Relative Property (T), simplices of invariant measures, and existentially closed models

Abstract

We prove a Bauer-Poulsen dichotomy theorem for simplices of invariant measures associated with permutation groups. More precisely, let $G$ be a transitive group of permutations of a countable set $\mathcal{S}$, and let $H$ be the stabilizer of a point of $\mathcal{S}$. Let $\overline{G}$ and $\overline{H}$ denote their closures in the topology of pointwise convergence. Assume the Polish group $\overline{H}$ has relative Property (T) in $\overline{G}$. Then the simplex $\mathcal{M}_\mathrm{inv}(2^\mathcal{S})$ of invariant probability measures for the induced action $G\curvearrowright 2^\mathcal{S}$ is Bauer if and only if $\overline{G}$ has Property (T), and is Poulsen otherwise. This addresses some examples and questions considered by Austin. We deduce this result from a more general model-theoretic statement of independent interest. To this end, we initiate the study of existentially closed models in affine logic.

Disclosure

“se; see Section 11. As we show subsequently in Section 13, under the hypothesis that the pair (G, H ) has Property (T), the main assumption of Theorem 1.3 is also satisfied by the theory PMPG/H This yields Theorem 1.1. Acknowledgments. AI assistants (ChatGPT and Gemini) were used primarily for proofreading and, in that context, contributed minor textual and mathematical correc- tions. They were also used for literature searches and stylistic editing. The mathemat- ical ideas and argum”

PDF page 6

Classification

Literature search

Multiplier

2

Verified

2026-08-23T05:00:55.520704Z

Structural counts