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Uniform Kazhdan Monsters and Schmidt's Property M

Primarily AI-generated textHuman understanding: some partsmath.DS — Dynamical Systemsmath.GR — Group Theory

Contributed by Mehdi Moradi ↗

Version 1 / Oct 07, 2026 / CC BY 4.0

Abstract

Let GG be an infinite finitely generated group whose proper subgroups are finite. We prove that a positive Kazhdan constant uniform over all finite generating sets forces every unit vector in every representation without invariant vectors to have a cofinite displacement gap. For ergodic probability-measure-preserving actions this gives \[ \liminf_{g\to\infty}\mu(gA\mathbin\triangle A) \geq \ku(G)^2\mu(A)(1-\mu(A)), \] and hence Schmidt's Property~MM. A tensor-square argument also gives a uniform gap from modulus one for diagonal matrix coefficients. Conditional on the graded-diagram package underlying the Osin--Sonkin construction, the theorem shows that their infinite finitely generated simple ICC group of bounded exponent has Property~MM. We include a corrected, constant-tracked proof of the Osin--Sonkin deduction from that package and conclude with questions about Ozawa's quasifinite Kazhdan quotients.

Provenance statement

OpenAI's GPT-5.6 Sol, used with ultra reasoning effort, assisted substantially in drafting this manuscript. Its role included organizing the literature review, formulating and revising the exposition, checking internal consistency, and preparing the \TeX{} source. An OpenAI assistant also helped prepare this review copy and check its bibliography and exposition. The main result is checked by the contributor, but the Osin--Sonkin construction is conditional. My suggested example was the family of Ozawa monsters, but the model failed to establish that.

Tools used

OpenAI
ChatGPTVersion 5.6 sol

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Version history

  1. v1Submitted by Mehdi MoradiInitial depositCurrentOct 07, 2026