Arthur Vigan

Maxime Lopez

Mona El Morsy

Eduard Muslimov

Alexandre Viret

Gérard Zins

Graham Murray

Anne Costille

Gilles P P L Otten

Ulf Seemann

M El Morsy

H Anwand-Heerwart

K Dohlen

P Blanchard

J Garcia

Y Charles

N Tchoubaklian

T Ely

M Phillips

J Paufique

J.-L Beuzit

M Houllé

J Costes

R Pourcelot

I Baraffe

R Dorn

M Jaquet

M Kasper

A Reiners

A Smette

L Blanco

L Pallanca

A Carlotti

É Choquet

D Mouillet

M N'diaye

Connecting SPHERE and CRIRES+ for the characterisation of young exoplanets at high spectral resolution: status update of VLT/HiRISE

Keywords: High-contrast imaging, High-spectral resolution, High-dispersion coronagraphy, VLT, Exoplanet, Characterisation, Spectroscopy

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Direct imaging allows to spatially separate and directly measure radiation from an exoplanet, which enables the spectral analysis of its atmosphere with minimized impact from the host star. Direct imagers such as SPHERE, [START_REF] Beuzit | SPHERE: the exoplanet imager for the Very Large Telescope[END_REF] GPI, [START_REF] Macintosh | First light of the Gemini Planet Imager[END_REF] and Send correspondence to A. Vigan (arthur.vigan@lam.fr). 1 arXiv:2207.06436v1 [astro-ph.IM] 13 Jul 2022 SCExAO 3 are designed to find and detect young planets around nearby stars, [START_REF] Macintosh | Discovery and spectroscopy of the young jovian planet 51 Eri b with the Gemini Planet Imager[END_REF][START_REF] Chauvin | Discovery of a warm, dusty giant planet around HIP 65426[END_REF][START_REF] Keppler | Discovery of a planetary-mass companion within the gap of the transition disk around PDS 70[END_REF] which can be used to study the population of young giant exoplanets through large direct imaging surveys. [START_REF] Vigan | The International Deep Planet Survey. I. The frequency of wide-orbit massive planets around A-stars[END_REF][START_REF] Galicher | The International Deep Planet Survey. II. The frequency of directly imaged giant exoplanets with stellar mass[END_REF][START_REF] Vigan | The VLT/NaCo large program to probe the occurrence of exoplanets and brown dwarfs at wide orbits. IV. Gravitational instability rarely forms wide, giant planets[END_REF][START_REF] Vigan | The SPHERE infrared survey for exoplanets (SHINE). III. The demographics of young giant exoplanets below 300 au with SPHERE[END_REF] However, current instruments are limited for the characterization of companions by their spectral resolution of R = λ/∆λ = 400 at most. [START_REF] Vigan | Exoplanet characterization with long slit spectroscopy[END_REF] High-dispersion spectrographs have been used to detect the thermal radiation of various giant exoplanets by relying on the large number of molecular lines detectable in the atmospheres of these planets, which can only be accessed at spectral resolutions above a few 10 000s. Important results have been obtained on both transiting and non-transiting planets, [START_REF] Snellen | The orbital motion, absolute mass and high-altitude winds of exoplanet HD209458b[END_REF][START_REF] Brogi | The signature of orbital motion from the dayside of the planet τ Boötis b[END_REF][START_REF] Birkby | Detection of water absorption in the day side atmosphere of HD 189733 b using ground-based high-resolution spectroscopy at 3.2µm[END_REF] as well as on spatially resolved low-mass companions. [START_REF] Snellen | Fast spin of the young extrasolar planet β Pictoris b[END_REF][START_REF] Schwarz | The slow spin of the young substellar companion GQ Lupi b and its orbital configuration[END_REF][START_REF] Hoeijmakers | Medium-resolution integral-field spectroscopy for high-contrast exoplanet imaging. Molecule maps of the β Pictoris system with SINFONI[END_REF] For the latter, the most well-known case is that of the directlyimaged exoplanet β Pictoris b [START_REF] Snellen | Fast spin of the young extrasolar planet β Pictoris b[END_REF] using the CRIRES instrument [START_REF] Kaeufl | CRIRES: a high-resolution infrared spectrograph for ESO's VLT[END_REF] on the ESO Very Large Telescope (VLT). This method has since provided more atmospheric detections and rotation speeds of young directly imaged companions, [START_REF] Schwarz | The slow spin of the young substellar companion GQ Lupi b and its orbital configuration[END_REF][START_REF] Hoeijmakers | Medium-resolution integral-field spectroscopy for high-contrast exoplanet imaging. Molecule maps of the β Pictoris system with SINFONI[END_REF][START_REF] Bryan | Constraints on the spin evolution of young planetary-mass companions[END_REF] providing unique insight into the properties of this population of objects. [START_REF] Bryan | Constraints on the spin evolution of young planetary-mass companions[END_REF] Measurements on young companions have been obtained with low-order adaptive optics systems like MACAO [START_REF] Arsenault | MACAO-VLTI: An Adaptive Optics system for the ESO VLT interferometer[END_REF] for the β Pictoris b result. With such systems, the atmospheric turbulence correction concentrates a moderate fraction of the energy in the PSF core (typically 50-60% in the K band [START_REF] Paufique | On-sky results of the adaptive optics MACAO for the new IR-spectrograph CRIRES at VLT[END_REF] ), but the level of the uncorrected halo is high. This means that the strongest limiting factor of the signal-to-noise ratio (S/N) of the planet signal close to the star remains the noise contributed by the uncorrected stellar halo. To significantly decrease the halo, it is necessary to rely on high-order adaptive optics known as extreme adaptive optics (ExAO), which can provide diffraction-limited images in the near-infrared (NIR) and therefore decrease the level of stellar halo (and noise) at the location of the planet. [START_REF] Kawahara | Spectroscopic Coronagraphy for Planetary Radial Velocimetry of Exoplanets[END_REF][START_REF] Snellen | Combining high-dispersion spectroscopy with high contrast imaging: Probing rocky planets around our nearest neighbors[END_REF][START_REF] Wang | Observing Exoplanets with High Dispersion Coronagraphy. I. The Scientific Potential of Current and Next-generation Large Ground and Space Telescopes[END_REF][START_REF] Mawet | Observing Exoplanets with High-dispersion Coronagraphy. II. Demonstration of an Active Single-mode Fiber Injection Unit[END_REF][START_REF] Vigan | Bringing high-spectral resolution to VLT/SPHERE with a fiber coupling to VLT/CRIRES+[END_REF] Diffraction-suppressing coronagraphs can also be used to further improve the contrast and decrease the level of stellar residuals at the location of the planet.

Until recently, no instrument dedicated to high-contrast imaging was equipped with means of characterizing detected companions at very high spectral resolutions. Conversely, none of the existing high-resolution spectrographs in the near infrared is equipped with either an ExAO system or with a coronagraph, which would facilitate the characterization of known companions. In recent years, several projects have proposed combining ExAO systems with existing mid-or highresolution spectrographs using single-mode fibers (SMF) either in the NIR, for example between NIRC2 and NIRSPEC at Keck (KPIC) [START_REF] Mawet | Observing Exoplanets with High-dispersion Coronagraphy. II. Demonstration of an Active Single-mode Fiber Injection Unit[END_REF][START_REF] Mawet | Keck Planet Imager and Characterizer: concept and phased implementation[END_REF] or between SCExAO and IRD at Subaru (REACH), [START_REF] Kawahara | Spectroscopic Coronagraphy for Planetary Radial Velocimetry of Exoplanets[END_REF][START_REF] Kawahara | Characterizing Earth-like Planets Using a Combination of High-Dispersion Spectroscopy and High-Contrast Instruments: Doppler-shifted Water and Oxygen Lines[END_REF][START_REF] Kotani | The infrared Doppler (IRD) instrument for the Subaru telescope: instrument description and commissioning results[END_REF] or in the visible, for example between SPHERE and ESPRESSO at the VLT. [START_REF] Lovis | Atmospheric characterization of Proxima b by coupling the SPHERE high-contrast imager to the ESPRESSO spectrograph[END_REF] For the NIR, the VLT offers a unique opportunity to achieve a similar feat by coupling the high-contrast imager SPHERE 1 with the recently upgraded high-resolution spectrograph CRIRES+, [START_REF] Dorn | The "+" for CRIRES: enabling better science at infrared wavelength and high spectral resolution at the ESO VLT[END_REF] which are now both available at the VLT/UT3. The coupling between these two flagship instruments has been proposed as the High-Resolution Imaging and Spectroscopy of Exoplanets project (HiRISE). [START_REF] Vigan | Bringing high-spectral resolution to VLT/SPHERE with a fiber coupling to VLT/CRIRES+[END_REF][START_REF] Otten | Direct characterization of young giant exoplanets at high spectral resolution by coupling SPHERE and CRIRES+[END_REF] SPHERE offers a unique ExAO system (SAXO) [START_REF] Fusco | High-order adaptive optics requirements for direct detection of extrasolar planets: Application to the SPHERE instrument[END_REF] that has demonstrated exquisite performance on-sky [START_REF] Sauvage | Wave-front sensor strategies for SPHERE: first on-sky results and future improvements[END_REF][START_REF] Petit | SPHERE eXtreme AO control scheme: final performance assessment and on sky validation of the first auto-tuned LQG based operational system[END_REF][START_REF] Milli | Performance of the extreme-ao instrument vlt/sphere and dependence on the atmospheric conditions[END_REF] and efficient coronagraphs; [START_REF] Carbillet | Apodized Lyot coronagraph for SPHERE/VLT. I. Detailed numerical study[END_REF][START_REF] Guerri | Apodized Lyot coronagraph for SPHERE/VLT: II. Laboratory tests and performance[END_REF] its infrared arm covers the Y, J, H, and K s bands. CRIRES+ is a refurbished and upgraded version of the CRIRES spectrograph, [START_REF] Kaeufl | CRIRES: a high-resolution infrared spectrograph for ESO's VLT[END_REF] operating in the Y, J, H, K, L, and M bands (0.9-5.3 µm) at R = 100 000 resolving power with a 0.2 -slit. It features three Hawaii-2RG 2k×2k detectors and a cross-disperser to cover more than 50% of the H band in a single observation. The large overlap in terms of spectral coverage between SPHERE and CRIRES+ is a key advantage in particular in the H and K bands, which contain strong molecular features for CO, CH 4 , H 2 O, or NH 3 . The high angular resolution and high-contrast capabilities of SPHERE at these wavelengths, combined with the high spectral resolution of CRIRES+, will provide a unique system capable of characterizing known directly imaged companions.

