EFFECTIVE DISCRETENESS RADIUS OF STABILISERS FOR STATIONARY ACTIONS

The paper proves an effective weak uniform discreteness bound for stabilizers of stationary actions (Theorem 1.2) with an explicit exponent δ, via a key convolution inequality for the discreteness radius IG and an averaging measure µs = ηK ∗ δs ∗ ηK, followed by a stationary pushforward to Sub(G) and a tail estimate (Theorem 8.1). The statements and steps are clearly visible in the text: the main bound ν({z : Gz ∩ Br ≠ {e}}) ≤ β r^δ for r < ρ and δ ≥ (3 ht(g) dimR G)−(rank(K)+1) are stated in the introduction and proved via Sections 6–8 (Theorem 1.2; Theorem 1.5; Theorem 8.1; δ lower bound in (6.21)) . The candidate’s solution reproduces the paper’s structure: push forward ν to θ = Stab∗ν on Sub(G), define the discreteness radius IG via a Zassenhaus neighborhood, use a contraction inequality for Aµ acting on I−δ, then apply stationarity and Markov to get the rδ tail bound; it also cites the same explicit δ bound. Two minor discrepancies: (i) the paper proves the key inequality for the specific compactly supported bi-K-invariant measure µs (not for every bi-K-invariant µ), and then uses that µ-stationarity is independent of µ to pass to µs ; the model states the inequality for any bi-K-invariant µ without justification. (ii) The model asserts θ is Ad(K)-invariant from bi-K-invariance of µ; this is not used in the paper and is not generally implied by stationarity. These issues are easily repaired (replace µ by µs using the Poisson boundary argument before applying the inequality), leaving a proof that is, in substance, the same as the paper’s.