If someone falls on a broken city sidewalk, they can sue the city for damages. Broken sidewalks may be caused by construction, or by tree roots, but if the city is aware of the defect and doesn't fix it, then they are liable.
There are four elements to a successful lawsuit:
The city had actual or constructive notice of the hazard.
The city failed to remedy the condition.
The condition caused injury to the plaintiff.
The plaintiff was not aware of the hazard or assumed the risk of injury.