In this paper we present the final proposed implementation of the system in Sect. 2, an assessment of the expected performance based on the final hardware in Sect. 3, and finally the current status of the project and its schedule in Sect. 4.

IMPLEMENTATION

Overview

The HiRISE coupling will be implemented on the VLT-UT3 between SPHERE and CRIRES+, which are located on Nasmyth platforms A and B, respectively. This is illustrated in Fig. 1. The system consists in three main sub-systems:

1. the fiber injection module (FIM) implemented inside of SPHERE. The role of the FIM is to pick-up the SPHERE scientific beam and inject the signal of a known exoplanet in a science fiber. The FIM will be described in Sect. 2.2;

2. the fiber bundle that connects SPHERE and CRIRES+ around the telescope. The bundle transmits the planetary and stellar signals up to the entrance of the spectrograph. The bundle will be described in Sect. 2.3;

Figure 1. Overview of the implementation at the telescope. SPHERE and CRIRES+ are both located on the VLT Unit Telescope 3 (UT3), on Nasmyth platforms A and B respectively. The bundle cannot be routed along structure that supports the primary mirror due to the altitude motion during science observations. The favored option is to route the bundle along the Nasmyth structures and beneath the floor. The route depicted here is not exactly the final one that has been studied and proposed by ESO.

3. and the fiber extraction module (FEM) implemented inside of CRIRES+. The role of the FEM is to reimage the output of the bundle's fibers in the VLT focal plane with the appropriate focal ratio. The FEM will be described in Sect. 2.4;

In addition to these main sub-systems, HiRISE also includes a dedicated electronics cabinet located on the SPHERE Nasmyth platform, and a dedicated control software developed following the VLT Software standards. These additional components will be described in Sect. 2.5.

Fiber injection module

The FIM is an additional sub-system implemented inside of the SPHERE enclosure, on the main CPI bench. It diverts the beam of the SPHERE/IFS towards a dedicated optical bench. The optical design with photon sharing is presented in Fig. 2 and the mechanical implementation is presented in Fig. 3. Images of the final hardware being integrated or tested in the lab are presented in Fig. 4.

Originally, the VLT/HiRISE system has been proposed for both the H and K bands. [START_REF] Vigan | Bringing high-spectral resolution to VLT/SPHERE with a fiber coupling to VLT/CRIRES+[END_REF] However, following the simulation study presented in [32], the expected performance in K-band was considered to be insufficient to justify the added complexity of implementing the K band in the system. Most of all, the K band requires the use of ZBLAN fibers that provide a high-tranmission at wavelengths longer than 1.6 µm, which corresponds to the typical wavelength cutoff of pure silica fibers. However, ZBLAN fibers are expensive and not easy to handle without suffering damages. Moreover, few industrial companies are ready to accept the risk to work with this type of fibers for manufacturing a long fiber bundle such as the one required for HiRISE. For these reasons, we decided to drop support for the K band and focus only on the H band using a pure silica fiber, reference 1310M-HP, manufactured by Coherent (formerly Nufern).

In the FIM, the pick-off mirror is immediately followed by a relay lens that reimages the telescope pupil on a piezo tiptilt stage. This tip-tilt is used to move the focal-plane image with respect to the fiber bundle and the tracking camera. The A retro-injection channel is included in the system to emit light from calibration fibers included in the fiber bundle and reimage them on the tracking camera on top of the astrophysical scene. These calibration fibers will be used for the acquisition procedure to center the planet's PSF on the science fiber, as detailed in [39]. Because the beam is not exactly collimated at the level of the dichroic filter, a corrector plate with a very long focal length was added in front of the corner cube to compensate for the small additional length between the dichroic and corner cube. The wavelength of the calibration source is at 1.3 µm, i.e. in the transition region of the dichroic filter.

The system is implemented on a vertical bench above the SPHERE/IFS beam. The pick-off mirror is introduced in the beam with an OWIS precision translation stage, which will be controlled from the SPHERE control software. Most mechanical parts are custom made from aluminum or Invar 36, manufactured by the mechanical workshop at the Georg-August-Universität in Göttingen. The other parts are off-the-shelf components from Thorlabs and Newport. The optics are a mix between off-the-shelf lenses and mirrors from Thorlabs and Edmund, and custom optics manufactured by Optique Fichou † (France).

The fiber bundle will be held in a multi-axis fiber optic positioner from Newport using a custom interface piece. The multi-axis positioner will be used for the fine positioning of the bundle with respect to the injection optics. The focus and tip and tilt of the bundle were used as compensator in the tolerancing analysis of the optical design, so these degrees of freedom are essential for a fine alignment of the system.

Fiber bundle

A schematic of the proposed design for the fiber bundle is presented in Fig. 5. The bundle is a single piece of hardware composed of three main sections. It features four science fibers (one dedicated to the planet) and ancillary fibers for calibration purposes. The bundle will be manufactured by Fibertech Optica ‡ (FTO) in Canada.

Section A is located on the SPHERE side and has a length of ∼3 m. Most of its length is located inside of the SPHERE enclosure. A part of section A is located outside of the enclosure because the ancillary calibration fibers get in/out of the bundle at the level of the interface between section A and B, and they connect to calibration sources located inside the HiRISE electronics cabinet. The input of the bundle features an hexagonal pattern of fibers that includes the four science fibers, one fiber connected to a fluxmeter, and four retro-injection fibers connected to a calibration source at 1.3 µm. The purpose of these five calibration fibers is further described in [39]. The input of the bundle will be directly AR-coated with a coating covering both the science and calibration wavelengths.

Section B is the long section of the bundle that goes around the telescope structure. It is more heavily protected because it will be exposed to the harsh environment of the telescope where it can be crushed or stepped on. It includes a central glass rod and a tensile element to maintain the length, minimize the mechanical stresses on the fibers inside, and protect the fibers from breaking. The route around the telescope has been investigated by the Paranal observatory and a steel cable has been pulled all the way to verify the routing feasibility and measure the exact length (75 m).

Figure 5. Global concept for the fiber bundle. The bundle is a single piece of hardware with no intermediate connectors in order to maximize the transmission of the astrophysical signal. It is divided in three sections: section A on the SPHERE side, section B around the telescope structure, and section C on the CRIRES+ side. Section B is more heavily protected than sections A and C, which lie in relatively protected environment inside of SPHERE and CRIRES+. The bundle includes 4 science fibers and a number of additional fibers used for target acquisition and for calibration. The bundle will be manufactured by Fibertech Optica (FTO) in Canada.

Finally, section C is located on the CRIRES+ side and has again a length of ∼3 m. In the output ferrule, the fibers are arranged in a double line. The main line includes the four science fibers plus two external calibration fibers that will be used during AIT; this line of fibers will be aligned along the slit of the spectrograph. The second line includes two calibration fibers, which will be used for the MACAO system to maintain the deformable mirror flat during the HiRISE observations. One of the two calibration fibers is a SMF identical to the science fibers, and the other is a multi-mode fiber (MMF) with a 50 µm core. Again, the output of the bundle will be directly AR-coated with a coating covering both the science and calibration wavelengths.

The interface parts between sections A, B and C have been specified to minimize impact during the installation of the bundle at the telescope. For the installation, it is foreseen to start the routing on the SPHERE side, which has both the main bundle and ancillary fibers, and then route section B and C step-by-step around the telescope. It was originally proposed to have connectors at the interfaces between sections to make the installation easier, but designing connectors with losses lower than 0.2 dB revealed to be extremely risky and costly, and the final level of losses could not be guaranteed.

Fiber extraction module

The FEM is implemented on the carrier stage unit (CSU) of the warm bench of CRIRES+. It is used to reimage the output of the science and calibration fibers in the VLT focal plane at F/15, which is the focal ratio expected by CRIRES+ for direct observations with the VLT. The opto-mechanical implementation of the FEM on the CSU is presented in Fig. 6.

Compared to the FIM, the FEM is a much simpler system with no active device to be controlled. Its role is to simply reimage the fibers in the VLT focal plane. To make the installation easier and avoid unnecessary bending of the fibers, a folding mirror is implemented in the FEM so that the bundle's ferrule is simply inserted vertically at the top of the FEM. Since it is installed in the visitor gas cell slot of CRIRES+, the FEM must be easily removable in a matter of minutes. It is held in place by 3 screws that can be accessed without removing the cover. The FEM can then be put back in place quickly using a set of calibrated shims and the precision pins available on the CSU.

