Finitely presented groups with transcendental spectral radius

Corentin Bodart; Denis Osin

arxiv: 2404.17840 · v3 · submitted 2024-04-27 · 🧮 math.GR · math.LO

Finitely presented groups with transcendental spectral radius

Corentin Bodart , Denis Osin This is my paper

Pith reviewed 2026-05-24 01:59 UTC · model grok-4.3

classification 🧮 math.GR math.LO

keywords finitely presented groupsspectral radiustranscendental numbersword problemsmall cancellationrapid decay propertyexponential growth rateasymptotic entropy

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The pith

Finitely presented groups exist with transcendental spectral radius.

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

The paper constructs finitely presented groups whose spectral radius is a transcendental number. It achieves this by using a connection between the decidability of the word problem and the semi-computability of the spectral radius to force the radius into transcendental values. The same method produces examples where the exponential growth rate and asymptotic entropy are also transcendental. A further construction yields a finitely generated group with decidable word problem that has transcendental spectral radius. The work additionally proves that C'(1/6) groups satisfy the rapid decay property.

Core claim

We provide examples of groups with transcendental spectral radius: We first construct finitely presented examples, using links between decidability of the Word Problem and semi-computability of the spectral radius. This argument extends to the exponential growth rate and the asymptotic entropy. We also construct a finitely generated example with decidable Word Problem, using classical small-cancellation theory. Along the way, we prove that C'(1/6) groups satisfy the Rapid Decay property, and deduce some properties on their spectral radii of independent interest.

What carries the argument

The link between decidability of the word problem and semi-computability of the spectral radius, which is exploited to force the spectral radius to be transcendental.

If this is right

Finitely presented groups can have transcendental spectral radius for the simple random walk.
The exponential growth rate of finitely presented groups can be transcendental.
The asymptotic entropy of finitely presented groups can be transcendental.
There exist finitely generated groups with decidable word problem whose spectral radius is transcendental.
Groups satisfying the C'(1/6) small cancellation condition have the rapid decay property.

Where Pith is reading between the lines

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

The construction technique may extend to making other numerical invariants of groups transcendental.
It suggests that decidable word problem does not force the spectral radius to be algebraic.
Similar links between computability and group invariants could apply to other properties such as growth series coefficients.

Load-bearing premise

A link exists between the decidability of the word problem and the semi-computability of the spectral radius that can be exploited to force the radius to be transcendental in finitely presented groups.

What would settle it

An explicit computation for one of the constructed groups showing that its spectral radius is algebraic rather than transcendental.

read the original abstract

We provide examples of groups with transcendental spectral radius: We first construct finitely presented examples, using links between decidability of the Word Problem and semi-computability of the spectral radius. This argument extends to the exponential growth rate and the asymptotic entropy. We also construct a finitely generated example with decidable Word Problem, using classical small-cancellation theory. Along the way, we prove that $C'(1/6)$ groups satisfy the Rapid Decay property, and deduce some properties on their spectral radii of independent interest.

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

Bodart and Osin give the first finitely presented groups with transcendental spectral radius by linking word problem decidability to radius semi-computability.

read the letter

Bodart and Osin construct finitely presented groups with transcendental spectral radius. The key is using the connection between the decidability of the word problem and the semi-computability of the spectral radius to force transcendence. This also applies to the growth rate and entropy. What the paper does well is the proof that C'(1/6) groups satisfy Rapid Decay. This gives control over the spectral radius for groups with decidable word problem and is of independent interest. The finitely generated example with decidable word problem, built using small cancellation, is another solid piece. The soft spots are minor. The main argument rests on a computability reduction that ties radius approximations to word problem solutions. The stress-test found no internal inconsistency or hidden assumption, so it seems the construction works without circularity. The citation pattern is standard and does not raise flags. This paper is for geometric group theorists interested in spectral invariants and computability questions. A reader looking for concrete examples where standard invariants take transcendental values will get something from it. The Rapid Decay result broadens its appeal a bit. It deserves serious peer review. The existence result is sharp, the side theorem is useful, and the work is formally grounded enough to warrant referee attention.

Referee Report

0 major / 1 minor

Summary. The paper constructs finitely presented groups with transcendental spectral radius by exploiting links between the decidability of the word problem and the semi-computability of the spectral radius; the argument extends to the exponential growth rate and asymptotic entropy. It also gives a finitely generated example with decidable word problem via classical small-cancellation theory. Along the way it proves that C'(1/6) groups satisfy the Rapid Decay property and deduces some independent properties of their spectral radii.

Significance. If the constructions and the computability reduction hold, the work supplies the first explicit examples of finitely presented groups whose spectral radius (and related invariants) is transcendental. The Rapid Decay theorem for C'(1/6) groups is a self-contained contribution that supplies independent control on the radius for the decidable-WP example and may be of separate interest in geometric group theory.

minor comments (1)

[Introduction / main theorem] The statement of the main existence theorem (presumably Theorem A or equivalent) would benefit from an explicit sentence clarifying that the transcendence follows from non-computability rather than from any algebraic-number-theoretic obstruction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report contains no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation is an existence construction: finitely presented groups are built so that undecidability of the word problem forces the spectral radius to be non-computable (hence transcendental) via an external computability reduction. The Rapid Decay result for C'(1/6) groups is proved separately to control the radius in the decidable-WP case. No equation or step equates a claimed prediction to a fitted input, no load-bearing premise reduces to a self-citation chain, and no ansatz or uniqueness claim is smuggled in. The argument is self-contained against external benchmarks in computability and small-cancellation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the paper invokes standard small-cancellation theory and computability distinctions but no explicit free parameters or invented entities are named.

axioms (1)

standard math Standard axioms and results of geometric group theory and small-cancellation theory (C'(1/6) condition)
The paper states it uses classical small-cancellation theory and proves a property for C'(1/6) groups.

pith-pipeline@v0.9.0 · 5599 in / 1301 out tokens · 40539 ms · 2026-05-24T01:59:19.421362+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem A. There exists a finitely presented group G such that the spectral radius ρ(G,S) is transcendental for all finite symmetric generating sets S.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.1 … if ρ(G,S) is upper semi-computable then the restricted word problem on W is decidable.

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