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Math-2359 · mathematics · Group theory

Strict Cost–$\ell^2$-Betti Inequalities for Free Group Actions

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Does there exist an infinite group with a free probability-measure-preserving action $\Gamma\curvearrowright(X,\mu)$ such that $\beta_1^{(2)}(\Gamma)<\operatorname{cost}(R_{\Gamma\curvearrowright(X,\mu)})-1$?

Formulation references: [S1], [S2].

Short English statement

Does there exist an infinite group with a free probability-measure-preserving action $\Gamma\curvearrowright(X,\mu)$ such that $\beta_1^{(2)}(\Gamma)<\operatorname{cost}(R_{\Gamma\curvearrowright(X,\mu)})-1$?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2359-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2359

The cost lower bound beta_1^(2)(G)+1 applies to every free p.m.p. action of the infinite main group. The higher-cost action is strictly above its other action’s cost and thus strictly above that lower bound, giving the record’s requested existence. Within these assumptions, this is a claimed affirmative answer to the stated question.

Strict Cost–\$
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ell^2\$-Betti Inequalities for Free Group Actions

OpenAI's claim

OpenAI presents a proof in the selected manuscript. A finitely generated amalgam is claimed to have free probability-preserving actions of different costs, refuting fixed price.

Mathematical statement of the claimed result

Result 1.
Let $A=F(a,b_1,\ldots,b_{99})$, $w=ab_1ab_2\cdots ab_{99}a$, $J=\langle b_1,\ldots,b_{99},w\rangle$, and \[ \Gamma=A*_J(J\times\langle t\rangle),\qquad\chi(a)=1,\quad\chi(b_i)=\chi(t)=0. \] Let $X$ be the free conull part of the Bernoulli shift $[0,1]^\Gamma$, and $Y_M=X\times\mathbb Z/M\mathbb Z$ with action $(x,r)g=(xg,r+\chi(g))$. Both carry their natural probability measures. Set $K=e^{96}96^{99}/95^{95}$, choose $0<\alpha<1/200$ with $K\alpha^3<1/2$, and put $\eta=\alpha/100$. The asserted orbit-equivalence-relation costs satisfy \[ \operatorname{Cost}(\Gamma\curvearrowright X)\ge1+\eta,\qquad \operatorname{Cost}(\Gamma\curvearrowright Y_M)\le1+99/M\quad(M>100). \] An integer $M>\max\{100,99/\eta\}$ therefore separates the costs of two essentially free probability-preserving actions of the same group.

Outline of the argument

Finite-height extensions of the amalgam yield graphings with costs approaching one. A relative graphing-compression inequality and Bernoulli finite-permutation models give a separate cost lower bound, supported by a planar word lemma; the two action constructions therefore have unequal costs.

Formal verification

Family 259 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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