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

Abstract

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.