Strong Topological Rokhlin Property: Finite-Index Ascent, Descriptive Complexity, and Effective Obstructions

Abstract

We give a finite symbolic characterization of the strong topological Rokhlin property in terms of globally realizable finite pattern systems. We use this characterization to prove finite-index ascent: if $H\leq G$ has finite index, $G$ is finitely generated, and $H$ has STRP, then $G$ has STRP. Consequently every finitely generated virtually free group has STRP, whereas among countable locally virtually free groups STRP holds exactly for the finitely generated ones. In the compact coding space $\operatorname{NSub}(F_ω)$ of countable groups, the STRP locus belongs to ${\boldsymbol Π}^0_4$, is ${\boldsymbol Σ}^0_3$-hard; the locus of finitely generated virtually free groups is ${\boldsymbol Σ}^0_3$-complete. For quotients $F_ω/N$ with $N$ recursively enumerable, every projectively isolated subshift has decidable finite tuple-language and contains an $N$-recursive configuration. This yields effective SFT obstructions to STRP, including subgroup and direct-product obstructions, and implies that $\mathrm{SL}_n(\mathbb Q)$ does not have \STRP for any $n\geq2$.