How Much Notice Is Required To Terminate Lease?

What ever the lease agreement states is what you go by, but my lease agreement says that I have to give them a 30 day written notice to terminate my lease agreement.

Most rental agreements state you give the landlord 30 day notice prior to moving out. Since you gave the management company a 90 day notification they had plenty of time to find another tenant after you moved out. They can not charge for any further rent after you move out. If they do then contact an attorney and have file a suit
against the landlord. Contact the tenant/landlord affairs for their
advice and get it in writing.

The notice requirements are typically set by code.  For instance, in California, the min. Amount of notice when there is no lease agreement, is set by the interval of payment. So if you pay rent once a month, then you need only give one month's notice; bi-weekly payment = 2 week notice.

If there is a lease agreement, then the agreement can set the amount of time while that lease is effective (so long as it complies with the code, or permitted exceptions).  Some states allow longer or shorter periods as an exception to the minimum notice requirements if they comply with certain provisions (i.e. The lease must indicate upon expiration that it will automatically renew unless x, y, z, and that provision appears in a larger font size, in bold, above the leasee's signature).  

Typically the relevant code section will also provide that if the required amount of notice in the rental agreement does not comply with the code, it is void.  Your lease agreement usually allows for such things by providing a

If the house i live is being auctioned off, how long do i have to move

I live in City of chicago, state of Illinois

You should give at least a 30 day notice of the lease agreement.