Is there AOG liability? From Wikipedia:


  Aircraft on Ground or AOG is a term in aviation maintenance indicating
  that a problem is serious enough to prevent an aircraft from flying.
  Generally there is a rush to acquire the parts to put the aircraft
  (A/C) back into service, and prevent further delays or cancellations
  of the planned itinerary.


I know AOG is a really serious issue for an airline. What I want to learn is the liability and how far it extends. For example; if AOG happens:


how long can this airplane stay in that airport? 
what is the airports role in this situation? Do they help them out or just charge the amount they stay on the ground?
how long can this airplane stay in that airport?


The aircraft is "AOG" so it cannot leave. It needs to stay as long as it takes to complete the necessary repair.


  what is the airports role in this situation? Do they help them out or just charge the amount they stay on the ground?


The airport usually would charge the airline the parking fee. The parking fee is a small cost of not having an aircraft available in revenue service.