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We classify biharmonic and harmonic homomorphisms$f:(G,g_1)\rightarrow(G,g_2)$ where $G$ is a connected and simply connectedthree-dimensional unimodular Lie group and $g_1$ and $g_2$ are left invariantRiemannian metrics.

Biharmonic and harmonic homomorphisms between Riemannian three dimensional unimodular Lie groups