My husband fell and broke his hand in front of our apartment can we sue our landlord

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I am familiar with Staten Island and I would suggest that you look at 3 factors:

1- Was this a city or housing authority project?..?There are strict deadlines
2- Was this a trip or slip? If a trip what was the defect,if any, that caused it.If a slip what substance caused it?
3-Did you ever notice this defect or substance before the accident?

Answering the above will help any attorney that you talk to