  We used the optical and near-infrared imagers located on the Liverpool, the
IAC80, and the William Herschel telescopes to monitor 18 M7$-$L9.5 dwarfs with
the objective of measuring their rotation periods. We achieved accuracies
typically in the range $\pm$1.5$-$28 mmag by means of differential photometry,
which allowed us to detect photometric variability at the 2$\sigma$ level in
the 50\% of the sample. We also detected periodic modulation with periods in
the interval 1.5$-$4.4 h in 9 out of 18 dwarfs that we attribute to rotation.
Our variability detections were combined with data from the literature; we
found that 65$\,\pm\,$18 $\%$ of M7$-$L3.5 dwarfs with $v\,$sin$\,i\ge30$ km
s$^{-1}$ exhibit photometric variability with typical amplitudes $\le$20 mmag
in the $I$-band. For those targets and field ultra-cool dwarfs with
measurements of $v\,$sin$\,i$ and rotation period we derived the expected
inclination angle of their rotation axis, and found that those with
$v\,$sin$\,i\ge30$ km s$^{-1}$ are more likely to have inclinations $\gtrsim40$
deg. In addition, we used these rotation periods and others from the literature
to study the likely relationship between rotation and linear polarization in
dusty ultra-cool dwarfs. We found a correlation between short rotation periods
and large values of linear polarization at optical and near-infrared
wavelengths.
