Given a fixed spectral resolution, what is the minimum detectable redshift? Given an spectral resolution $R$ (for example $R=1000$), what is the minimum redshift which I can measure? How is this related to the minimum velocity among galaxies? If this velocity is related with the galaxies' peculiar velocities, could I distinguish isolated galaxies from galaxies in overdensities?
I know that $R= \lambda/\Delta\lambda$, but from here I'm getting nowhere.
Any advice or help would be appreciated.
 A: Your question isn't answerable with the parameters you have defined.
The minimum discernable redshift depends on the resolving power of the spectrograph, the stability of the spectrograph (let's ignore that), how well it is wavelength calibrated, the signal-to-noise ratio of the spectrum, and the spectrum of the galaxy (in particular, the nature of the features in the spectrum from which the redshift are measured).
I can give you a rule of thumb based on measurements of stellar spectra - which is that it is very hard to do better than about a twentieth of a resolution element without adopting special techniques (like iodine cells etc).
For a resolving power of 1000, the resolution element in velocity units is 300 km/s, so the smallest possible redshift you could find would be about 15 km/s, which corresponds to $z = 5\times 10^{-5}$.
I stress that this assumes you have an extremely stable, well-calibrated spectrograph and very high signal-to-noise on a galaxy with plenty of sharp spectral features. If any of these conditions are not met then you will do worse.
However, that isn't what limits your ability to detect a small cosmological redshift. The limit is not due to your detection technology, but because cosmological redshift and Doppler shift due to a peculiar velocity are indistinguishable. Since peculiar velocities within groups and clusters of galaxies can be hundreds of km/s, the minimum redshift you can attribute to cosmological expansion is around 1000 km/s and even then will have a large fractional uncertainty. Given a Hubble constant of 70 km/s / Mpc, this means you are limited to galaxies more distant than 12 Mpc or so.
Hence your resolving power is fine to resolve the internal motions of galaxy groups and clusters.
