We study the properties of the Heisenberg antiferromagnet with spatially anisotropic nearest-neighbour exchange couplings on the kagome net, i.e. with coupling J in one lattice direction and couplings J along the other two directions. For J/J > 1, this model is believed to describe the magnetic properties of the mineral volborthite. In the classical limit, it exhibits two kinds of ground states: a ferrimagnetic state for J/J < 1/2 and a large manifold of canted spin states for J/J > 1/2. To include quantum effects self-consistently, we investigate the Sp(N) symmetric generalisation of the original SU(2) symmetric model in the large-N limit. In addition to the dependence on the anisotropy, the Sp(N) symmetric model depends on a parameter kappa that measures the importance of quantum effects. Our numerical calculations reveal that in the kappa-J/J plane, the system shows a rich phase diagram containing a ferrimagnetic phase, an incommensurate phase, and a decoupled chain phase, the latter two with short- and long-range order. We corroborate these results by showing that the boundaries between the various phases and several other features of the Sp(N) phase diagram can be determined by analytical calculations. Finally, the application of a block-spin perturbation expansion to the trimerised version of the original spin-1/2 model leads us to suggest that in the limit of strong anisotropy, J/J >> 1, the ground state of the original model is a collinearly ordered antiferromagnet, which is separated from the incommensurate state by a quantum phase transition.
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