  We characterize groups admitting Anosov representations into
$\mathsf{SL}(3,\mathbb R)$, projective Anosov representations into
$\mathsf{SL}(4,\mathbb R)$, and Borel Anosov representations into
$\mathsf{SL}(4,\mathbb R)$. More generally, we obtain bounds on the
cohomological dimension of groups admitting $P_k$-Anosov representations into
$\mathsf{SL}(d,\mathbb R)$ and offer several characterizations of Benoist
representations.
