The bright star $pi$ Men was chosen as the first target for a radial velocity follow-up to test the performance of ESPRESSO, the new high-resolution spectrograph at the ESOs Very-Large Telescope (VLT). The star hosts a multi-planet system (a transiting 4 M$_oplus$ planet at $sim$0.07 au, and a sub-stellar companion on a $sim$2100-day eccentric orbit) which is particularly appealing for a precise multi-technique characterization. With the new ESPRESSO observations, that cover a time span of 200 days, we aim to improve the precision and accuracy of the planet parameters and search for additional low-mass companions. We also take advantage of new photometric transits of $pi$ Men c observed by TESS over a time span that overlaps with that of the ESPRESSO follow-up campaign. We analyse the enlarged spectroscopic and photometric datasets and compare the results to those in the literature. We further characterize the system by means of absolute astrometry with Hipparcos and Gaia. We used the spectra of ESPRESSO for an independent determination of the stellar fundamental parameters. We present a precise characterization of the planetary system around $pi$ Men. The ESPRESSO radial velocities alone (with typical uncertainty of 10 cm/s) allow for a precise retrieval of the Doppler signal induced by $pi$ Men c. The residuals show an RMS of 1.2 m/s, and we can exclude companions with a minimum mass less than $sim$2 M$_oplus$ within the orbit of $pi$ Men c). We improve the ephemeris of $pi$ Men c using 18 additional TESS transits, and in combination with the astrometric measurements, we determine the inclination of the orbital plane of $pi$ Men b with high precision ($i_{b}=45.8^{+1.4}_{-1.1}$ deg). This leads to the precise measurement of its absolute mass $m_{b}=14.1^{+0.5}_{-0.4}$ M$_{Jup}$, and shows that the planetary orbital planes are highly misaligned.
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