arxiv: 2604.15803 · v1 · submitted 2026-04-17 · 🧮 math.GR

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

Jvbin Yao

Pith reviewed 2026-05-10 07:40 UTC · model grok-4.3

classification 🧮 math.GR

keywords entropy on homogeneous spacespair rapid decaysubexponential Lorentz controlsubgroups of SL_n(Z)strongly relatively hyperbolic groupsRényi entropy ratesrandom walks on coset spaces

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The pith

For SL_n(Z) with n at least 3, the pair rapid decay property for (G,H) holds exactly when H has finite index in G.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends entropy calculations from groups to homogeneous spaces G/H by showing that asymptotic Shannon entropy agrees with a spectral-radius quantity c(G,H;μ) for measures of finite entropy and finite moments, while lower and upper Rényi entropy rates converge to the Shannon value as the parameter approaches 1. It introduces subexponential Lorentz control on pairs (G,H) and derives classification results for finitely generated subgroups H that make (G,H) satisfy pair rapid decay or belong to SLC_subexp. A complete criterion is obtained when G is strongly relatively hyperbolic, together with explicit classifications in several hyperbolic settings. The strongest result equates three conditions for G equal to SL_n(Z) with n at least 3: membership in SLC_subexp, possession of pair rapid decay, and finite index of H in G.

Core claim

The asymptotic Shannon entropy on the homogeneous space G/H agrees with the spectral-radius quantity c(G,H;μ) for measures with finite entropy and suitable finite moments; the lower and upper asymptotic Rényi entropy rates converge to the Shannon entropy as α approaches 1 from above; for finitely supported measures a spectral-radius formula holds for the Rényi rates with α between 1 and 2, implying continuity at α=1; and, when G is strongly relatively hyperbolic, (G,H) belongs to SLC_subexp or has pair rapid decay if and only if H satisfies an explicit structural condition, which for G=SL_n(Z) n≥3 reduces to H having finite index.

What carries the argument

Pair rapid decay for the pair (G,H) on the homogeneous space G/H, which controls the decay of convolution operators and matrix coefficients on the coset space and is equivalent to subexponential Lorentz control under the stated hypotheses.

If this is right

Entropy rates of random walks on G/H can be read off from the spectral radius whenever the measure has finite entropy and moments.
For SL_n(Z) with n≥3 the only subgroups H yielding pair rapid decay are those of finite index.
In strongly relatively hyperbolic groups the structural condition for subexponential Lorentz control completely determines which subgroups produce pair rapid decay.
Rényi entropy rates for finitely supported measures on G/H are continuous at α=1 and equal the spectral-radius expression.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Infinite-index subgroups of higher-rank lattices are expected to produce strictly positive entropy gaps on the corresponding homogeneous spaces.
The equivalence may extend the known rigidity of finite-index subgroups in SL_n(Z) to decay properties of associated random walks.
Explicit classifications in hyperbolic settings suggest that pair rapid decay detects virtual normality or commensurability in those cases.

Load-bearing premise

The entropy agreement and Rényi convergence results require measures with finite entropy and finite moments, while the complete classification criterion requires the ambient group to be strongly relatively hyperbolic.

What would settle it

An explicit subgroup H of infinite index in SL_3(Z) for which the pair (SL_3(Z),H) still satisfies pair rapid decay, or a probability measure of finite entropy on such a pair whose asymptotic entropy on G/H differs from the spectral radius c(G,H;μ).

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper extends the entropy framework to homogeneous spaces and provides an equivalence classification for subgroups of SL_n(Z) with pair rapid decay.

