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The rotation-activity connection explains stellar activity in terms ofrotation and convective overturn time. It is well established in stars ofspectral types F--K and in M-type stars of young clusters. It is notestablished in field M-dwarfs, because they rotate very slowly, and detectingrotation periods or rotational line broadening is a challenge. In fieldM-dwarfs, saturation sets in below v_rot = 5 km/s, hence they are expected topopulate the non-saturated part of the rotation-activity connection. This workfor the first time shows intrinsically resolved spectral lines of slowlyrotating M-dwarfs and makes a first comparison to estimates of convectivevelocities. By measuring rotation velocities in a sample of mostly inactiveM-dwarfs, the unsaturated part of the rotation-activity connection is followedinto the regime of very low activity. Spectra of ten M-dwarfs are taken at aresolving power of R = 200 000 in the molecular FeH absorption band. Theintrinsically narrow lines are compared to model calculations and rotationalbroadening is measured. In one star, an upper limit of vsini = 1 km/s wasfound, significant rotation was detected in the other nine objects. Allinactive stars show rotation below or equal to 2 km/s. In the two active starsAD Leo and YZ CMi, rotation velocities are found to be 40-50% below the resultsfrom earlier studies. The rotation activity connection holds in field early-Mstars, too. Activity and rotation velocities of the sample stars are well inagreement with the relation found in earlier and younger stars. The intrinsicabsorption profiles of molecular FeH lines are consistent with calculationsfrom atomic Fe lines. Investigation of FeH line profiles is a very promisingtool to measure convection patterns at the surfaces of M-stars.Comment: accepted by A&A, 11 pages [abstract abridged

The narrowest M-dwarf line profiles and the rotation-activity connection at very slow rotation