The FEM does not include active elements, but it must be able to deal with the MACAO adaptive optics system of CRIRES+, which is located downstream in the optical train. One of the difficulties of the MACAO system is that its deformable mirror cannot be maintained flat in open-loop on the timescale of hours, which are foreseen for the observations with HiRISE. To cope with that limitation, the fiber bundle include two calibration fibers (1 SMF, 1 MMF) located nearby the science fibers, which will be fed by a broadband source and will serve as an AO guide star for the MACAO system. Only one of the two will be used, but to keep some flexibility two different types of fibers were included in the design of the bundle: the SMF will be totally unresolved so that it truly resembles a stellar object, but the transmitted flux might be too low for MACAO; the MMF will be slightly resolved with its large 50 µm core, but it will provide a much higher flux. The two options will be tested during the commissioning. Figure 6. Left: Overview of the implementation of the FEM of the CSU at the entrance of CRIRES+. Right: Detailed view of the opto-mechanical design of the FEM with a cut through the module. The interface part at the top that will hold the ferrule of the fiber bundle is not the final one in this drawing. The full module is about 15 cm-long and 8 cm-wide.

Electronics and control

The FIM in HiRISE includes several active and passive devices, which must be interfaced with the HiRISE workstation for operating the system. All the controllers are located in the HiRISE electronics cabinet (HiCab), which will be installed underneath the SPHERE instrument on the Nasmyth platform. To simplify the implementation, the HiRISE workstation is directly embedded within the HiCab in the form of an industrial computer running VLT Software (CentOS Linux 7.7 + VLT common software 2020). The HiRISE workstation is connected to the devices and sensors through different means (Fig. 7). The iPORT frame grabber from Pleora Technologies is connected to the workstation using a dedicated 10GB/s network interface and shortrange optical link. The tip-tilt stage controller (E272 from PI) and the connected power switch (iBoot-G2) are connected on the private HiRISE instrument network through a Cisco switch. Finally, the other devices and sensors are connected to a Beckhoff PLC computer, which is itself connected to the private HiRISE network to be accessible from the workstation. Pictures of the hardware in the integration clean room at LAM are presented in Fig. 8.

Nasmyth platform

Control room

Finally, a dedicated control software and interface is in development for the control of the system. The main interface is presented in Fig. 9. The HiRISE control software follows the same logic and organization as all VLT instruments in order to make the future operations as transparent as possible for the observatory staff. In particular, it is foreseen to develop dedicated templates for the main calibrations of FIM, as well as for the target centering, which corresponds to the accurate placement of the planet's PSF on the science fiber. The target (star) acquisition will be performed by the SPHERE system, and the scientific data acquisition by the CRIRES+ system.

EXPECTED PERFORMANCE

The end-to-end simulations of [32] have clearly demonstrated that the final detection performance of HiRISE is driven by the overall transmission of the system. The transmission budget has been updated based on the final optical components and AR coatings. The complete budget is presented in Fig. 10.

For SPHERE and CRIRES+ we rely on values that have been measured either in the lab or on sky, which have been kindly provided by the two instrument consortia. For the lenses in the FIM and the FEM, we use the AR coatings information provided by the optics manufacturers. For the mirrors, purchased from Edmund, we use a value 0.98 for the reflectivity at all wavelengths.

One of the important budget contributors is the fiber bundle, and in particular the coupling efficiency of the planet's PSF into the science fiber. For the fused silica fiber itself, we use the attenuation values provided in [40] and scaled to the attenuations provided at 1300 and 1500 nm by Coherent for the 1310M-HP fiber ( 10. Top: Individual components in the transmission budget for HiRISE, based on the final hardware (FIM, FEM), on simulations (fiber coupling), on lab measurements and theoretical values (fiber transmission), on measurements provided by the instrument consortia (SPHERE, CRIRES+), and on theoretical values (sky, telescope). See Sect. 3 for more detailed explanations. Bottom: Expected endto-end transmission of the system based on the transmission budget. The plot includes a comparison with the values published in [32], which were the current baseline for the development of the project. the values from [40]). For the coupling efficiency we use the HiRISE simulation model [START_REF] Otten | Direct characterization of young giant exoplanets at high spectral resolution by coupling SPHERE and CRIRES+[END_REF] to compute the overlap integrals between the planet's PSF electric field and the field transmitted by the science fiber. For this computation, we include ExAO residuals (for a 0.8 seeing), SPHERE non-common path aberrations measured with ZELDA, [START_REF] N'diaye | Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor. II. Concept validation with ZELDA on VLT/SPHERE[END_REF][START_REF] Vigan | Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor. III. On-sky validation in VLT/SPHERE[END_REF][START_REF] Vigan | Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor. IV. Temporal stability of non-common path aberrations in VLT/SPHERE[END_REF][START_REF] Vigan | pyZELDA: Python code for Zernike wavefront sensors[END_REF] and the wavefront errors of the FIM based on the Zemax model with as-built optics. We also include a centering error of ∼8 mas (∼0.2 λ/D at 1.6 µm) on the science fiber, which corresponds to a realistic pointing error after centering the planet on the science fiber. [START_REF] Morsy | Validation of strategies for coupling exoplanet PSFs into single-mode fibres for high-dispersion coronagraphy[END_REF] The updated transmission budget shows a small gain of ∼20% at 1.6 µm compared to the values previously reported in [32]. This is encouraging for the final performance of HiRISE, which should hopefully be better than anticipated from previous end-to-end simulations. An update of the astrophysical simulations is currently on-going in preparation for the on-sky validation. The gains are mainly due to the high quality of the AR coatings for the custom optics, the quality of the coatings for the FIM dichroic, and the use of highly transmissive fused silica fibers. The current model does not take into account the transmitance of the glasses used for the custom lenses. This may have a slight negative impact that will be evaluated in the near future.

A significant contributor to the transmission budget is the SPHERE DICH-K dichroic, which is used to send the K band light to IRDIS and the Y JH bands to the IFS and HiRISE. This dichroic shows a strong decrease of its transmission starting at 1.6 µm, which is clearly visible in the top panel of Fig. 10. As part of the SPHERE+ project, [START_REF] Boccaletti | SPHERE+: Imaging young Jupiters down to the snowline[END_REF] an updated version of that dichroic could be considered to improve the transmission budget of HiRISE and the SPHERE IFS. Another possibility to improve the global transmission would be to switch to a dedicated fiber-fed spectrograph such as the VIPA concept, [START_REF] Bourdarot | Experimental test of a 40 cm-long R=100 000 spectrometer for exoplanet characterisation[END_REF] which recently demonstrated a transmission of the order of 40% in H-band on sky Carlotti et al. (this conference).

PROJECT STATUS AND SCHEDULE

HiRISE has followed a step-by-step approach for its validation by ESO. First, the science case for the instrument was submitted to ESO's observing programmes committee (OPC) in September 2019, which expressed a strong scientific support for the project at the end of 2019 and offered the technical aspects to be evaluated by the Paranal observatory and the ESO's scientific and technical committee (STC). Then, a visitor instrument proposal containing a full technical description was submitted to the STC in April 2020, leading to a validation by the STC and ESO's council at the end of 2020 and the acceptance of HiRISE as a visitor instrument for the VLT. The project will now hold a final review of the project with the observatory at the end of July 2022. This final review is not a final design review in the common sense of the term but a final validation of the technical aspects and interfaces that have already been thoroughly discussed between the project and the observatory. As a result, the procurement of the hardware has already started since 2020 (from long lead-time elements) and is now almost complete.

The HiRISE project is therefore entering its main AIT phase in Europe. At the time of writing (end of June 2022), all the hardware for the FIM has been received at LAM, which is the most complex part of the whole system. The bundle is expected to be delivered by FTO in October 2022, and the mechanical parts of the FEM will start to be manufactured over the summer. A "SPHERE simulator" is currently being aligned to deliver a beam with the same properties as the SPHERE IFS beam in which the FIM will be placed. In parallel, a fine pre-assembly of the FIM and its optics will be performed using metrology. Then, the SPHERE simulator and FIM will be aligned relative to each other so that the final optical alignment of the FIM can be performed. Because the fiber bundle will arrive later in the year and because the bundle will not easily be usable to assess the optical quality, we anticipate the use of a Phasics near-infrared wavefront sensor in the injection focal plane to evaluate the optical quality of the alignment. The procedure for the validation of the full system including the bundle and the FEM is still under study.

The current schedule foresees an installation of the system at the VLT between February and March 2023. Although this schedule may shift in case of major issues during AIT in Europe, slots have already been allocated in the observing schedule for period P110. These slots will be dedicated to the installation, which requires the telescope to be put offline for some days and nights, and to the on-sky validation of the system to assess the final performance. Assuming a reasonable level of performance, scientific observations could start at the beginning of P111 in April 2023.

HiRISE is a visitor instrument, which means that it will not be offered for observations to the community as a regular facility instrument. However, anyone interested to observe with HiRISE is cordially invited to contact the project to discuss opportunities. The project anticipates the submission of a large observing programme to observe most known sub-stellar companions observable from Paranal but additional programmes in collaboration with the HiRISE project team are a highly desirable possibility.

Figure 2 .Figure 3 .

 23 Figure 2. Optical design and photon sharing of the FIM located in SPHERE. The science channel transmits only the H band for the injection into the science fiber. The tracking channel transmits the Y and J bands for acquisition and monitoring with the tracking camera. All the optics are coated with appropriate AR-coatings. Figure borrowed from [39].

Figure 4 .

 4 Figure 4. Pictures of some of the final FIM hardware. Left: pre-assembly of the FIM mechanical structure. Top right: custom dichroic filter. Bottom right: FLI C-RED 2 tracking camera and its Pleora frame grabber.

Figure 7 .

 7 Figure 7. Devices and network implementation of HiRISE. The HiRISE workstation is directly embedded in the HiRISE electronics cabinet to make the installation of the system as easy as possible.

