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You can prevent your neighbor from acquiring title by adverse possession, and an easement by prescription if you give him permission to use that part of your property on which his fence is located. By giving him permission, he will not meet the requirements set forth in the New York Real Property Actions and Proceedings Law ("RPAPL") Section 501 that the possession be "adverse, hostile, and notorious". And the period of time required for acquiring title by adverse possession if 10 years and can be found in the New York Civil Practice Law and Rules ("CPLR") Section 212
I have reprinted those sections for your reference below:
New York Real Property Actions and Proceedings Law Section 501 states the following:
§ 501. Adverse possession; defined.
For the purposes of this article:
1. Adverse possessor. A person or entity is an "adverse possessor" of
real property when the person or entity occupies real property of
another person or entity with or without knowledge of the other's
superior ownership rights, in a manner that would give the owner a cause
of action for ejectment.
2. Acquisition of title. An adverse possessor gains title to the
occupied real property upon the expiration of the statute of limitations
for an action to recover real property pursuant to subdivision (a) of
section two hundred twelve of the civil practice law and rules, provided
that the occupancy, as described in sections five hundred twelve and
five hundred twenty-two of this article, has been adverse, under claim
of right, open and notorious, continuous, exclusive, and actual.
3. Claim of right. A claim of right means a reasonable basis for the
belief that the property belongs to the adverse possessor or property
owner, as the case may be. Notwithstanding any other provision of this
article, claim of right shall not be required if the owner or owners of
the real property throughout the statutory period cannot be ascertained
in the records of the county clerk, or the register of the county, of
the county where such real property is situated, and located by
New York Civil Practice Law and Rules ("CPLR")
§ 212. Actions to be commenced within ten years. (a) Possession
necessary to recover real property. An action to recover real property
or its possession cannot be commenced unless the plaintiff, or his
predecessor in interest, was seized or possessed of the premises within
ten years before the commencement of the action.
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