Cusp Density and Commensurability of Non-arithmetic Hyperbolic Coxeter Orbifolds

For three distinct infinite families $\left(R_m\right)$ , $\left(S_m\right)$ , and $\left(T_m\right)$ of non-arithmetic 1-cusped hyperbolic Coxeter 3-orbifolds, we prove incommensurability for a pair of elements $X_k$ and $Y_l$ belonging to the same sequence and for most pairs belonging two different ones. We investigate this problem first by means of the Vinberg space and the Vinberg form, a quadratic space associated to each of the corresponding fundamental Coxeter prism groups, which allows us to deduce some partial results. The complete proof is based on the analytic behavior of another commensurability invariant. It is given by the cusp density, and we prove and exploit its strict monotonicity.