Figure 8 .

 8 Figure 8. Programmable logic controller (PLC) for the FIM, which controls some of the devices: calibration sources, temperature probes, flux meter and pick-off mirror translation stage. Other important devices such as the tip-tilt mirror and the tracking camera are controlled directly by the workstation through a private network.

Figure 9 .

 9 Figure 9. Graphical user interface of the FIM control panel for HiRISE. The HiRISE instrument software is entirely developed following the VLT Software standards.

  which are very slightly worse than

			1.0			Transmission budget
			0.8
	Transmission	0.4 0.6
			0.2
		1400 0.0 0.06 0.07	1450 Sky Telescope	1500 SPHERE CPI 1550 FIM Wavelength [nm] 1600 1650 Fiber (coupling) Fiber (transmission) End-to-end transmission	1700 Design as-built 1750 FEM CRIRES+ Otten et al. (2021) 1800
		0.05
	Transmission	0.03 0.04
		0.02
		0.01
		1400 0.00	1450	1500	1550 Wavelength [nm] 1600 1650	1700	1750	1800
	Figure

* https://www.fresnel.fr/ † https://www.optique-fichou.com/ ‡ https://fibertech-optica.com/

ACKNOWLEDGMENTS

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme, grant agreements No. 757561 (HiRISE) and 678777 (ICARUS), from the Commission Spécialisée Astronomie-Astrophysique (CSAA) of CNRS/INSU, and from the Action Spécifique Haute Résolution Angulaire (ASHRA) of CNRS/INSU co-funded by CNES, and from Région Provence-Alpes-Côte d'Azur under grant agreement 2014-0276 (ASOREX).


Camille Eslan

Céline Sandrine Costa

Vu Vial

Ha Hoang

Pham

Vial

Value components in equestrian self-organization

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Theo Soriano

David Novo

Guillaume Prenat

Gregory Di Pendina

Pascal Benoit

MemCork: Exploration of Hybrid Memory Architectures for Intermittent Computing at the Edge

As the memory can occupy up to 95% of the silicon area for some MCUs, the power it consumes represents the dominant part of the total circuit consumption [START_REF] Patrigeon | Ultra low-power integrated systems for the internet-ofthings[END_REF]. An MCU is designed to run simple applications with one or more tasks. They include a very simple memory hierarchy with memories integrated in a flat address space (no cache). In general, the architecture includes at least three memories: a working one based on SRAM technology, a storage one based on Flash technology and a Read-Only Memory (ROM) that contains the boot code. Depending on the application parameters, such as the duration of the inactive phase, the right compromise must be found between keeping the volatile memories powered and saving and restoring their contents in non-volatile memory so that they can be switched off.

Emerging non-volatile memory technologies (NVM) such as FRAM, RRAM or MRAM have grown in maturity and their integration into MCU architecture has been the subject of a vast amount of research. These memories have SRAMlike performance while having the capability to hold saved data even if the power is turned off. Accordingly, they open new opportunities in the design of ultra-low-power MCUs for standby power critical applications and for energy harvestingbased systems. Non-volatile MCUs that rely on the use of emerging NVM are presented as interesting alternatives to SRAM/Flash-based MCUs. Prior works include MCUs implemented with FRAM [START_REF] Zwerg | An 82 µA/MHz microcontroller with embedded FeRAM for energy-harvesting applications[END_REF] [3] [4] [START_REF] Liu | A 130-nm ferroelectric nonvolatile system-onchip with direct peripheral restore architecture for transient computing system[END_REF], RRAM [START_REF] Su | A 462GOPs/J RRAM-based nonvolatile intelligent processor for energy harvesting IoE system featuring nonvolatile logics and processing-in-memory[END_REF] [7] [START_REF] Wu | A 43pJ/cycle non-volatile microcontroller with 4.7 µs shutdown/ wake-up integrating 2.3-bit/cell resistive RAM and resilience techniques[END_REF] and MRAM [START_REF] Sakimura | A 90nm 20MHz fully nonvolatile microcontroller for standby-power-critical applications[END_REF] [10] [START_REF] Tahoori | A universal spintronic technology based on multifunctional standardized stack[END_REF]. Emerging NVMs are used to provide instant on/off capabilities to MCUs with near-zero power consumption during the inactive phases. These works have investigated the possibility of replacing SRAM and Flash with a single universal memory, or a mix of SRAM/Emerging NVM memories. Although these studies have succeeded in showing the advantages of using emerging NVM or hybrid solutions in specific applications, they usually present a particular solution that is difficult to generalise to different application and technology contexts. This non-systematic approach is costly in terms of design time and does not allow an easy and rapid exploration of different size/performance/power consumption trade-offs.

Our goal is to find the best memory architecture based on a data mapping that takes advantage of the different properties of the available technologies. We present MemCork, a framework for memory architecture and data mapping exploration in intermittent MCU applications. Based on an evaluation on a technology-agnostic FPGA prototype, it allows exhaustively exploring the possible data mapping and memory architecture combinations to find the most energy-efficient solution. To show the relevance of our approach, we use MemCork to evaluate two different intermittent applications and show that different application characteristics can influence the best design choice for the memory architecture and the data mapping.

Abstract-Microcontroller units (MCUs) are often used in Internet of Things nodes that operate intermittently. Such nodes alternate active and inactive phases under strict energy constraints. Typically, the memory system has a significant impact on overall MCU energy consumption. Memory accesses and memory leakage power often dominate the consumption of active and inactive phases, respectively. Emerging Non-Volatile Memory (NVM) technologies have recently enabled the design of nonvolatile MCUs that can significantly r educe e nergy consumption during inactive phases. However, replacing all memories with emerging NVMs is not necessarily the best solution, as it often results in dynamic power overhead during active phases. Instead, a hybrid memory architecture that combines volatile and non-volatile technologies is a promising alternative. However, designing hybrid memory MCUs is challenging because the technology that best fits a d ata s egment d epends o n i ts access pattern during execution (e.g., program memory experiences mostly reads while the stack alternates reads and writes). For a given intermittent application, our goal is to find the best memory architecture based on a data mapping that takes advantage of the different properties of the available memory technologies. To this end, we present MemCork, a tool for hybrid memory architecture exploration in intermittent computing devices.

Based on an instrumented execution on a technology-agnostic FPGA prototype, our tool exhaustively explores the possible data mapping and memory architecture combinations to find the most energy-efficient solution. We evaluate MemCork on two representative intermittent applications and find a customised memory architecture and data mapping that reduces energy consumption by up to 23% compared to a fully NVM solution.

Index Terms-IoT, intermittent computing, design exploration, MRAM, FPGA emulation, wireless sensor node, edge computing, tiny ML

I. INTRODUCTION

For many applications, electronic systems operate intermittently. This is particularly the case in the context of the Internet of Things (IoT). IoT nodes can be used for sensing, actuating, processing, receiving, and/or transmitting data, but these functions are only executed periodically or following a trigger event, meaning that the device remains inactive for the rest of the time. Nodes include computing and storage resources embedded in a Microcontroller Unit (MCU) and usually operate in energy-constrained environments. To avoid unnecessary energy consumption during the inactive phases, IoT nodes can operate under different energy-saving modes, the most aggressive one being the total cut-off of the power supply. This powering down can be intended or not, depending on available energy management policies. Although cutting the power supply is the most radical solution for limiting energy consumption, it is also the one that penalises the most in terms of performance and reactivity.

II. MCU DESIGN AND MAIN CHALLENGES

The design of an energy-efficient MCU implementation of an intermittent IoT application is a challenging task. It includes the definition of an architecture based on a set of memory technologies and the corresponding data and instruction mapping. MCU architecture and compilation flow. As shown in Fig. 1, most MCUs include a Reduced Instruction Set Computer (RISC) CPU core, a set of task-specific logic circuits such as Direct Memory Access (DMA) or Wake-Up Controller, peripherals to communicate with other components, Volatile Memories (VM) and NVM. NVM is usually based on a ROM for the boot code, and a storage memory (Flash or EEPROM) for the program and read-only data. The VM (SRAM) is used to store data during program execution. At the software level, the source code is compiled and then linked to generate the executable, using a dedicated toolchain. We focus on the linker script, which declares and defines the memories and their corresponding address ranges. At compile time, this script maps instructions and data to the memories by grouping them into typically five sections. The .text section contains the application code as a sequence of instructions, the read-only data and the default values of the application global variables initialised to specific values other than zero. The application global variables are also stored in the .data section where default values are initialised from .text section at startup and can be modified during execution. In the same way, the .bss section contains all uninitialised variables and is initialised to zero at startup. The .heap is used for dynamic memory allocation and the .stack for temporary data such as function parameters, return address and local variables. For both of them, their size is defined by the programmer in the linker script. As depicted in Fig. 1, the .text section is stored in NVM, while .data, .bss, .heap and .stack are located in VM. Main challenges. The first challenge is the definition and design of high-level intermittent application and memory models. Accordingly, we define an intermittent application model that covers a wide range of possible applications by dividing them into several phases and defining the activity and operating modes of the memories in each phase. Regarding high-level memory energy models, we define and build them for different technologies and sizes in a consistent way for activation-based evaluation. For the mainstream technologies, we rely on information extracted from datasheets and memory compilers, while for MRAM we perform electrical simulations and characterisation of complete memory banks. The second challenge comes from the fact that in the case of an eventbased IoT node, memory size and access pattern depend on the application type and on its environment. For example, a typical temperature sensor node application will sample data and do some processing (possibly min/max, average) before transmitting the data. This type of application requires a small memory and generates sporadic requests. However, if we take the example of an intelligent application based on machine learning, involving significantly more processing and the storage of many constant values, the memory needs and activity are considerably higher. Furthermore, in eventbased applications, the wake-up probability depends on the environment in which the system is located. Regardless of the application, the memory usage will be strongly correlated with the wake-up frequency. All these parameters have an impact on the activity of the memory and therefore on its energy. To address this challenge, we propose MemCork, a framework for intermittent MCU application evaluation. We use an application evaluation platform allowing a finegrained tracking of the memory activity, performed on a technology-agnostic FPGA prototype to characterise various applications running in real-time in a real-life environment. Finally, the third challenge relates to the design exploration of all the possible memory architectures and data mapping options considering multiple memory technologies for a given application. Accordingly, in MemCork, we introduce a memory architecture and data mapping exploration tool that takes advantage of the different properties of memory technologies to find a custom architecture and data mapping for minimal memory energy consumption.

