pith. sign in

arxiv: 2604.08490 · v1 · submitted 2026-04-09 · 🧮 math.GR · math.PR

Small entropy doubling for random walks and polynomial growth

Guy Blachar This is my paper

Pith reviewed 2026-05-10 17:18 UTC · model grok-4.3

classification 🧮 math.GR math.PR
keywords random walksentropypolynomial growthvirtually nilpotent groupsapproximate groupsGromov theoremdoubling conditions
0
0 comments X

The pith

A bounded entropy increase from scale n to 2n forces a finitely generated group to be virtually nilpotent.

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

The paper proves that for any finitely supported symmetric random walk on a group, an entropy inequality H(R_{2n}) ≤ H(R_n) + log K at large enough n implies the group is virtually nilpotent, with explicit dependence on K and the minimal step probability. This supplies an entropy-based test for polynomial growth that parallels the small-doubling condition in Gromov's theorem. A reader cares because entropy is computable from the walk and directly detects the algebraic structure without measuring volume growth. The argument adapts an entropy form of the Balog–Szemerédi–Gowers theorem to locally compact groups and combines it with approximate-group structure theorems. As a consequence, groups that are not virtually nilpotent must exhibit a uniform superlogarithmic entropy gap.

Core claim

If a finitely supported symmetric random walk R_n on a group satisfies H(R_{2n}) ≤ H(R_n) + log K at some sufficiently large scale n, then the group is virtually nilpotent. The proof proceeds by adapting Tao's entropy Balog–Szemerédi–Gowers argument to unimodular locally compact groups and invoking structural results on approximate groups. The resulting bounds on the nilpotency class and growth depend on K and on μ_min, the smallest positive probability in the support. The same hypothesis yields entropy criteria for polynomial growth and shows that non-virtually-nilpotent groups have random-walk entropy that grows faster than any fixed superlogarithmic function.

What carries the argument

The small-entropy-doubling condition H(R_{2n}) ≤ H(R_n) + log K on the random-walk measures R_n, which limits information growth and forces the support to behave like an approximate subgroup of polynomial growth.

If this is right

  • The group admits a nilpotent subgroup of finite index whose growth rate is bounded in terms of K and μ_min.
  • Polynomial growth can be certified by computing or estimating entropy of the walk at two scales.
  • Any non-virtually-nilpotent group has random-walk entropy that eventually exceeds every superlogarithmic function by a positive constant.
  • The same entropy condition implies that the support of R_n forms an approximate group of bounded doubling.

Where Pith is reading between the lines

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

  • Numerical simulation of random walks on a finitely presented group could in principle decide virtual nilpotency when the entropy gap is large enough.
  • The result suggests analogous entropy-doubling criteria may exist for other algebraic properties such as amenability or soficity.
  • One could ask whether the same conclusion holds for non-symmetric or infinitely supported walks, or for walks on non-discrete locally compact groups.

Load-bearing premise

The entropy inequality holds at some sufficiently large but finite scale n for a finitely supported symmetric random walk.

What would settle it

A concrete counter-example would be a finitely generated group that is not virtually nilpotent together with an explicit finitely supported symmetric probability measure whose entropy satisfies H(R_{2n}) ≤ H(R_n) + log K at arbitrarily large n.

read the original abstract

Gromov's theorem states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent. A key ingredient in its proof is the small doubling property. In this work, we study entropy analogues of this property for random walks on groups. We show that if a finitely supported symmetric random walk $R_n$ satisfies \[ \mathrm{H}(R_{2n}) \le \mathrm{H}(R_n) + \log K \] at some sufficiently large scale $n$, then the underlying group is virtually nilpotent, with bounds depending on $K$ and $\mu_{\min}$. Our approach adapts Tao's entropy Balog--Szemer\'edi--Gowers argument to unimodular locally compact groups, combined with structural results on approximate groups. As applications, we obtain entropy-based criteria for polynomial growth. We also deduce an entropy gap phenomenon: if $G$ is not virtually nilpotent, then the entropy of random walks on $G$ grows faster than a universal superlogarithmic function.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript proves an entropy analogue of Gromov's theorem: if a finitely supported symmetric random walk R_n on a unimodular locally compact group satisfies H(R_{2n}) ≤ H(R_n) + log K at a sufficiently large scale n (depending on K and μ_min), then the group is virtually nilpotent. The proof adapts Tao's entropy Balog-Szemerédi-Gowers inequality to convolution powers on such groups and combines it with structural results on approximate groups. Applications include entropy-based criteria for polynomial growth and an entropy gap: non-virtually nilpotent groups have random-walk entropy growing faster than a universal superlogarithmic function.

