On Euler characteristic and fundamental groups of compact manifolds

Let $M$ be a compact Riemannian manifold, $\pi :\tilde{M}\to M$ be the universal covering and $\omega$ be a smooth $2$-form on $M$ with ${\pi }^{\ast }\omega$ cohomologous to zero. Suppose the fundamental group ${\pi }_1(M)$ satisfies certain radial quadratic (resp. linear) isoperimetric inequality, we show that there exists a smooth $1$-form $\eta$ on $\tilde{M}$ of linear (resp. bounded) growth such that ${\pi }^{\ast }\omega =d\eta$. As applications, we prove that on a compact Kahler manifold $(M,\omega )$ with ${\pi }^{\ast }\omega$ cohomologous to zero, if ${\pi }_1(M)$ is $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ or automatic (resp. hyperbolic), then $M$ is Kahler non-elliptic (resp. Kahler hyperbolic) and the Euler characteristic $(-1{)}^{\frac{{\dim }_{\mathbb{R}}M}{2}}\chi (M)\ge 0$ (resp. $>0$).