruples: we triangulate F into tetrahedra, fix a basepoint, and then replace each tetrahedron by the quadruple of matrices that map the basepoint to its four corners. The vertices of the tetrahedra are actually on one ∆-orbit, because we have chosen F to be an ideal polyhedron (all of its vertices are cusps). If O is a principal ideal domain, then these vertices are even on one Γ-orbit, because there is just one SL 2 (O)-orbit of cusps in this case.

Explicit determination of the homology classes

In this section, we outline an algorithm to compute a chain of matrix quadruples satisfying Assumption 2.11, for an imaginary quadratic number field k with ring of algebraic integers O.

We call a polyhedron in hyperbolic 3-space H an ideal polyhedron, if all of its vertices are cusps. Our algorithm inputs an ideal polyhedron F tessellated by finitely many fundamental domains for the action of SL 2 (O) on H, such that the facets of F occur in pairs with an SL 2 (O)-identification between them. Yasaki [START_REF] Yasaki | Hyperbolic tessellations associated to Bianchi groups, Algorithmic number theory[END_REF] has conceived and implemented an algorithm which, for all O, produces ideal polyhedra satifying this condition. He has obtained explicit polyhedra for all discriminants of absolute value smaller than 100, and all imaginary quadratic fields of class number 1 and 2. So we know that our input object exists and is practically available for our computations. We equip the Borel-Serre boundary of hyperbolic 3-space H with a coordinate system by taking the cusp in C ∪ {∞} as the first coordinate, and the position on the attached real plane, expressed as a number in C, as the second coordinate.

Before we proceed we need some preliminaries. Recall that SL 2 (C) acts on the boundary of hyperbolic 3-space by fractional linear transformations, via the formula

a b c d • z = az + b cz + d .
The SL 2 (O)-orbits of P 1 k for this action correspond to the ideal class group of k, with the principal ideals corresponding to the orbit of ∞. Lemma 4.1 (Flöge). Let α β in P 1 k be any cusp with (α, β) not a principal ideal. Then its stabiliser in P SL 2 (O) is isomorphic to the Abelian group (with respect to natural addition)

A := c ∈ O | c α β ∈ O and c α 2 β 2 ∈ O by the map c → ± 1 -c α β c( α β ) 2 -c 1 + c α β .
Proof. Given in [2, 9.3]. Note that Flöge used the action formula a b c d •z = az-b cz-d .

Corollary 4.2. Let α β in P 1 k be any cusp with (α, β) not a principal ideal, and let

M α,β = α -1 β β 0 . Then M -1 α,β conjugates the stabiliser of α β in SL 2 (O) to the group ± 1 O α 2 ∩ O β 2 0 1 .
Proof. We perform the conjugation using the explicit shape of the cusp stabiliser given in Flöge's lemma above, finding 0

1 β -β α ± 1 -c α β c( α β ) 2 -c 1 + c α β α -1 β β 0 = ± 1 c β 2 0 1 .
Then by Lemma 4.1, the cusp stabiliser gets conjugated to the group ±

1 1 β 2 A 0 1 , with A := c ∈ O | c α β ∈ O and c α 2 β 2 ∈ O . We have A β 2 = O α 2 ∩ O αβ ∩ O β 2 , which equals O α 2 ∩ O β 2
by considering the factorisations of the fractional ideals for O.

We now once and for all fix representatives (α, β) of the ideal class group of k, and matrices M α,β as above. (For the principal ideal we take (1, 0) and M 1,0 = id 2 the 2 × 2-identity matrix.) Let C 1 , . . . , C h (with h the class number of k) be the matrices thus obtained. Then every cusp in P 1 k is of the form gC i • ∞ for some g in SL 2 (O), with i uniquely determined by the SL 2 (O)-orbit of the cusp.

(2) By Poincaré's theorem on fundamental polyhedra [START_REF] Poincaré | [END_REF], the properties of F give rise to a finite index subgroup Γ of SL 2 (O), such that F is a fundamental domain for Γ, strict in its interior, and such that inside hyperbolic space, the polyhedron F has boundary zero modulo the action of Γ. Denoting the stabiliser in Γ of the cusp ∞ by Γ ∞ , this allows us to write dα = s j=1

A j β j in Z[∆ × ∆ × ∆] Γ with β j in Z[Γ ∞ × Γ ∞ × Γ ∞ ] Γ∞ ,
where Γ ∞ is clearly contained in U . As F is an ideal polyhedron, the orbit space F/Γ is a hyperbolic manifold, and hence the image of Γ in P SL 2 (O) is torsion-free. By the construction of the Borel-Serre compactification [START_REF] Serre | Le problème des groupes de congruence pour SL2[END_REF], the boundary of F modulo Γ consists of one 2-torus at each cusp. Tori are closed surfaces that admit no boundary, so for all j we obtain d(β j ) = 0.

