What does 'thrust up the trap' mean in context?
At Thirty-fourth Street young Richard quickly thrust up the trap and ordered the cabman to stop.
Mammon and the Archer by O. Henry
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It may refer to a sliding window in the cab that separates the passenger compartment from the driver for privacy. It would have to be slid aside to talk to the cab driver.
Here is a definition of "trapdoor" that may fit:
American Heritage Dictionary "trapdoor"
"A hinged or sliding door in a floor, roof, or ceiling."
In this case, it would be a sliding door in the compartment separator.
Here is a description of such a door in a horse-drawn cab:
Wikipedia "hansom cab" design
"The passengers could give their instructions to the driver through a trap door near the rear of the roof."