Unitary representations of the isometry groups of Urysohn spaces

Rémi Barritault, Colin Jahel, Matthieu Joseph

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Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves that every continuous unitary representation of G = Isom(U_Δ) is dissociated (via Katětov approximation by oligomorphic subgroups) and then classifies irreducibles as inductions from setwise stabilizers of finite subsets with kernels the pointwise stabilizers; moreover, every representation splits as a direct sum of irreducibles (Theorems 4.12 and 4.13, announced as Theorem 1.1) . The model’s solution mirrors these steps and cites the same Mackey-type classification for dissociated representations (Theorem 3.9) and the no-algebraicity/weak elimination-of-imaginaries input used in the paper’s proofs .

Referee report (LaTeX)

\textbf{Recommendation:} no revision

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The classification of unitary representations of Isom(U\_Δ) for all countable Δ is proved cleanly via dissociation and an approximation scheme by oligomorphic subgroups, yielding a full Mackey-type description and complete reducibility. The approach dovetails with model-theoretic tameness (no algebraicity, weak elimination of imaginaries) and extends prior work beyond finite Δ. The paper also delivers consequential corollaries (type I, property (T), ergodic rigidity). Key steps—Theorem 1.1 (see 4.13), Theorem 4.12, and Theorem 3.9—form a coherent pipeline and match the model's outline   .