Note that it is possible to obtain an explicit description of the group Γ involved in the above proof [START_REF] Riley | Applications of a computer implementation of Poincaré's theorem on fundamental polyhedra[END_REF], but that this is outside of the aims of the present paper.

We shall now show that the classes obtained in Algorithm 4.4 give rise to non-zero classes in K 3 (O)/torsion = K 3 (k)/torsion ≃ Z under the method discussed in Remark 2. [START_REF] Zagier | Dedekind Zeta Functions and the Algebraic K-theory of Fields[END_REF].

We observe that K M 3 (k) is 2-torsion for any number field k by [START_REF] Weibel | Algebraic K-theory of rings of integers in local and global fields[END_REF]Theorem 8] (and in fact trivial if k is totally imaginary). Therefore K 3 (k) ind /torsion = K 3 (k)/torsion. As stated in loc. cit., the rank of K 3 (k) is given by the number of infinite places, so that for an imaginary quadratic field k we have K 3 (k) ind /torsion = K 3 (k)/torsion ≃ Z. Proof. That α satisfies the conditions in Assumption 2.11 is stated in Proposition 4.5. According to Lemma 2.10(2), Proposition 2.12 applies to α with t = 1.

Clearly Remark 2.6 applies, so it only remains to show that the resulting element in B(k)/torsion ⊆ p(k)/torsion is non-trivial. But the dilogarithm D : C ♭ → R applied to the generators of Z[C ♭ ] satisfies all the relations imposed when creating p(k) or B 2 (k). Moreover, if an ideal tetrahedron in H has vertices a 0 , a 1 , a 2 and a 3 in P 1 C , then its hyperbolic volume equals cr 3 (a 0 , a 1 , a 2 , a 3 ). (For a discussion of those facts see, e.g., [13, §2].)

But according to Proposition 2.12, the only (g 0 , g 1 , g 2 , g 3 ) in α that contribute to the image in p(L) are those for which the four cusps

Q i = g i • [1, 0] in P 1 C are distinct. Applying the dilogarithm to the resulting generator [cr 3 (Q 0 , Q 1 , Q 2 , Q 3
)] in p(L) then gives the volume of the corresponding ideal tetrahedron in H in the triangulation of H obtained by collapsing the boundary in the Borel-Serre bordification to single points at the cusps. Under this process all other (g 0 , g 1 , g 2 , g 3 ) collapse to degenerate tetrahedra, so in total we obtain the volume of the triangulation, which is non-zero.

Examples

In this section, we consider some explicit examples for the machinery developed in Sections 2 and 4, by applying our algorithm to two cases where the ring of integers is Euclidean. For O Euclidean, Cremona [1] has given a coordinate description of an ideal polyhedron in hyperbolic 3-space fulfilling our algorithm's input condition, and we use the coordinates printed there. 5.1. The Eisenstein integers case. We consider the case of the Eisenstein integers, namely the ring O Then the base triangles of the triangulation of the Borel-Serre compactification F of F for the action of Γ are given in Figure 5.1. Recall that the ideal tetrahedra of our triangulation are spanned as the convex hulls of a base triangle together with the basepoint. The set of cusps in the boundary of F is {0, 1, ρ, ∞}. We will make use of the following matrices in Γ, which map some (but not necessarily all) of the elements of this set of cusps back into this set. The tetrahedra of the triangulation, specified by these base triangles, of F then give rise to the following chain of matrix quadruples α: Here the quadruple (1, CA, BA, AB) specifies the only tetrahedron of non-zero volume. Letting those matrices act on [1,0] and taking the cross-ratio of the resulting four points yields -1/(ρ-1) = ρ, so that the element we obtain in ker(δ 2,k ) is [ρ]. Note that in this triangulation we use only Γ, not the full group SL 2 (O), and we can strengthen the discussion after Algorithm 4.

