Uniform Kazhdan Monsters and Schmidt's Property M
Let \(G\) 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~\(M\). A tensor-square argument also gives a uniform gap from modulus one for diagonal matrix coefficients. Conditional on the graded-diagram…
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