III. MODELLING ASSUMPTIONS

In this section, we define a set of parameters to model and evaluate the energy of intermittent computing MCUs to enable hybrid memory architecture exploration. To this end, we first define an intermittent application model that covers a wide range of possible applications by dividing them into several phases. We approximate the overall energy as a function of key parameters that capture the activity and operating modes of each memory in each phase. We then build parametric memory models for different technologies and sizes to enable an activation-based energy consumption estimation. Fig. 2 illustrates a possible MCU power profile of an IoT node running an intermittent application. The system is inactive by default and performs event-driven tasks. Events are depicted as flags in the picture; they can be periodic or aperiodic. Before transitioning from an active to an inactive phase, the node can secure the data stored in VM with a transfer to NVM (backup). In this case, it is necessary to add hardware and software support for backup and restore (the reverse operation) of volatile data. Based on these basic assumptions, we define several timing parameters to characterise the application profile. We consider the system in an evaluation time window denoted as T eval = T a + T i , where T a and T i are the sums of all the active (t a ) and inactive (t i ) phases, respectively. We further split t a into an execution phase t ru , a backup phase t ba and a restore phase t re . Thus, t a and t i depend on the frequency of events and the duration of the corresponding phases.

A. Intermittent application model

Regarding the energy modeling, we define the MCU energy during T eval as E mcu , which corresponds to the sum of the consumption during T a (E active ) and T i (E inactive ). E active is the sum of the energy of each active phase (e active ) and E inactive is the sum of each inactive phase (e inactive ). Thus, e active = e restore +e run +e backup . E mcu is also the sum of the energy of the memory E mem and the remaining logic E logic . However, as justified in Section I, we assume E logic ≪ E mem . Thus, we approximate the consumption of the MCU to that of all its memories, i.e., E mcu ≈ E mem . The memory energy during any phase is E mem = E dyn + E stat , with E dyn being the sum of the dynamic energy of each memory (e dyn ) and E stat the sum of their static energy. We define the dynamic energy of a memory as e dyn = (n rd ×E rd )+(n wr ×E wr ), n rd and n wr being the number of read and write accesses, E rd and E wr the read and write energy, respectively. We define backup and restore content as B br = B vm with B vm being the size of data and/or instructions stored in a VM. For backup activity, we consider the reading of B vm in each of the VM and the writing of B br in the destination NVM. For restore activity, we consider the reversed process. Backup and restore phases duration t ba and t re depend directly on the bandwidth of the source and destination memories and on the amount of data to be transferred B br . We assume that the backup and restore procedures are executed by a dedicated device such as a DMA.

For static energy, we assume a typical memory level power management, i.e., power modes can be controlled independently for each memory. We also assume On and Off modes. Memories can only be accessed in On mode, which corresponds to a static leakage power of P statOn . The Off mode allows a lower static leakage power P statOf f but leads to the loss of the VM content. The static energy of an execution phase of duration t phase is e stat = P stat × t phase . For P stat during a run phase, we consider all memories that contain data or instructions as On while other memories are Off (e.g., empty memory or NVM used only for backup). Concerning backup and restore, we consider all VM and NVM involved in the process as On while other memories are Off. Finally, during an inactive phase all memories are Off.

B. Memory models

To calculate the energy for different memories, we model each technology in terms of the parameters defined in the previous subsection: E rd , E wr , P statOn , P statOf f . As we explore memory sizing, we choose to express these parameters as a function of the memory size. We collect data for different memory sizes of each technology and approximate each parameter as an affine function ax + b with x being the memory size, and a and b technology-dependent constants.

Parameters were extracted from commercial MCU datasheets for 1MB and 2MB Flash memory with a 90nm technology node, which was scaled down to 28nm. The same information was obtained for two 28nm FD-SOI SRAM (16kB and 64kB) built with the single-port high-density compiler supporting forward body bias (SPHD-BODYBIAS) from STMicroelectronics.

Emerging NVM, such as MRAM, are less mature than mainstream technologies. As a result, we have no access to datasheet information or memory compilers. Instead, we resort to the design of a full memory at the transistor level and its characterisation to extract the mentioned parameters. We choose to design a high-density 32-bit word single bank with an SRAM-like interface. The chosen CMOS process is 28nm FDSOI from STMicroelectronics and the considered MRAM processes are STT and SOT, with MTJ diameters of 28nm.

The first step is to calibrate the compact MTJ models of the two MRAM technologies for electrical simulations. For each of them, we calibrate the model to fit the state of the art for an MTJ diameter of 35nm, and then extrapolate to 28nm. Based on these compact models, we can design and characterise 128kB STT and SOT memories. Both memories include 1024 rows and columns. Since the word size is 32 bits, we have 32 words per row, resulting in a 32-to-one multiplexing to select the word. The address is thus composed of 15 bits: 10 to select the row and 5 to select the word in the row. We characterise the bit cell by evaluating the parasitic resistance and capacitance of the access lines depending on the memory size. This is evaluated by drawing a very simplified layout of the memory array to approximate the size of the access rows and parasitics. These parasitics are injected in the critical path of the memory, which contains the bit cell, made of the MTJ and access transistor, the writing drivers, reading amplifiers and access transistors such as activation transistors and multiplexing. Thus, the critical path is representative of the worst-case scenario for any bit cell and allows an exhaustive set of electrical simulations using standard electrical simulators such as Spectre from Cadence. We use these simulations to ensure the operation of the memory and extract the main parameters in terms of writing and reading currents, as well as timing data. Full memory operation is then performed and validated using a "fast-spice" simulator, such as UltraSim from Cadence, to reduce the simulation runtime and cover a large number of cases. From these other simulations, we can extract write and read energies, the On and Off static power due to leakage of a 128kB memory for both STT and SOT as shown in Table I. To extend these estimations to other memory sizes, we build linear functions of the MRAM parameters. We assume that P statOn and P statOf f increase linearly with the memory size as all powered transistors contribute to leakage. For E rd and E wr we assume constant values because the bit-cells selected in a memory access remain constant for any memory size.

IV. MEMCORK

Now that we have defined the application and memory models, we look more closely at the memory exploration tool. In this section, we present MemCork, a framework for memory architecture and data mapping exploration in intermittent MCU applications. First, we propose an evaluation methodology based on a fine-grained tracking of the memory activity performed on a technology-agnostic FPGA prototype. It enables to retrieve both compile-time and run-time application parameters of real-time intermittent applications running in a real-life environment. Second, we combine the extracted activity information with the high-level models introduced in Section III in an exploration tool. This tool allows us to exhaustively explore the possible hybrid memory architectures and data mappings to select the most energy-efficient solution. We follow a methodology similar to [START_REF] Soriano | An FPGA-based emulation platform for edge computing node design exploration[END_REF], where an FPGAbased emulation platform for edge computing node design exploration is used to track the memory activity of a real execution. It includes a wireless sensor node prototype for architecture evaluation and exploration under real-life conditions. The prototype is based on a Nexys video FPGA development board connected to a radio module (with Bluetooth low energy and LoRa) and sensors (inertial measurement unit, microphone, temperature, humidity, and camera). It also integrates a power management unit for battery voltage regulation. As illustrated in Fig. 3, an open-source 32b RISC-V MCU architecture with various peripherals is embedded into the FPGA.

A. Application evaluation

We extend the platform with an enhanced monitor, capable of retrieving run-time information and to evaluate the activity of memories with a fine granularity. Monitoring memory addresses individually is not realistic as it would generate excessively large traces. Thus, we change the activity monitoring to record at the section granularity (see Section II). The monitoring infrastructure performs two tasks: (1) configuration of the FPGA monitors based on the application compile-time parameters (sizes of the sections), and (2) recording of application run-time parameters (obtained by running the application on the platform during T eval ). The monitor, configured from software, tracks n rd and n wr , the number of reads and writes in several parameterizable sections. It also measures T i as described in Section III. Once configured, the monitor is nonintrusive.

It must be noticed that the application execution on FPGA does not include backup and restore phases, as they depend on the memory technology. Their execution is only considered in the subsequent hybrid memory exploration step (Section IV-A). Since they result in an extra activity at the beginning (back-up), and the end (restore) of the inactive phases, it requires a minimum inactive duration t i between two run phases to be feasible.

B. Hybrid memory exploration

• Compile-time parameters As illustrated in Fig. 4, this exploration tool uses five inputs; first, a library containing the properties and energy models of each available memory technology (detailed in Section III); second, M , the set of memories, with their corresponding technology and size defined by the user; third, N , the set of sections and their properties; fourth, the compile-time parameters that contain information about sections such as their sizes; finally, the output of the monitor with the resulting run-time parameters (detailed in Section IV-A).

