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;μ)) for measures with finite entropy and suitable finite moment, and that the lower and upper asymptotic Rényi entropy rates converge to the Shannon entropy as (α\downarrow1). For finitely supported measures, we also obtain a spectral-radius formula for the asymptotic Rényi entropy rates (h_α(X,μ)), (α\in(1,2]), and hence continuity at (α=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.

• The paper presents explicit spectral formulas for Shannon and Rényi entropy rates using weighted interpolation of quasi-regular representations.
• It demonstrates that pairing a group with a subgroup under relaxed subexponential Lorentz control conditions yields novel analytic and rigidity results.
• It systematically classifies subgroups across various group settings, revealing that only finite-index or specific cyclic subgroups satisfy the pair rapid decay property.

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

Analytic Framework: Pair Rapid Decay and Entropy

This paper extends the analytic theory of the (pair) rapid decay (RD) property to the setting of homogeneous spaces $G/H$ and develops its consequences for random walks, entropy theory, and harmonic analysis. Unlike classical RD—which requires the group itself to satisfy RD—a pair $(G, H)$ may possess the "pair RD" property even when $G$ fails RD, provided the quasi-regular representation on $\ell^2(G/H)$ admits sufficient off-diagonal decay.

Key technical advances include a weighted interpolation estimate for quasi-regular representations acting in $\ell^q(G/H)$, interpolating between $\ell^1$ and the strong pair RD estimate on $\ell^2$. This machinery yields several novel entropy-theoretic results for random walks on $G/H$:

• For probability measures $\mu$ on $G$ with finite entropy and appropriate moment conditions, the lower and upper asymptotic Shannon entropies for trajectories on $(G, H)$0 not only coincide, but agree with a spectral quantity $(G, H)$1 (defined via the $(G, H)$2-operator spectral radii of convolution powers).
• The lower and upper Rényi entropy rates converge to the Shannon entropy rate as the order parameter $(G, H)$3.
• For finitely supported $(G, H)$4, the asymptotic Rényi entropy rates $(G, H)$5 (for $(G, H)$6) admit an explicit spectral radius formula, and continuity holds at $(G, H)$7.

The dependence on genuine polynomial decay is shown to be somewhat superfluous for these analytic results: the authors introduce "subexponential Lorentz control" ($(G, H)$8), where the polynomial growth envelope for RD can be relaxed to a much weaker, subexponential control. The arguments are robust to such relaxations, which enables application in contexts where polynomial RD fails but subexponential estimates persist.

Rigidity and Classification: Which Subgroups Yield Pair Rapid Decay?

A major focus is a systematic classification of subgroups $(G, H)$9 where the pair $G$0 satisfies pair RD or lies in $G$1. Earlier partial results (e.g., for co-amenable subgroups) are substantially strengthened and generalized. The work is partitioned across various ambient group classes:

1. (Strongly) Relatively Hyperbolic Groups: The pair $G$2 has pair RD if and only if $G$3 has polynomial growth (relative to the induced length function).
2. Free, Surface, and Hyperbolic Groups: Explicit structure theorems describe exactly which finitely generated $G$4 yield pair RD or subexponential estimates. For $G$5 ($G$6), or the fundamental group of a genus $G$7 surface, these are precisely the finite index, infinite cyclic, or trivial subgroups.
3. Hyperbolic 3-Manifolds: For $G$8 with $G$9 a closed hyperbolic manifold, $\ell^2(G/H)$0 if and only if $\ell^2(G/H)$1 is virtually cyclic or has nontrivial normal core in $\ell^2(G/H)$2.
4. Higher-Rank Lattices: For $\ell^2(G/H)$3 ($\ell^2(G/H)$4), all three conditions—subexponential Lorentz control, pair RD, and $\ell^2(G/H)$5 having finite index in $\ell^2(G/H)$6—are equivalent.

These results rest on combination of analytic and geometric techniques: interpolative estimates, fine growth arguments, group-theoretic rigidity, and utilization of subgroup structure theorems from Kleinian, relatively hyperbolic, and arithmetic group theory literature. The authors also analyze limit sets, (relative) quasiconvexity, and normal core phenomena.

Strong Numerical and Structural Consequences

Numerical consequences: Whenever $\ell^2(G/H)$7 has pair RD (resp. subexponential Lorentz control), the Shannon and Rényi entropy rates for random walks on $\ell^2(G/H)$8 agree with operator-derived spectral radii—this grants strong explicit formulas for key asymptotic invariants.

Structural dichotomy: Particularly in the arithmetic context ($\ell^2(G/H)$9 for $\ell^q(G/H)$0), the classification is rigid: only finite-index subgroups $\ell^q(G/H)$1 can support the analytic machinery. This is in stark contrast to hyperbolic or relatively hyperbolic contexts, where (for example) s-normality and more subtle normal-core phenomena can admit some infinite-index $\ell^q(G/H)$2.

These findings furnish counterexamples to possible generalizations (e.g., free products and amalgams), and disprove claims made in the earlier literature (notably, in the strong version of a Theorem in [Chatterji and Zarka, v1]). Detailed analysis of Rips constructions and arithmetic subgroups also highlights the delicate balance between algebraic and geometric group properties for the status of RD or subexponential control.

Implications and Future Directions

Theoretical implications: The interpolation methods and entropy identities establish the robustness of entropy and random-walk theory under relaxation from polynomial to subexponential decay, but also reveal the prohibitive restrictions when ambient groups lack significant expansion properties. The explicit subgroup classification results lay groundwork for further computation and rigidity theory in group representations and harmonic analysis on homogeneous spaces.

Applications: The results yield sharp entropy comparison and continuity theorems for random walks on homogeneous spaces, with implications for geometric group theory, ergodic theory, probability, and the operator algebraic study of group actions.

Perspectives: The authors pose structural questions about whether, in non-elementary hyperbolic groups, the only infinite-index $\ell^q(G/H)$3 for which $\ell^q(G/H)$4 may support analytic entropy identities and subexponential Lorentz control are those with nontrivial normal core—i.e., those "lifted" from proper quotients. Future work could focus on understanding the extent to which quotient-lifting mechanisms exhaust all such pairs.

Conclusion

This paper advances the analytic and structural theory of rapid decay and entropy for homogeneous spaces, providing comprehensive entropy rate formulas, continuity properties, and explicit subgroup classifications for a variety of group classes. The interplay between analytic control, subgroup structure, and geometric phenomena—clarified via an overview of interpolation methods, rigidity theory, and geometric group methods—drives the rigid dichotomies and sharp consequences established. These techniques and results set the stage for deeper investigation into analytic and probabilistic invariants on nontrivial homogeneous spaces and their ties to group-theoretic rigidity and growth.