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The bright star $\pi$ Men was chosen as the first target for a radialvelocity follow-up to test the performance of ESPRESSO, the new high-resolutionspectrograph at the ESO's Very-Large Telescope (VLT). The star hosts amulti-planet system (a transiting 4 M$_\oplus$ planet at $\sim$0.07 au, and asub-stellar companion on a $\sim$2100-day eccentric orbit) which isparticularly appealing for a precise multi-technique characterization. With thenew ESPRESSO observations, that cover a time span of 200 days, we aim toimprove the precision and accuracy of the planet parameters and search foradditional low-mass companions. We also take advantage of new photometrictransits of $\pi$ Men c observed by TESS over a time span that overlaps withthat of the ESPRESSO follow-up campaign. We analyse the enlarged spectroscopicand photometric datasets and compare the results to those in the literature. Wefurther characterize the system by means of absolute astrometry with Hipparcosand Gaia. We used the spectra of ESPRESSO for an independent determination ofthe stellar fundamental parameters. We present a precise characterization ofthe planetary system around $\pi$ Men. The ESPRESSO radial velocities alone(with typical uncertainty of 10 cm/s) allow for a precise retrieval of theDoppler 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 using18 additional TESS transits, and in combination with the astrometricmeasurements, we determine the inclination of the orbital plane of $\pi$ Men bwith high precision ($i_{b}=45.8^{+1.4}_{-1.1}$ deg). This leads to the precisemeasurement of its absolute mass $m_{b}=14.1^{+0.5}_{-0.4}$ M$_{Jup}$, andshows that the planetary orbital planes are highly misaligned.

A precise architecture characterization of the $\pi$ Men planetary system