We propose an algorithm that explores all possible memory architecture and section mapping combinations for the evaluated application, with m = |M | the number of memories, and n = |N | the number of sections. A memory M j can store |M j | bytes, and can be read-only (R), or read/write (R+W). |N i | is the size in bytes of a section N i . A feasible mapping is an application f : N → M where all sections can be mapped to the available memories. One or more N i sections can be mapped into a memory M j provided that R+W N i sections are only mapped to a R+W M j memory. Based on its inputs, the tool generates S, the set of solutions (feasible mappings) with cardinal |S| ≤ m n , depending on the constraints of the different sections and memory, i.e., type (R or R+W) and size.

Based on the section sizes and on the required backup space, the tool computes the dimension of each memory for each solution of the S set. A user-defined upper and lower limit is assumed for each memory technology. According to the generated section mapping, it determines the minimal memory depth that minimises the total consumption. A memory size is valid only if it is within the range allowed in the corresponding memory technology library. Once calculated for each solution, the tool extracts S ′ , the subset of S containing all solutions for which the memory size is valid.

Finally, based on the run-time parameters obtained with the monitor described above, the solutions of the set S ′ are evaluated following the energy modelling methodology proposed in Section III. As mentioned in Section IV-A, the memory activity and duration related to backup and restore is introduced at this stage. The energy of each memory for each application phase during the evaluation window is calculated.

The output of the tool is the solution set F , which is a subset of the set S ′ . It only includes the solutions where execution is feasible, i.e., complies with the minimum t i required for backup and restore.

V. RESULTS

Recent studies have succeeded in showing the advantages of using emerging NVM or hybrid solutions in specific applications. MemCork is a systematic and automated exploration tool, able to cover a wide range of applications. To demonstrate this capability, we define two applications with very different characteristics: a simple light application representative of temperature-like sensor nodes, and a much more complex heavy application based on machine learning inference programmed in Lite for Microcontrollers. For both applications we assume a periodic wake-up with a period T and an identical execution phase at each wake-up. In this particular case, T eval = T is sufficient to evaluate the application.

Table II shows the application parameters for both applications. For the light one, the code is very simple resulting in a .text section smaller than 10kB and the amount of raw data generated by the sensor is very low (4B). The data section is empty while the .bss section is a bit more than 5kB because it contains all the user-level input and output buffers of communication peripherals. For the heavy application, the code is significantly more complex integrating multiple libraries (TensorFlow Lite + KissFFT), which results in a significantly larger .text section (236kB). Furthermore, the input raw data have a maximum size of 32kB, the associated features use 2kB and the TensorFlow Lite model uses 20kB. Concerning run-time parameters, the light application t ru is 17ms versus the 3.53s for the heavy application. Table II shows that in both cases, reads in .text section represent a large part of the overall memory activity (from 83% to 98%). For the light application, there is no activity in the .text section as it is empty and the application does not use dynamic allocation resulting in no activity in the .heap section. The .bss and .stack sections are the most active after the .text section. This is because it contains the input and output buffers of the peripherals: in this type of application, the use of the peripherals represents a large part of the activity. Concerning the heavy application, the .heap and .stack sections show a high activity due to the complexity of the application. We also observe a certain activity in the .bss section. Indeed, this section contains an array of about 10kB generated by TensorFlow and called Tensor Arena, which is used to store input, output, and intermediate arrays during inference. Finally, we notice that the activity in the .data section is very low because the application uses very rarely initialised global variables. We used MemCork to study the influence of application parameters such as the wake-up period, section size and activity on the memory architecture and section mapping. The read/write accesses for each section were retrieved for the considered application, over the chosen evaluation window. The results are summarised in Fig. 5: the top bar graphs for the light application and the bottom ones for the heavy application. In this experiment, we considered 7 memories in the set M: one Flash (green), two SRAM (dark blue), two STT (pink) and two SOT (light blue); and 6 sections in the set N: .text (dark purple), .data (purple), .bss (dark pink), .heap (orange), .stack (yellow), and backup (white). Because of the page write granularity of Flash memory, it can only contain read-only sections and be used for backup. The exploration was performed on this set for both applications with different wake-up periods T i : 0.1 second, 1 second, and 1 day for the light applications; 5 seconds, 10 minutes and 1 day for the heavy application. For each run, we extracted for analysis, among thousands of generated mappings, three relevant solutions: HYB (hybrid), NV (fully non-volatile) and BEST (optimal solution). The hybrid solution uses a SOT memory for .text section and backup content plus an SRAM memory for all other sections. The fully non-volatile solution uses a single SOT memory for all sections. We use these two solutions as a baseline to represent typical microcontroller architecture with SOT replacing Flash and both Flash and SRAM. Finally, the optimal solution is the one that results in the lowest average power consumption during T eval . Each line of the graph represents a N → M mapping solution on the left side, and the associated average memory power consumption on the right. For example, the first line of Fig. 5(a) represents the light application mapped in a hybrid architecture where .text section and backup content are located in a 16kB SOT memory while .bss, .heap and .stack sections are located in an 8kB SRAM. The corresponding average power for a 100ms period is 0.2 mW.

We observe for both applications, on the lowest considered duty cycles (1 day period), that the best solution relies on a Flash for the .text section. In the absence of precise figures, it should be noted here that we have not taken into account the static consumption P statOf f of the Flash: we can therefore assume that this is an idealised consumption for this technology. This encourages the choice of Flash for very low duty cycle applications, even though its read energy is higher than that of other technologies. For the light application, for a 100ms and 1s period the BEST solution selected by MemCork was a hybrid solution with .text section in an 8kB SRAM and all other sections and backup content in two 8kB SOT. This solution allows up to a 23% reduction in average energy consumption compared to a fully non-volatile solution. Considering the heavy application, for a 5s period, the best solution is a 256kB SOT for .text, .data and .bss sections, a 64kB SRAM for .heap and .stack and a 64kB SOT for content backup/restore. This solution allows a 15% reduction in average memory power compared to a fully nonvolatile approach. It combines the low active static power of SRAM in On mode during active phase and the low static power of SOT in Off mode. For a 10 min period, for the BEST solution, .heap and .stack sections are now directly mapped to the 64kB SOT and the use of a 64kB SRAM is no longer optimal. The resulting optimised fully non-volatile solution uses two non-volatile memories that are smaller than the single memory of the traditional solution, allowing a 12% reduction of the power consumption. For such a small duty cycle, this solution takes full advantage of the low inactive static power of SOT technologies. The solutions generated by the memory mapper are optimised to avoid large unused memory segments, which reduces the aggregate size of the memories.

VI. RELATED WORKS

To the best of our knowledge, this is the first work to propose a systematic and automated exploration framework of hybrid memory MCU with custom data placement. Prior works target architecture exploration and/or data placement for cache-based System-on-Chip (SoC) with hybrid memory technologies. These works also rely on a methodology for the evaluation of performance, energy, and/or area of hybrid memory-based SoCs [13] [14]. In [START_REF] Malladi | FAME: A fast and accurate memory emulator for new memory system architecture exploration[END_REF] and [START_REF] Chang | Performance impact of emerging memory technologies on big data applications: A latency-programmable system emulation approach[END_REF], the authors use an FPGA emulation method to evaluate the performance of hybrid memory architecture. However, these works target high-performance computing architectures and applications, which differ significantly in the intermittent edge computing domain targeted in this work. Patrigeon et al. [START_REF] Patrigeon | FlexNode: a reconfigurable internet of things node for design evaluation[END_REF] developed a memory technology exploration methodology that targets MRAM integration in ultra-low power MCU architecture. They also use an FPGA to emulate a complete microcontroller architecture in a real-world environment and evaluate memory consumption for different technologies using activation-based estimations. Assuming a two-section memory (one section for the application code and one section for the rest), they observe that a hybrid configuration with MRAM for code and SRAM for data is 26% more energy efficient than a traditional Flash + SRAM approach. However, this work only considers very basic applications and low complexity benchmarks. Furthermore, they only perform technology exploration for two memories with a constant size (128kB for code and 16kB for data) regardless of the application size.

VII. CONCLUSION

We introduced MemCork, a systematic and automated framework for hybrid memory architecture and data mapping exploration. This tool allows identifying the most energyefficient solution for a given application based on a finegrained tracking of the memory activity, performed on a technology-agnostic FPGA prototype running applications in real-time and in a real-life environment. We used MemCork on two typical applications to study the influence of the application type and its environment on the most energyefficient solution. MemCork found automatically solutions for both types of applications, reducing memory consumption by up to 23% compared to a naive memory architecture based on a single NVM. We expect MemCork to encourage the use of emerging memory technologies in the design of low-power MCU for intermittent computing at the edge.

Fig. 1 .

 1 Fig. 1. Typical microcontroller compilation flow and architecture

Fig. 2 .

 2 Fig. 2. Intermittent computing MCU power profile

Fig. 3 .

 3 Fig. 3. Evaluation platform

Fig. 4 .

 4 Fig. 4. MemCork exploration flow

Fig. 5 .

 5 Fig. 5. Memory architecture (left) and corresponding average power (right) of a hybrid (HYB), a fully non-volatile (NV) and the optimal (BEST) solutions proposed by MemCork depending on different wake-up periods for the light (top) and heavy (bottom) applications

ACKNOWLEDGMENT

The authors acknowledge the support of the French National Research Agency (ANR), under grant ANR-19-CE24-0017 (NV-APROC project).


