  We investigate consequences of the M\"obius invariance of the BFKL
Hamiltonian and of the triple Pomeron vertex. In particular, we show that the
triple Pomeron vertex in QCD, when restricted to the large $N_c$ limit and to
the space of M\"obius functions, simplifies and reduces to the vertex used in
the Balitsky-Kovchegov (BK) equation. As a result, the BK equation for the
dipole density appears as a special case of the nonlinear evolution equation
which sums the fan diagrams for BFKL Green's functions in the M\"obius
representation. We also calculate the corrections $O(1/(N_c^2-1)$ to the triple
Pomeron vertex in the space of M\"obius functions, and we present a
generalization of the BK-equation in the next-to-leading order approximation in
the $1/N_c$ expansion.