-3 := O k for k = Q( √ -3). Let ρ := 1+ √ - 3 2
A := 0 -1 1

  We denote the class of a ⊗ b in ∧2 L * with (a) ∧(b). It is easily verified that (a) ∧(b) + (b) ∧(a) is trivial. For three non-zero points p 0 , p 1 and p 2 in L 2 with distinct images in P 1 L , we define cr 2 (p 0 , p 1 , p 2 ) in ∧2 L * by the rules: • cr 2 (gp 1 , gp 2 , gp 3 ) = cr 2 (p 1 , p 2 , p 3 ) for every g in GL 2 (L); • cr 2 ((1, 0), (0, 1), (a, b)) = (a) ∧(b). Because cr 2 ((0, 1), (1, 0), (a, b)) = (b) ∧(a) and cr 2 ((1, 0), (a, b), (0, 1)) = (-ab -1 ) ∧(b -1 ) = -(a) ∧(b), we see that cr 2 is alternating. It is also clear that if we scale one of the p i by λ in Λ then cr 2 (p 0 , p 1 , p 2 ) changes by a term (λ) ∧(c) with c in L * . Let (2.1) ∧2 L * / Λ ∧L * = ∧2 L * (λ) ∧(c) with λ in Λ and c in L * .

Theorem 4 . 6 .

 46 Let k be an imaginary quadratic number field with ring of algebraic integers O, and let α in Z[∆×∆×∆×∆] ∆ be a class obtained from Algorithm 4.4. Then Proposition 2.12 applies to α with Λ = Λ = O * . Multiplying the resulting element in ker(δ 2,k ) by a = lcm(2, | Λ|) as in Remark 2.6, we obtain a non-zero element in K 3 (k)/torsion.

.

  The ideal polyhedron F specified for O -3 in [1] admits as symmetry group the group of order 12 generated by 1 -1 1 0 and 1 ρ -1 ρ 0 . Therefore the group Γ discussed in the proof of Proposition 4.5 is a normal subgroup of index 12 in SL 2 (O -3 ), such that the quotient of SL 2 (O -3 ) by Γ is that symmetry group.

Figure 5 . 1 .

 51 Figure 5.1. Base triangles for O -3 . Each vertex is specified by the matrix which moves the point (∞, 0) into it.

2 ,.

 2 The matrices B and C are in the cusp stabiliser Γ ∞ ; concatenating them after the matrix A gives the moves from the cusp at ∞ to the cusps at the corners of F ; and we generate them with the three basic moves A • ∞ = 0, B • 0 = ρ and C • 0 = 1. Combining with words in B and C concatenated before the matrix A, allows us to express all of the vertices of the base triangles as images of the basepoint b = (∞, 0). The coordinates of the vertices of the base triangles in Figure5.1 are then the following:(∞, ρ) = B • b, (∞, 1) = C • b, (0, 0) = A • b, (0, 1) = AC • b, (0, ρ) = AB • b, (1, 0) = CA • b,(1, 1)= CAC • b, (1, ρ) = CAB • b, (ρ, 0) = BA • b, (ρ, 1) = BAC • b, (ρ, ρ) = BAB • b.

( 1 ,

 1 CAC, C, CAB) + (1, CAB, C, BAC) + (1, BAC, C, B) + (1, BAC, B, BAB) +(1, BA, BAC, BAB) + (1, BA, BAB, B) + (1, AB, BA, B) + (1, A, AB, B) +(1, A, AC, AB) + (1, A, C, AC) + (1, AC, C, CA) + (1, CA, C, CAC) + (1, CA, CAC, CAB) +(1, CA, CAB, BAC) + (1, CA, BAC, BA) + (1, CA, BA, AB) + (1, CA, AB, AC).

  4 and Proposition 4.5. Namely, because the image of Γ in P SL 2 (O) is torsion-free by the proof of Proposition 4.5, we can use Λ = Λ = {±1} and ∆ = GL 2 (k)/{±id 2 } in Theorem 4.6. Then a = 2, and the non-trivial element we obtain in B(k)/torsion ≃ K 3 (k)/torsion is 2[ρ].5.2. The case of discriminant-7. Let ω := 1+ √ -7 2 in k = Q( √ -7). The ideal polyhedron F specified for O -7 in[1] admits as symmetry group the group of order 6 generated by

Acknowledgements. We are grateful to the Irish Research Council for funding a work stay of the second author in Amsterdam, and the De Brun Centre for Computational Homological Algebra for funding a work stay of the first author in Galway. The initial research for this paper was strongly propulsed by these two meetings of the authors. We would like to thank Herbert Gangl and Philippe Elbaz-Vincent for useful conversations.