Adrien Le Boudec
email: adrien.le-boudec@ens-lyon.fr

Nicolás Matte Bon
email: mattebon@math.univ-lyon1.fr

SOME TORSION-FREE SOLVABLE GROUPS WITH FEW SUBQUOTIENTS

Some torsion-free solvable groups with few subquotients

Adrien Le Boudec, Nicolás Matte Bon

Introduction

A solvable group G has finite rank if there is k ě 1 such that all finitely generated (fg) subgroups of G are generated by at most k elements. The class of groups of finite rank is stable under the operations of taking subgroups and quotients, and hence under taking subquotients. Recall that a group Q is a subquotient of G if there are subgroups K, H of G with K H and Q » H{K. The simplest example of a fg solvable group of infinite rank is the wreath product C p Z. The following celebrated theorem of P. Kropholler asserts that looking at the metabelian subquotients of a fg solvable group G suffices to detect that G has infinite rank.

Theorem ( [START_REF] Kropholler | On finitely generated soluble groups with no large wreath product sections[END_REF]). If G is a fg solvable group of infinite rank, then G admits a subquotient isomorphic to C p Z for some prime p.

One of the motivating questions of the present article is whether there exists a "torsionfree version" of this theorem. More precisely, if G is a fg torsion-free solvable group of infinite rank, does G always admit a fg torsion-free metabelian subquotient of infinite rank ? We answer this question in the negative:

Theorem 1.1. The group G below is a fg 3-step solvable group with the following properties:

(1) G is torsion-free;

(2) G contains a normal subgroup that is free abelian of infinite rank;

(3) every finitely generated torsion-free metabelian subquotient of G is virtually abelian.

Even before considering subquotients, if G is a fg n-step solvable group (n ě 3) with a given property P, it is natural to ask whether G admits fg metabelian subgroups that Date: June 14, 2022. Supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency. retain P. Natural examples are provided by "P " not being virtually abelian", or "P " not being virtually nilpotent". A positive answer to that kind of questions generally allows to reduce the study of certain problems about fg solvable groups to fg metabelian groups, which are generally much more tractable. When P is the property of having inifnite rank, the following consequence of Theorem 1.1 shows that the answer is negative. The existence of groups with this property is also new.

Corollary 1.2. The group G is a fg torsion-free solvable group of infinite rank such that every fg metabelian subgroup of G is virtually abelian.

Recall that a solvable group has finite torsion-free rank if it admits a series with abelian quotients G i {G i`1 satisfying dim Q pG i {G i`1 b Qq ă 8. Another natural question that arises from P. Kropholler's theorem is to ask whether every fg solvable group of infinite torsion-free rank admits fg metabelian subquotients of infinite torsion-free rank. This was answered in the negative by P. Kropholler, who constructed counter-examples in [START_REF]A note on the cohomology of metabelian groups[END_REF]. Despite having infinite torsion-free rank, these examples admit rather large subgroups containing torsion elements. The study of the class of solvable groups admitting no metabelian subquotient of infinite torsion-free rank is the subject of the recent work [START_REF] Jacoboni | Soluble groups with no Z Z sections[END_REF] of Jacoboni-Kropholler. Theorem B in [START_REF] Jacoboni | Soluble groups with no Z Z sections[END_REF] shows these groups enjoy strong structural restrictions.

On the construction. Our family of groups from Theorem 1.1 is very much inspired by the construction of solvable groups of P. Kropholler [START_REF]A note on the cohomology of metabelian groups[END_REF], Brieussel [START_REF] Brieussel | About the speed of random walks on solvable groups[END_REF] and Brieussel-Zheng [START_REF] Brieussel | Speed of random walks, isoperimetry and compression of finitely generated groups[END_REF]. The constructions from [START_REF] Brieussel | About the speed of random walks on solvable groups[END_REF][START_REF] Brieussel | Speed of random walks, isoperimetry and compression of finitely generated groups[END_REF] take as inputs several parameters, including a sequence of groups ∆ n " xs n , t n y coming with a specified pair of generators; and produce solvable groups G such that on the one hand even if the groups ∆ n are very small (for instance finite or virtually abelian), the group G might be much larger; and on the other hand good choices for the ∆ n allow some control on the algebraic structure of G. It was already observed in [START_REF] Jacoboni | Soluble groups with no Z Z sections[END_REF] that a construction close to [START_REF] Brieussel | About the speed of random walks on solvable groups[END_REF] yields solvable groups of infinite torsion-free rank which do not admit metabelian subquotients of infinite torsionfree rank (like the groups in [START_REF]A note on the cohomology of metabelian groups[END_REF]). The construction has the property that in order to avoid metabelian subquotients of infinite torsion-free rank, it is necessary that the groups ∆ n have torsion elements, and as a consequence the groups G from [START_REF] Brieussel | About the speed of random walks on solvable groups[END_REF] and [START_REF] Jacoboni | Soluble groups with no Z Z sections[END_REF] have torsion elements. The delicate aspect of our construction is to ensure torsion-freeness of the group G while keeping the other properties intact.

Recall that the FC-center F CpGq of a group G is the subgroup of G consisting of elements having a finite conjugacy class. A subgroup of G is termed FC-central if it is contained in the FC-center of G. Recall also that a group is locally finite if all finitely generated subgroups are finite.

In the sequel we denote by L the class of groups G that admit a normal subgroup L such that L is locally finite and abelian, and G{L is cyclic. We note that the class L is stable under passing to subgroups and under taking quotients.

Lemma 2.1. Let G be a finitely generated group that is torsion-free. If G admits a finitely generated abelian normal subgroup A such that A is FC-central in G and G{A is in L, then G is virtually abelian.

Proof. Since A is finitely generated, its centralizer C G pAq has finite index in G. Hence upon replacing G by a finite index subgroup, we can assume that A is central in G. Let N be a normal subgroup of G containing A and t P G such that G " N ¸xty and N {A is abelian. Since A is central in G, N is nilpotent. Since N is torsion-free, the quotient N {ZpN q of N by its center is also torsion-free [Rob96, 5.2.19]. But here N {ZpN q must be locally finite, so it follows that N {ZpN q is trivial. So we deduce that N is abelian. Now given x P N , there exists k ě 1 such that x k lies in the center of G. Hence x k " tx k t ´1 " ptxt ´1q k . But x and txt ´1 commute since N is abelian, so rt, xs k " 1. Since G is torsion-free, we infer that x commutes with t. Since x was arbitrary in N and G " N ¸xty, this shows that G is abelian.

We also denote by C the class of groups G such that G{F CpGq belongs to L. Proposition 2.2. Let G be a finitely generated group that belongs to C. If G is torsion-free and metabelian, then G is virtually abelian.

Proof. Since metabelian groups satisfy the maximal condition on normal subgroups [START_REF] Hall | Finiteness conditions for soluble groups[END_REF], the subgroup F CpGq is finitely generated. The group G being torsion-free, F CpGq is also abelian [Rob96, 14.5.9]. Hence it follows that F CpGq is a finitely generated abelian group, and Lemma 2.1 then implies that G is virtually abelian.

We deduce the following.

Proposition 2.3. If G is a group in C, then every finitely generated torsion-free metabelian subquotient of G is virtually abelian.

Proof. Let K be a fg torsion-free metabelian subquotient of G. The class C being stable under taking subgroups and quotients, K belongs to C. The statement then follows from Proposition 2.2.

Notation.

In the sequel we denote by C k " Z{kZ " t0, . . . , k ´1u. Let D " @ a, b | a 2 " b 2 " 1 D be the infinite dihedral group, and Z the infinite cyclic subgroup of D generated by ab, so that D " Z ¸xay " Z ¸xby. We denote by D 1 the derived subgroup of D, which is the index two subgroup of Z generated by pabq 2 . 2.3. The groups. Choose two increasing sequences pd n q and pk n q of positive integers such that k n ě 2d n . For n ě 1, let H n " D kn ¸Ckn , where the action of C kn on D kn is a cyclic permutation of the factors.

Let ∆ " Z ˆśn H n . We denote by π 0 : ∆ Ñ Z the projection from ∆ to the first factor Z, and for all n ě 1 π n : ∆ Ñ H n is the projection from ∆ to H n .

We define two elements s, t P ∆ as follows:

' Let σ 0 be a generator of the first factor Z of ∆. For n ě 1, we denote by σ n the element of H n that belongs to the normal subgroup D kn of H n , and that is defined by σ n p1q " a, σ n pd n q " b, and σ n pjq " id for every j P t1, . . . , k n u such that j ‰ 1, d n . We denote by s the element of ∆ defined by π 0 psq " σ 0 and π n psq " σ n for all n ě 1. ' t is the element of ∆ defined by π 0 ptq " 0, and π n ptq is the generator 1 of C kn for all n ě 1.

Definition 2.4. We denote by G the subgroup of ∆ generated by s and t.

The group G depends on pd n q and pk n q, but to simplify we omit pd n q and pk n q from the notation.

Notation 2.5. In the sequel for i P Z, we write s i " t i st ´i.

Lemma 2.6. The element s 2 is central in G.

Proof. s 2 belongs to the first factor Z of ∆ since π n psq has order 2 for all n ě 1.

Lemma 2.7. For all i ě 0 the following hold:

(1) π n prs, s i sq belongs to pD 1 q kn for all n ě 1.

(2) If n is such that d n ´1 " i then the d n -coordinate of π n prs, s i sq is equal to pbaq 2 , and all other coordinates of of π n prs, s i sq are trivial.

(3) for all n such that d n ´1 ą i, π n prs, s i sq is trivial.

Proof. 1 is clear since π n ps i q belongs to pDq kn for all n ě 1. When d n ´1 " i, the first statement of 2 is a simple computation, and the second statement follows from the fact that the supports of π n psq and π n ps i q intersect only at coordinate d n ´1 " i in view of the inequality k n ě 2d n . 3 is the true for the same reason.

