Context : The rotation-activity connection explains stellar activity in terms of rotation and convective overturn time .
It is well established in stars of spectral types F–K as well as in M-type stars of young clusters , in which rotation is still very rapid even among M-dwarfs .
The rotation-activity connection is not established in field M-dwarfs , because they rotate very slowly , and detecting rotation periods or rotational line broadening is a challenge .
In field M-dwarfs , saturation sets in below v _ { \mathrm { rot } } = 5 km s ^ { -1 } , hence they are expected to populate the non-saturated part of the rotation-activity connection .

Aims : This work for the first time shows intrinsically resolved spectral lines of slowly rotating M-dwarfs and makes a first comparison to estimates of convective velocities .
By measuring rotation velocities in a sample of mostly inactive M-dwarfs , the unsaturated part of the rotation-activity connection is followed into the regime of very low activity .

Methods : Spectra of ten M-dwarfs are taken at a resolving power of R = 200 000 at the CES in the near infrared region where molecular FeH has strong absorption bands .
The intrinsically very narrow lines are compared to model calculations of convective flows , and rotational broadening is measured .

Results : In one star , an upper limit of v \sin { i } = 1 km s ^ { -1 } was found , significant rotation was detected in the other nine objects .
All inactive stars show rotation below or equal to 2 km s ^ { -1 } .
In the two active stars AD Leo and YZ CMi , rotation velocities are found to be 40–50 % below the results from earlier studies .

Conclusions : The rotation activity connection holds in field early-M stars , too .
Activity and rotation velocities of the sample stars are well in agreement with the relation found in earlier and younger stars .
The intrinsic absorption profiles of molecular FeH lines are consistent with calculations from atomic Fe lines .
Investigation of FeH line profiles is a very promising tool to measure convection patterns at the surfaces of M-stars .