Suppose that a $d$-dimensional torus $T$ acts smoothly and effectively on an $n$-dimensional closed manifold $M$. What conditions on $d$ and $n$ imply that $\pi_1(M)$ must be infinite?

Consider the case where $d=n-1$. Within Section 4 of this paper by Grove and Ziller, it is shown that if $n\geq 4$ and $T^{n-1}$ acts smoothly and effectively on $M^n$, then $\pi_1(M)$ is infinite. In contrast for $n=2$, there is of course the $S^1$-action on $S^2$, and for $n=3$, there is the $T^2$-actions on $S^3$ and lens spaces.

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