Proposition 2.8. The subgroup F CpGq is free abelian of infinite rank, and Q " G{F CpGq is isomorphic to C 2 Z.

Proof. Let R " À ně1 pD 1 q kn . So R is the subgroup of ś n H n consisting of elements having all their coordinates in pD 1 q kn , and only a finite number of these coordinates are non-trivial. Set A " xσ 2 0 y ' R. We claim that G contains A. Since π n psq " σ n has order two for all n ě 1 and π 0 ps 2 q " σ 2 0 , it is enough to see that R lies inside G, or equivalently that G contains pD 1 q kn for all n ě 1. Since π n ptq permutes transitively the factors of D kn , it is actually enough to see that G contains the copy of D 1 located at the d n -coordinate in D kn . Arguing by induction on n, we see that this follows from Lemma 2.7.

We note that A is normal in G, as it is actually normalized by the entire group ∆. We shall now check that A lies in F CpGq. Since s 2 in central in G (Lemma 2.6), it is enough to check that pD 1 q kn is FC-central in G for all n ě 1. For a fixed n ě 1, the subgroup G n of G consisting of elements g such that π n pgq P pD 1 q kn is a finite index subgroup of G since D 1 has finite index in D and C kn is finite. Moreover G n commutes with pD 1 q kn because D 1 is abelian, so pD 1 q kn indeed lies in the FC-center of G.

We shall now verify that Q " G{A is isomorphic to C 2 Z. If we denote by s i the image of s i in Q, then s i has order two for all i, and it follows from Lemma 2.7 that rs, s i s is trivial for all i. This implies that the map from Q to C 2 Z that sends s to the Dirac function at the identity and t to a generator of Z induces a surjective group homomorphism. Moreover we easily see that no s i belongs to the subgroup generated by the s j for j ă i. This implies that Q Ñ C 2 Z is an isomorphism. And since C 2 Z has trivial FC-center, we also see that A " F CpGq.

We deduce the following: Corollary 2.9. Every finitely generated torsion-free metabelian subquotient of G is virtually abelian.

Proof. This follows from Propositions 2.3 and 2.8.

Torsion-freeness.

The construction of the group G above is a variation of Brieussel's construction in [START_REF] Brieussel | About the speed of random walks on solvable groups[END_REF], which involves three sequence p n q, pd n q and pk n q, and the groups H n " D kn n ¸Ckn , where D n are finite dihedral groups. This also leads to FC-central extensions of C 2 Z, but with the property that F CpGq is locally finite (and hence torsion). With the hope to avoid the appearance of torsion, in our definition the finite dihedral groups have been replaced by the infinite dihedral group D, and the extra factor Z has been added in order to make the generator s of infinite order. But at this point torsion-freeness of the group G is very much not guaranteed. Observe that the projection of G to each factor H n contains many elements of finite order (for instance π n ps i q P H n is an involution for every n and every i). One might expect that by taking appropriate commutators of such elements (so as to arrange in particular the projection to the factor Z to be trivial), it should be easy to find elements of finite order in G. And indeed for most choices of the sequences pd n q, pk n q, the group G will contain many elements of finite order. Perhaps surprisingly, it turns out that for the specific choice d n " n `1, the group G is torsion-free. The goal of this paragraph is to prove this assertion.

Definition 2.10. We denote ϕ a : D Ñ C 2 the epimorphism such that ϕ a paq " 1 and ϕ a pbq " 0 (here and below we use additive notation for C 2 " t0, 1u, so that 1 denotes its generator). Similarly we denote ϕ b : D Ñ C 2 the epimorphism such that ϕ b pbq " 1 and ϕ b paq " 0.

Thus ϕ a pgq is given by the parity of the number of a's in any word representing g, and similarly for ϕ b pgq. We will use the following easy lemma.

Lemma 2.11. Let γ P D be such that ϕ a pγq " ϕ b pγq " 1. Then γ has infinite order.

Proof. Let ϕ : D Ñ C 2 ˆC2 be given by ϕ " pϕ a , ϕ b q. Recall that an element g P D has order 2 if and only if it is conjugate either to a or to b, thus ϕpgq P tp1, 0q, p0, 1qu. Thus the condition ϕpγq " p1, 1q tells at once that γ is not trivial (since ϕpγq is not-trivial) and not an involution, and therefore has infinite order. Proposition 2.12. Suppose that d n " n`1 for all n ě 1. Then the group G is torsion-free.

Proof. We retain the above notation, so that A " F CpGq and Q " G{A. Given g P G non-trivial, we show that g has infinite order. Write g as a word in s ˘1, t ˘1 and, by a standard argument, replace this word by an equivalent word in the free group generated by s, t to write g as g " s 1 i 1 ¨¨¨s k i k t m , with i j , i , m P Z. Note that an expression of g of the form above is far from being unique (in particular, there might be repetitions in the indexes i j ).

If m ‰ 0, then the projection of g to Q has infinite order, and thus so does g. Therefore we can assume that m " 0. Similarly we note that if ř k j"1 j ‰ 0, then π 0 pgq P Z is non-trivial, and thus again g has infinite order. Therefore we can suppose that g has the form

(1)

g " s 1 i 1 ¨¨¨s k i k , j P Z, ÿ j j " 0.

For every P Z let αp q P C 2 be given by αp q " ÿ j : i j " j (mod 2).

Note that the projection of g to G{A » C 2 Z coincides with the configuration α P ' Z C 2 . If the latter is the trivial configuration, we have g P A, and thus g has infinite order by Proposition 2.8. Therefore we can assume that this is not the case, so that there exists some such that αp q " 1. On the other hand by definition of α we have

ÿ jPZ αpjq " k ÿ i"1
k " 0 (mod 2), so we deduce that the set E " t P Z : αp q " 1u is non-empty and has even cardinality. Thus |E| ě 2. Let m and M be respectively the smallest and largest element of E. Upon conjugating g by t ´m, we can suppose that m " 0, while M ą 0. To show that g has infinite order, we will show that the projection π M pgq P H M has infinite order. To this end note that for every i P Z the element π M ps i q belongs to the normal subgroup D k M of H M , and is equal to the element r i P D k M given by (2)

r i pxq " $ & % a if x " i `1 pmod k M q b if x " M `i `1 pmod k M q e otherwise.
From (1) it follows that π M pgq is equal to the element f P D k M given by f " r 1 i 1 ¨¨¨r k i k . We wish to show that the coordinate in f pM `1q P D is an element of infinite order of D by applying Lemma 2.11. To this end, note that ϕ b pf pM `1qq is determined by the parity of the number of bs that appear in the expression f pM `1q " r 1 i 1 pM `1q ¨¨¨r k i k pM `1q. By (2), we have r j pM `1q " b if and only if i j P k M Z. Thus

ϕ b pf pM `1qq " ÿ j : i j Pk M Z j " ÿ Pk M Z αp q (mod 2).
Now recall that 0 and M are respectively the minimum and maximum values of P Z such that αp q " 1 (mod 2). Since k M ą 2M `1 ą M , it follows that in the previous sum there is exactly one term equal to 1 (mod 2), namely αp0q. Therefore ϕ b pf pM `1qq " 1. A similar reasoning shows that ϕ a pf pM `1qq "

ÿ j : i j Pk M Z`M j " ÿ Pk M Z`M αp q (mod 2),
and again, in the last sum there is exactly one term equal to 1 (mod 2) namely, αpM q. Thus ϕ a pf pM `1qq " 1. By Lemma 2.11, f pM `1q has infinite order. Therefore so does f " π M pgq, and so does g.

Conclusion.

The proof of Theorem 1.1 is now complete, as it follows from Proposition 2.12, Proposition 2.8 and Corollary 2.9. 2.6. Final comments. Our initial motivation for wondering about the existence of (nonvirtually abelian) fg torsion-free solvable groups with every fg metabelian subgroup virtually abelian, grew out from the work [START_REF] Boudec | Growth of actions of solvable groups[END_REF]. There we prove that if G is a fg torsion-free metabelian group, then the growth of every faithful action of G (as defined in [START_REF] Boudec | Growth of actions of solvable groups[END_REF]) is at least quadratic, provided G is not virtually abelian. We also conjecture that the same holds for torsion-free solvable groups. Since the desired property passes to overgroups, the metabelian case automatically implies that the conjecture holds for the solvable groups that contain a fg metabelian subgroup that is not virtually abelian. The examples presented in this note show that this does not include all torsion-free solvable groups.

Finally we mention that another motivation for a better understanding of the class of torsion-free solvable groups comes from the study of random walks and isoperimetry. Pittet and Saloff-Coste showed that on any torsion-free solvable group of finite rank which is not virtually nilpotent, the return probability of the simple random walk has the slowest possible decay for a group of exponential growth, namely expp´n 1 3 q [PSC03]. They conjectured that, conversely, this decay should characterise groups of finite rank among torsion-free solvable groups [START_REF] Saloff-Coste | Analysis on Riemannian co-compact covers[END_REF]. This conjecture is naturally related to the theory surrounding Kropholler's theorem and its possible generalisations, as the return probability does not decrease when passing to subquotients. But the groups constructed here seem unlikely to be counterexamples to the conjecture.

Acknowledgments. We are very grateful to Peter Kropholler for decisive discussions regarding this problem, and for his comments on a preliminary version of this work. We thank Jérémie Brieussel and Tianyi Zheng for a conversation on the return probability on the groups constructed here.

Proofs

2.1. Torsion-free metabelian subquotients. The purpose of this paragraph is to prove Proposition 2.3 below, which will be used to check item 3 in Theorem 1.1. That proposition follows from standard arguments. We include proofs for completeness.