Note that g

with the g i in SL 2 (O), implies that i 1 = i 2 , and with i = i 1 = i 2 we have that C -1 i g -1 2 g 1 C i is in (4.3) u µ 0 u -1 with u in O * and µ in I for some fractional ideal I of k. Namely, for the orbit of ∞ this is clear because the corresponding C i is the 2 × 2-identity matrix; and for a cusp C i • ∞ corresponding to a non-principal ideal this follows from Corollary 4.2, using that O * = {±1}, because the only two imaginary quadratic fields with O * = {±1} are Q( √ -3) and Q( √ -1), which have class number 1. Take F as at the beginning of this section, i.e., it is an ideal polyhedron tessellated by finitely many fundamental domains for the action of SL 2 (O) on H, such that the facets of F occur in pairs with an SL 2 (O)-identification between them. Let p 1 , . . . , p s be the cusps in the boundary of F . Fix elements A 1 , . . . , A s in GL 2 (k) of the form gC i with g in SL 2 (O) such that p j = A j • ∞. On the plane attached to the cusp p j , we induce the origin and axes for the second coordinate by transport with the matrix A j of the origin and axes on the plane at the cusp ∞. Denote the group {± 1 µ 0 1 | µ ∈ O} by U . We then have the following algorithm. • Triangulate the surface of F . This is always possible since F is a polyhedron and hence its surface is homeomorphic to a 2-sphere. As F is an ideal polyhedron, we can choose all of the vertices of the triangulation to lie again at the cusps; and we do so. • Omit the triangles that touch the basepoint b, in order to obtain a set of base triangles.

• For each base triangle, record the tetrahedron that it spans together with b. collection of tetrahedra is the triangulation that we want. Second and final step: Express F as a chain of matrix quadruples, such that each of the ideal tetrahedra is specified by the quadruple of matrices that map the basepoint b to its four corners. We obtain these matrices by concatenating

• a U -move in the plane attached by the Borel-Serre construction at the cusp ∞ with • the fixed matrix A i moving the cusp ∞ to the cusp where the i-th corner of the tetrahedron is located.

We observe that the matrices chosen here are unique up to multiplication by ±id 2 . Moreover, if we have g i • b = hg j • b for moves g i and g j , and h in SL 2 (O), then g -1 i hg j is in the group in (4.3) and stabilises b, hence is of the form u 0 0 u -1 with u in O * . Taking, in the notation of Section 2, Λ = Λ = O * , and ∆ = GL 2 (k)/ Λ • id 2 , we have well-defined quadruples (g 0 , g 1 , g 2 , g 3 ) of classes in ∆, and the geometric identification of points under the action of SL 2 (O) now gives rise to identification of the classes in ∆ of the corresponding chosen matrices. Finally, we notice that from (4.3) it is also clear that the quadruples are all in P 3 (k).

We now show that the resulting element α in P 3 (k) satisfies Assumption 2.11.

Proposition 4.5. The formal sum α of quadruples of matrix classes produced by Algorithm 4.4 has the following properties.

(1) It is a sum of (g 0 , g 1 , g 2 , g 3 ) with each g i in some coset A j i U .

(2) In P 2 (k) ∆ , we can write dα = s j=1 A j β j for β j in the kernel of

Proof.

(1) The second step of Algorithm 4.4 places the matrices g i into some coset A j i U . Therefore the group Γ discussed in the proof of Proposition 4.5 is a normal subgroup of index 6 in SL 2 (O -7 ), such that the quotient of SL 2 (O -7 ) by Γ is that symmetry group. The set of cusps in the boundary of our ideal polyhedron F is {0, 1, ω, ω 2 , ω+1 2 , ∞}. We will make use of the following matrices, which map some (but not necessarily all) of the elements of this set of cusps back into this set.

The matrices V and W are not in SL 2 (O), so we do not obtain an element of H 3 (Γ; Γ 1 , . . . , Γ s ) along the way as was the case with the element α for the Eisensteinian integers. But this is no problem, because for Theorem 4.6, we only need the matrices to be in ∆ ∞ 1,1 . Concatenating words in V and W after the matrix S gives the moves from the cusp at ∞ to the cusps at the corners of F ; and we generate them with the three basic moves S • ∞ = 0, V • 0 = ω 2 and W • 0 = 1 2 . The matrices V 2 and W 2 are in the cusp stabiliser Γ ∞ ; combining with words in V 2 and W 2 concatenated before the matrix S, allows us to express all of the vertices of the base triangles as images of the basepoint b = (∞, 0). The coordinates of the vertices of the base triangles in Figure 5.2 are then the following:

The tetrahedra of the triangulation, specified by these base triangles, then give the following α:

(1, SW 2 , V W S, V SW 2 ) + (1, SW 2 , W 2 SV 2 , V W SW 2 ) + (1, V SW 2 , V W S, V 2 S)