Support rings, Polishability, and the SIN property

Let T be a finite nontrivial group and I countable. We prove that a Borel subgroup P ≤ T^I is Polishable whenever its realized supports form a Polishable Boolean ring ℬ and each principal-support group P_A, A ∈ ℬ, has an exact coordinate frame over T[ℬ|A]. This includes groups locally contained in finite direct powers of a finite quasisimple group. In that case, P admits a compatible Polish SIN topology exactly when its support ring admits a compatible Polish additive topology with a downward-closed neighborhood base. Solid support topologies also give a global frame and a complete bi-invariant metric. For abstract groups locally contained in finite powers of a finite nonabelian simple group S, the same SIN criterion uses the Boolean ring of principal normal subgroups; its locally compact instances are S^J × D, with J at most countable and D countable discrete. We further prove that principally trivial Boolean-ring covers preserve Polishability for every finite deck group. The proofs use frame-independent category and Polish group extensions. Explicit counterexamples delimit the coefficient, coordinate-projection, intrinsic and finite-deck hypotheses.

GitHub repository: https://github.com/eulogica/support-rings-sin