How many units is the apartment building? Is it rent stabilized? If you commenced a holdover last month and the tenant is asking you to check on his lights last week, I suspect he's up to something. Has he filed any complaints ? How did you commence the holdover? How and in what manner did you serve him his notice to quit? Without the assistance of an attorney? (a given, I speculate, but I am asking just to be sure).
Not knowing more information, I'd be afraid to offer any advice or suggestions yet.
My answer is for general purposes only and is not not intended to establish an attorney-client relationship, nor is it advice upon which you should act or rely. But, if you really want me to tell you something upon which you can actually rely: don't eat yellow snow.
If you have not done this before, or at least seen a summary proceeding done, you would be well served to hire an attorney. It is a very technical process and if not done correctly your case can be dismissed and you will have to start over. The fees you pay will be well worth the time and effort you will save.
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The expectation is low that your lawsuit complies with the requirements of law, and because you have so many questions about this form of proceeding, you have not stumbled on all the information freely offered by the court.
You will not be entitled to a judgment for past rent unless you provided specific information in the notice of petition that in addition to the request for a judgment for possession, you were including a request for a money judgment for all rent in arrears as well as for the use and occupancy that became due to you following the expiration of the lease. So look at your notice of petition and see whether you made that notice of a request for a money judgment. Look also at your petition and see whether you made a request for a sum certain money judgment for unpaid rent as well as for the use and occupancy through the date of judgment.
In certain circumstances the acceptance by the landlord of use and occupancy paid after the commencement of the lawsuit will not have an affect on the underlying claim for a judgment for possession. If you provide a receipt for "rent" that may be construed as a reviving of the tenancy and will end your case against the tenant. If the tenant provides a check that notes that it is payment for rent, consult an attorney.
The fact that the apartment is in terrible condition may have an impact on your ability to hold on to the security deposit or a part of the security deposit to repair conditions of damage caused by the tenant that are beyond allowed ordinary wear and tear. The condition of the apartment should not have a bearing on the actual prosecution of the lawsuit, unless the lease was terminated due to the tenant maintaining the apartment in terrible condition and after a notice by you to correct that breach of lease, the tenant did not do so and you terminated the lease.
If you did not serve a notice of termination of the month to month tenancy and let the time run on the notice then your filing of a holdover petition may be premature. When a tenant with a lease converts to a month to month tenancy the landlord must serve a written thirty day notice of tenancy termination. The law requires that the notice is served in the same manner as a notice of petition and petition. The end date for the termination notice must be the same as the end date of a rent month (usually the last day of the month if the rent due date is the first of the month.). if you filed your petition in March 2013, then the month to month tenancy end date (as stated by you in the petition as the day the term expired) would be February 28, 2013, if you served a proper notice of termination of tenancy in January 2013 and provided at least thirty days notice.
Read more about the owner's summary proceeding for eviction at:
Read about a tenant's defense strategies at:
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.