Was Rome at one time a "safe haven" for runaway slaves?
Roman law in regard to slaves was quite severe. A Roman master could treat his slave as he pleased -- sell him, beat him, or put him to death. However, morality and practical matters caused most masters to restrain their hands. After all slaves cost money, so few were incline to ruin their investment.
A runaway slave could not be harbored by Roman law. A master could pursue his slave at anytime and Roman authorities were required to give the master aid in recovering the slave. Runaway slaves could not be sold while on the lam. Hence, Roman law did everything possible to cause a slave to be returned to his master when found. Of course, a slave who tended to run away became less valuable on the market, leading to masters being less than kind to a returning runaway. A runaway status could not be hidden because runaway slaves were branded on the forehead when found.
Because of the high number of slaves within the Roman society, popular penalty for a runaway slave tended to be of the severest kind. So, no, Rome would not be a safe haven for a runaway slave. However, being one of the largest cities in the world at that time, it would be a great place in which to get lost among the crowds. A newcomer in a small town stands out. A newcomer in a large city is just one of too many.