  The physical interpretation of lattice QCD simulations, performed in a small
volume, requires an extrapolation to the infinite volume. A method is proposed
to perform such an extrapolation for three interacting particles at energies
above threshold. For this, a recently formulated relativistic $3\to 3$
amplitude based on the isobar formulation is adapted to the finite volume. The
guiding principle is two- and three-body unitarity that imposes the imaginary
parts of the amplitude in the infinite volume. In turn, these imaginary parts
dictate the leading power-law finite-volume effects. It is demonstrated that
finite-volume poles arising from the singular interaction, from the external
two-body sub-amplitudes, and from the disconnected topology cancel exactly
leaving only the genuine three-body eigenvalues. The corresponding quantization
condition is derived for the case of three identical scalar-isoscalar particles
and its numerical implementation is demonstrated.
