This paper deals with sequences of random variables belonging to a
fixed chaos of order q generated by a Poisson random measure on a Polish space.
The problem is investigated whether convergence of the third and fourth moment
of such a suitably normalized sequence to the third and fourth moment of a centred
Gamma law implies convergence in distribution of the involved random variables.
A positive answer is obtained for q = 2 and q = 4. The proof of this four moments
theorem is based on a number of new estimates for contraction norms. Applications
concern homogeneous sums and U-statistics on the Poisson space.