We derive the asymptotic expansions of the wave function of three particles having equal mass with finite-range interactions and infinite or zero two-dimensional scattering length colliding at zero energy and zero orbital angular momentum, from which a three-body parameter
D is defined. The dimension of
D is length squared, and we call
D three-body scattering area. We find that the ground-state energy per particle of a zero-temperature dilute Bose gas with these interactions is approximately $\frac{{\ensuremath{\hbar}}^{2}D}{6m}{\ensuremath{\rho}}^{2}$, where $\ensuremath{\rho}$ is the number density of the bosons and
m is the mass of each boson. Such a zero-temperature Bose gas is stable at
D>0 in the thermodynamic limit, and metastable at
D<0 in the isotropic harmonic trap if the number of bosons is less than ${N}_{\mathrm{cr}}\ensuremath{\approx}3.6413\sqrt{\frac{\ensuremath{\hbar}}{m\ensuremath{\omega}|D|}\phantom{{}^{1}}}$, where $\ensuremath{\omega}$ is the angular frequency of the harmonic trap. If the two-body interaction supports bound states,
D typically acquires a negative imaginary part, and we find the relation between this imaginary part and the amplitudes of the pair-boson production processes. We derive a formula for the three-body recombination rate constant of the many-boson system in terms of the imaginary part of
D.