The two towers, Apex and Ceyane, together have 857 apartments. Of these, about 600 flats have already been sold. The towers are a part of Supertech's Emerald Court project.
"We have filed an SLP (Special Leave Petition) in the Supreme Court," Supertech Chairman and Managing Director R K Arora told PTI.
In the petition, he said, the company has contested that the two towers were constructed as per approved building plans and there was no violation.
The affected apartment buyers have been protesting against the demolition and some of them have formed a group and approached the Supreme Court independently.
On April 11, the Allahabad High Court ordered the demolition of the two buildings and the refund of money to apartment buyers.
The high court judgement was passed while allowing a writ petition of the Emerald Court Owners Resident Welfare Association, which alleged that the approval and construction of the two towers was "in complete violation of the UP Apartment Acts".
The complex comprises residential premises and shopping establishments.
The court also ordered that money be refunded to all those who had invested in the two towers "with 14 per cent interest compounded annually".
The petitioner had claimed the Noida Authority had given permission to raise the height of the two towers, which were earlier supposed to have only 24 floors, "without maintaining the mandatory distance of 16 metres) from an adjoining building block, making it "unsafe, apart from blocking air and light.
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