read the letter

The one or two things to know are that the authors have extended the entropy framework for rapid decay to homogeneous spaces G/H, proving agreement of asymptotic Shannon entropy with a spectral radius quantity and convergence of Renyi rates, and that they use this to classify subgroups with the pair rapid decay property, culminating in an equivalence for SL_n(Z). What is new is the application to homogeneous spaces, the definition of subexponential Lorentz control for pairs, and the specific equivalence result that for G = SL_n(Z), n greater than or equal to 3, the conditions of being in SLC_subexp, having pair rapid decay, and H having finite index in G are all equivalent. They also give a complete criterion in the strongly relatively hyperbolic case and explicit classifications in hyperbolic settings. The paper does well in separating the SL_n(Z) result from the general hyperbolic one and deriving it via entropy agreement and spectral formulas for Renyi rates. The argument uses standard assumptions about measures and shows implications in both directions. The soft spots are proportionate to the specialized nature. The entropy agreement and Renyi convergence rely on measures with finite entropy and suitable finite moments, which is reasonable but limits the scope. The complete criterion assumes the group is strongly relatively hyperbolic. The derivations seem to follow from the entropy agreement and spectral-radius formulas without circularity, as the stress-test indicates no internal inconsistency or hidden assumptions. The citation pattern seems to build on prior work without self-referential issues. This paper is for researchers in geometric group theory who are interested in rapid decay properties and their connections to operator algebras and random walk behavior. It would be valuable for someone studying classifications of subgroups in hyperbolic groups or arithmetic groups like SL_n(Z), as it organizes some phenomena with concrete equivalences. A reader focused on entropy methods on groups and spaces will appreciate the spectral radius formulas and continuity results. The work shows clear thinking through its structured approach to the classifications. I believe it deserves a serious referee because the central claims are evidentially sharp enough with the provided framework, even if some technical parts might need clarification in review. My recommendation is to send this to peer review rather than desk reject.

Referee Report

0 major / 2 minor

Summary. The paper extends the entropy framework for groups with rapid decay to homogeneous spaces G/H. It proves agreement of the asymptotic Shannon entropy on G/H with the spectral-radius quantity c(G,H;μ) for measures with finite entropy and suitable finite moments, convergence of lower/upper asymptotic Rényi entropy rates to the Shannon entropy as α↓1, and spectral-radius formulas for the Rényi rates h_α(X,μ) when μ is finitely supported. The authors introduce subexponential Lorentz control for pairs (G,H) and obtain classification results for finitely generated subgroups H≤G with the pair rapid decay property or belonging to SLC_subexp, including a complete criterion in the strongly relatively hyperbolic case, explicit classifications in several hyperbolic settings, and the equivalence for G=SL_n(Z), n≥3, of (G,H)∈SLC_subexp, pair rapid decay, and finite index of H in G.

Significance. If the results hold, this work meaningfully extends entropy and rapid-decay techniques to homogeneous spaces, supplying new tools for analyzing random walks on G/H and for classifying subgroups via analytic properties. The equivalence theorem for SL_n(Z) directly links finite-index subgroups to pair rapid decay and subexponential Lorentz control, which is a concrete, falsifiable connection between algebraic and spectral invariants. The derivations rely on external spectral-radius quantities and standard group-theoretic assumptions with no free parameters or circular definitions.

minor comments (2)

[Abstract] Abstract: the technical terms 'pair rapid decay', 'SLC_subexp', and 'subexponential Lorentz control' appear without a one-sentence gloss or forward reference; adding a brief parenthetical definition or citation to the introduction would improve accessibility.
[Abstract] The abstract states that explicit classifications are obtained 'in several hyperbolic settings' but does not name the groups; listing one or two concrete examples (e.g., free groups or surface groups) would clarify the scope without lengthening the abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the detailed summary of our results, and the recommendation for minor revision. No specific major comments or criticisms are provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations rely on external spectral quantities and standard assumptions

full rationale

The paper extends entropy frameworks to homogeneous spaces G/H using finite-entropy measures and moment conditions, deriving agreement between asymptotic Shannon entropy and the spectral-radius quantity c(G,H;μ), plus Rényi convergence. The equivalence for SL_n(Z) (n≥3) between SLC_subexp, pair rapid decay, and finite index is obtained via homogeneous-space entropy agreement and spectral-radius formulas, without reducing any central claim to a fitted parameter or self-referential definition. Definitions of pair rapid decay and subexponential Lorentz control are introduced explicitly and implications are shown bidirectionally using external group-theoretic tools. No load-bearing step collapses by construction to its inputs; the argument remains independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Based solely on the abstract; the paper relies on standard assumptions from geometric group theory and random walk theory. It introduces one new concept (subexponential Lorentz control) without independent evidence outside the definitions.

axioms (1)

domain assumption Standard properties of groups, homogeneous spaces, and random walks with finite entropy and finite moments hold.
Invoked for the entropy agreement and convergence statements.

invented entities (1)

subexponential Lorentz control for pairs (G,H) no independent evidence
purpose: To enable classification of subgroups with pair rapid decay or membership in SLC_subexp.
New notion defined in the paper for the classification problems.

pith-pipeline@v0.9.0 · 5548 in / 1453 out tokens · 63876 ms · 2026-05-10T07:40:12.415340+00:00 · methodology

discussion (0)

Reference graph

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