Rule 5.2.2 says this (the same text appears in both the original and All Aboard Games rules):
5.2.2 Laying track and buying tokens is optional, but a corporation that is active must have a running route: A route that connects one of its Rail Yards or Stations to another of its Rail Yards or Stations.
Does purchasing track rights allow a company to meet this requirement, or is a company required to have two of its own tokens placed on the map?
One possibility is that you would check for a route during the run trains step, and thus renting track rights would fulfill the requirement. Another possibility is that you check for a route during the track/token laying step (in which rule 5.2.2 appears), and thus renting track rights which appears in a later step would not fulfill the requirement.