Topological obstruction for Anosov representations into $\mathrm{PO}(p,q)$

We establish new restrictions on the virtual cohomological dimension of word hyperbolic groups admitting Anosov representations into $\mathrm{PO}(p,q)$ ($p>q\ge 2$) or the exceptional Lie group ${\mathrm{G}}_2^{\mathrm{\prime }}$. Our main result states that, if $P_k$ is the parabolic subgroup of $\mathrm{PO}(p,q)$ stabilizing an isotropic $k$-plane in ${\mathbb{R}}^{p,q}$, a group admitting a $\{P_1,P_q\}$-Anosov representation into $\mathrm{PO}(p,q)$ has virtual cohomological dimension strictly less than $p$. As a result, a ${\mathbf{H}}^{p,q-1}$-convex cocompact subgroup of $\mathrm{PO}(p,q)$ cannot act properly discontinuously and cocompactly on ${\mathbf{H}}^{p,q-1}$. The main theorem also implies that a group admitting a $P_{\alpha }$-Anosov representation into the exceptional Lie group ${\mathrm{G}}_2^{\mathrm{\prime }}$ has virtual cohomological dimension strictly less than $4$, where $\alpha$ is a short restricted root of ${\mathrm{G}}_2^{\mathrm{\prime }}$. As an application, we prove that the homogeneous space ${\mathrm{G}}_2^{\mathrm{\prime }}\mathrm{/}\mathrm{SU}(2,1)$ does not admit any compact quotient.