  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.
