Reflections on branched covers

A flag complex $L$ determines a locally CAT(0) cube complex $P_L$ with the links of all vertices isomorphic to $L$. The fundamental group of $P_L$ is the commutator subgroup of the right-angled Coxeter group $W_L$. We observe that an edge contraction $L\to L\mathrm{/}e$ determines a $2$-fold branched cover $P_L\to P_{L\mathrm{/}e}$ (up to a homotopy equivalence). We use this branched cover to give an inequality between the ${\mathbb{F}}_2$-$L^2$-Betti numbers of $W_L$ and $W_{L\mathrm{/}e}$. This reduces an ${\mathbb{F}}_2$-version of the Singer Conjecture for right-angled Coxeter groups to minimal flag triangulations of even dimensional spheres. We also use this inequality to compute the $L^2$-Betti numbers of $W_L$ when $L$ is a flag triangulation of the projective plane or the torus.