Rigidity of compact holomorphic curves in compact complex parallelizable manifolds $\Gamma\backslash \mathrm{SL}(2,\mathbb C)$ and its geometric applications

Let $\Gamma\subset\mathrm{SL}(2,\mathbb C)$ be a cocompact lattice and $X=\Gamma\backslash\mathrm{SL}(2,\mathbb C)$ the associated compact complex parallelizable manifold. We show that any non-constant holomorphic map $f\,:\,M \rightarrow X$ from a compact Riemann surface $M$ into $X$ decomposes as $f=t\circ h\circ \alpha$, where $\alpha\,:\,M\rightarrow \mathrm{Alb}(M)$ is the Albanese map, $h\,:\,\mathrm{Alb}(M) \rightarrow X=\Gamma\backslash \mathrm{SL}(2,\mathbb C)$ has its image in a maximal torus $T=\Gamma\cap A\backslash A\cong \Bbb Z\backslash \mathbb C^*$ in $X$ ($A$ being a maximal torus in $\mathrm{SL}(2,\mathbb C)$) defining an algebraic group homomorphism $h:\mathrm{Alb}(M) \rightarrow T=(A\cap\Gamma)\backslash A$, and finally $t$ is a right translation by some element of $\mathrm{SL}(2,\mathbb C)$. The proof is based on Bishop's measure theoretic criterion of analyticity of sets combined with a simple observation in hyperbolic geometry.
I will discuss some applications of this rigidity.