On the fundamental groups of subelliptic varieties

We show that the fundamental group of any smooth subelliptic variety is finite. Moreover, it is also proved that every finite group can be realized as the fundamental group of a smooth subelliptic variety. As a consequence, it follows that there exists a smooth subelliptic variety homotopy equivalent to the $n$-sphere if and only if $n>1$. This result can be considered as a negative answer to the algebraic version of Gromov's problem on the homotopy types of Oka manifolds.