Subgroup mixing in Baumslag-Solitar groups

The paper proves two main theorems: (i) for any symmetric, generating measure with bounded support on BS(m,n), the conjugation action is topologically μ-mixing on K∞, and on every KN when |m|=|n|; (ii) if |m|≠|n|, there exists a finitely supported, symmetric, generating μ for which the action is not topologically μ-mixing on any finite phenotype piece KN. These are stated as Theorems 1.2 and 1.1, respectively, and proved via ends convergence on the Bass–Serre tree (Cartwright–Soardi, with the amenability hypothesis verified for BS(m,n)) and a key deterministic “gluing” lemma (Lemma 3.9) in the mixing direction, and via a biased t vs t^{-1} random walk plus a p-adic growth estimate (Proposition 3.1 and Theorem 3.3) in the non-mixing direction . The model’s Part (2) departs crucially from the paper: it posits zero-mean increments E[Y]=0 yet claims “drift to +∞ along a typical sample path” and uniform-in-time failure of mixing, whereas the paper requires a bias μ(t)≠μ(t^{-1}) to obtain a positive-probability event on which partial sums stay positive for all times, yielding uniform non-mixing at each large time step (Theorem 3.3) . In Part (1), the model also relies on an unjustified “any fixed reduced subword appears many times” claim; the paper instead uses ends convergence, escape of compacts, independence, and Lemma 3.9 to guarantee the needed concatenations with high probability (Equations (8)-(9) and the final gluing step) .