It depends. The general rule is that the landlord is not liable to the lessee for injuries arising from a condition of the property. But, there are lots of exceptions. Once is for a latent defect which the landlord knew or should have known existed, which caused the injury. You need to discuss the specific facts of your situation, and how and why you fell, with a lawyer.
I'm sorry to hear about your hand injury. The compensation you may be entitled to receive for the injuries and damages you sustained will depend on many factors. For instance, what did you trip or fall over? Was the slab allowed to exist in a dangerous condition? Did your landlord know about any dangerous conditions? I would suggest doing a search for an Avvo lawyer near you who has a rating of 7.0 or higher and get all your questions answered. You may also want to go to www.atla.org and look for a lawyer in your area. Feel free to read our free report about how to find, interview and hire the right lawyer for you. It's posted at www.jacksonwilson.com
While I don't have a 7.0 rating or higher (and I don't know why with 12 years of experience and multiple six-figure verdicts), there may be one yet unexplored avenue. While you may not be able to establish liability against your landlord, many insurance policies have "medical coverage" provisions that will pay medical bills only (no pain and suffering, lost wages, etc.) regardless of liability. So don't be shy in making a claim against your landlord's insurance policy.
As my brethren stated above, your liability claim against yur landlord depends on the facts regarding how you fell and whether the landlord had prior notice of a dangerous condition. And you may have a medpay claim under your landlord's policy which does not require a findng of liability.