Inspired by this question about whether dark matter is matter, noting that dark matter tends to be clumped in galaxies near the center and less so on the edges, accepting that many (most?) galaxies have large black holes at their center, and theoretically, black holes have 'infinite' gravity, could black holes actually be the dark matter that holds galaxies together?
Put another way, how do we model the 'infinite' gravity of a black hole when considering the dynamics of a galaxy?
Best Answer
Do the black holes at the center of galaxies account for the experimental results that prompted the introduction of dark matter? No
The primary piece of evidence that originally sparked the idea of dark matter is the rotation curve of galaxies. We found that galaxies don't rotate like the luminous matter suggests it should rotate. Specifically, given some estimates of enclosed mass at some radius from the center of a given galaxy, galaxies were found to rotate faster at given radii than expected. In other words, the luminous matter didn't seem to account for all the mass within any given radius. This lead to the idea that galaxies are permeated by a "dark matter" that isn't luminous. For this idea to work though, the dark matter needs to permeate the galaxy, it can't all be concentrated at the center of the galaxy like the central black hole (to really explain why this is one would have to get into a bit more details into rotation curves and how the expectations differ from observations).
Other pieces of evidence, like the dynamics which occurs when two galactic clusters collide also wouldn't be accounted for by galactic central black holes. See e.g. the bullet cluster.
Is it possible that the dark matter is made up of many smaller black holes? Possible, but not likely.
At one point in time, there was conjecture that the dark matter consisted of (moderate sized) black holes and other compact objects which have low luminosity. This was the MACHO theory (MAssive Compact Halo Objects). But this theory has largely fallen out of favor.
As Ben points out in a comment, another candidate might be primordial black holes, but their abundance appears to be too low to be good candidates at this time.