Significance. If the central claim holds, the work is a meaningful contribution to geometric group theory and probability on groups. It translates the small-doubling implication of Gromov's theorem into an entropy-doubling setting for random walks, with explicit dependence on K and μ_min, and extends the setting to unimodular locally compact groups. The adaptation of Tao's entropy BSG argument and the derivation of both polynomial-growth criteria and an entropy-gap phenomenon are clear strengths; the argument is presented as a direct combination of existing structural theorems without circularity.

minor comments (2)
  1. [Introduction] The introduction should explicitly define the notation R_n as the n-fold convolution power of the underlying measure μ before stating the main inequality.
  2. [§5] In the statement of the entropy-gap result, the superlogarithmic function should be specified more precisely (e.g., by giving its explicit form or growth rate) rather than left as 'universal superlogarithmic'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of the entropy-doubling result and its applications, and the recommendation of minor revision. We have reviewed the report carefully.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central claim adapts Tao's external entropy Balog-Szemerédi-Gowers inequality to convolution powers on unimodular locally compact groups, then applies independent structural results on approximate groups to convert the small entropy doubling assumption into control on approximate subgroups and virtual nilpotency. The scale n is quantified explicitly in terms of K and μ_min; finite support is used only to guarantee positive minimal mass and atomicity. No self-citations are load-bearing, no parameters are fitted to the target conclusion, and no step reduces by definition or renaming to its own inputs. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Based on abstract only; no explicit free parameters beyond the given K and μ_min in the statement. Axioms are standard properties of entropy and group theory. No invented entities. Full details unavailable.

free parameters (2)
  • K
    Bound on entropy doubling; appears as input parameter in the hypothesis rather than fitted.
  • μ_min
    Minimum probability mass; appears as input parameter in the hypothesis.
axioms (2)
  • standard math Entropy satisfies standard subadditivity and other information-theoretic inequalities.
    Invoked implicitly in adapting Tao's entropy BSG argument.
  • domain assumption The group is unimodular locally compact.
    Stated as the setting for the adaptation.

pith-pipeline@v0.9.0 · 5469 in / 1402 out tokens · 36389 ms · 2026-05-10T17:18:32.929379+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [1]

    The structure of approximate groups

    Emmanuel Breuillard, Ben Green, and Terence Tao. The structure of approximate groups. Publ. Math. Inst. Hautes ´Etudes Sci., 116:115–221, 2012

    work page 2012

  2. [2]

    Emmanuel Breuillard and Matthew C. H. Tointon. Nilprogressions and groups with moderate growth.Adv. Math., 289:1008–1055, 2016

    work page 2016

  3. [3]

    Isop´ erim´ etrie pour les groupes et les vari´ et´ es.Rev

    Thierry Coulhon and Laurent Saloff-Coste. Isop´ erim´ etrie pour les groupes et les vari´ et´ es.Rev. Mat. Iberoamericana, 9(2):293–314, 1993

    work page 1993

  4. [4]

    Cover and Joy A

    Thomas M. Cover and Joy A. Thomas.Elements of information theory. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, second edition, 2006

    work page 2006

  5. [5]

    Timothy Gowers, Ben Green, Freddie Manners, and Terence Tao

    W. Timothy Gowers, Ben Green, Freddie Manners, and Terence Tao. On a conjecture of Marton.Ann. of Math. (2), 201(2):515–549, 2025

    work page 2025

  6. [6]

    Timothy Gowers, Ben Green, Freddie Manners, and Terence Tao

    W. Timothy Gowers, Ben Green, Freddie Manners, and Terence Tao. Marton’s conjecture in abelian groups with bounded torsion.Ann. Fac. Sci. Toulouse Math. (6), 35(1):1–33, 2026

    work page 2026

  7. [7]

    Groups of polynomial growth and expanding maps.Inst

    Mikhael Gromov. Groups of polynomial growth and expanding maps.Inst. Hautes ´Etudes Sci. Publ. Math., (53):53–73, 1981

    work page 1981

  8. [8]

    V. A. Ka ˘imanovich and A. M. Vershik. Random walks on discrete groups: boundary and entropy.Ann. Probab., 11(3):457–490, 1983

    work page 1983

  9. [9]

    Symmetric random walks on groups.Trans

    Harry Kesten. Symmetric random walks on groups.Trans. Amer. Math. Soc., 92:336–354, 1959

    work page 1959

  10. [10]

    A finitary version of Gromov’s polynomial growth theorem

    Yehuda Shalom and Terence Tao. A finitary version of Gromov’s polynomial growth theorem. Geom. Funct. Anal., 20(6):1502–1547, 2010

    work page 2010

  11. [11]

    Soardi and Wolfgang Woess

    Paolo M. Soardi and Wolfgang Woess. Amenability, unimodularity, and the spectral radius of random walks on infinite graphs.Math. Z., 205(3):471–486, 1990

    work page 1990

  12. [12]

    Product set estimates for non-commutative groups.Combinatorica, 28(5):547– 594, 2008

    Terence Tao. Product set estimates for non-commutative groups.Combinatorica, 28(5):547– 594, 2008

    work page 2008

  13. [13]

    Sumset and inverse sumset theory for Shannon entropy.Combin

    Terence Tao. Sumset and inverse sumset theory for Shannon entropy.Combin. Probab. Com- put., 19(4):603–639, 2010

    work page 2010

  14. [14]

    Inverse theorems for sets and measures of polynomial growth.Q

    Terence Tao. Inverse theorems for sets and measures of polynomial growth.Q. J. Math., 68(1):13–57, 2017

    work page 2017

  15. [15]

    Romain Tessera and Matthew C. H. Tointon. Small doubling implies small tripling for balls of large radius.Discrete Anal., pages Paper No. 9, 9, 2025

    work page 2025

  16. [16]

    Matthew C. H. Tointon. Commuting probabilities of infinite groups.J. Lond. Math. Soc. (2), 101(3):1280–1297, 2020. Universit´e Paris-Dauphine – CEREMADE, Place de Lattre de Tassigny, 75016 Paris, France Email address:guy.blachar@gmail.com

    work page 2020