Topological restrictions on Anosov representations

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