The Fr\"olicher spectral sequence of certain solvmanifolds

We show that the Fr"olicher spectral sequence of a complex parallelizable solvmanifold is degenerate at $E_2$-term. For a semi-direct product $G={\text{C}}^n{⋉}_{\varphi }N$ of Lie-groups with lattice $\mathrm{\Gamma }={\mathrm{\Gamma }}^{\mathrm{\prime }}⋉{\mathrm{\Gamma }}^{\mathrm{\prime }\mathrm{\prime }}$ such that $N$ is a nilpotent Lie-group with a left-invariant complex structure and $\varphi$ is a semi-simple action, we also show that, if the Fr"olicher spectral sequence of the nilmanifold $N\mathrm{/}{\mathrm{\Gamma }}^{\mathrm{\prime }\mathrm{\prime }}$ is degenerate at $E_r$-term for $r\ge 2$, then the Fr"olicher spectral sequence of the solvmanifold $G\mathrm{/}\mathrm{\Gamma }$ is also degenerate at $E_r$-term.