Entropy on Homogeneous Spaces and Classification Results for Subgroups with the Pair Rapid Decay Property

We study pair rapid decay for homogeneous spaces $G/H$ and its applications to random walks and subgroup structure. The entropy framework for groups with rapid decay is extended to homogeneous spaces, proving that the asymptotic Shannon entropy on $G/H$ agrees with a spectral-radius quantity $c(G,H;\mu)$ for measures with finite entropy and suitable finite moment, and that the lower and upper asymptotic R\'enyi entropy rates converge to the Shannon entropy as $\alpha\downarrow1$. For finitely supported measures, we also obtain a spectral-radius formula for the asymptotic R\'enyi entropy rates $h_\alpha(X,\mu)$, $\alpha\in(1,2]$, and hence continuity at $\alpha=1$. We further introduce the notion of subexponential Lorentz control for pairs $(G,H)$ and study the associated classification problems for finitely generated subgroups $H\le G$ for which $(G,H)$ has pair rapid decay or belongs to $\mathbf{SLC}_{\mathrm{subexp}}$. We obtain a complete criterion in the strongly relatively hyperbolic case and explicit classifications in several hyperbolic settings. We also show that for $G=\mathrm{SL}_n(\mathbb Z)$, $n\ge3$, the conditions $(G,H)\in \mathbf{SLC}_{\mathrm{subexp}}$, pair rapid decay, and finite index of $H$ in $G$ are equivalent.