Strong ergodicity phenomena for Bernoulli shifts of bounded algebraic dimension

Abstract: The algebraic dimension of a Polish permutation group $Q\leq \mathrm{Sym}(\mathbb{N})$ is the smallest $n\in\omega$, so that for all $A\subseteq \mathbb{N}$ of size $n+1$, the orbit of every $a\in A$ under the pointwise stabilizer of $A\setminus{a}$ is finite. We study the Bernoulli shift $P\curvearrowright \mathbb{R}^{\mathbb{N}}$ for various Polish permutation groups $P$ and we provide criteria under which the $P$-shift is generically ergodic relative to the injective part of the $Q$-shift, when $Q$ has algebraic dimension $\leq n$. We use this to show that the sequence of pairwise $*$-reduction-incomparable equivalence relations defined in [KP21] is a strictly increasing sequence in the Borel reduction hierarchy. We also use our main theorem to exhibit an equivalence relation of pinned cardinal $\aleph_1^{+}$ which strongly resembles the equivalence relation of pinned cardinal $\aleph_1^{+}$ from [Zap11], but which does not Borel reduce to the latter. It remains open whether they are actually incomparable under Borel reductions. Our proofs rely on the study of symmetric models whose symmetries come from the group $Q$. We show that when $Q$ is "locally finite" -- e.g. when $Q=\mathrm{Aut}(\mathcal{M})$, where $\mathcal{M}$ is a locally finite countable structure with no algebraicity -- the corresponding symmetric model admits a theory of supports which is analogous to that in the basic Cohen model.