强拓扑Rokhlin性质：有限指标上升、描述复杂度与有效障碍
Strong Topological Rokhlin Property: Finite-Index Ascent, Descriptive Complexity, and Effective Obstructions
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中文总结 AI 辅助
本文通过有限符号刻画证明强拓扑Rokhlin性质在有限指标下上升，并刻画其描述复杂度，进而给出有效SFT障碍，表明SL_n(Q)无此性质。
中文摘要 AI 辅助
我们给出了强拓扑Rokhlin性质（STRP）的一个有限符号刻画，该刻画基于全局可实现有限模式系统。利用这一刻画，我们证明了有限指标上升性质：若$H\leq G$具有有限指标，$G$是有限生成的，且$H$具有STRP，则$G$也具有STRP。因此，每个有限生成的几乎自由群都具有STRP，而在可数局部几乎自由群中，STRP恰好对有限生成的群成立。在可数群的紧致编码空间$\operatorname{NSub}(F_\omega)$中，STRP轨迹属于${\boldsymbol \Pi}^0_4$类，且是${\boldsymbol \Sigma}^0_3$-困难的；有限生成几乎自由群的轨迹是${\boldsymbol \Sigma}^0_3$-完全的。对于商群$F_\omega/N$（其中$N$是递归可枚举的），每个投影隔离子位移具有可判定的有限元组语言，并包含一个$N$-递归构型。这产生了STRP的有效SFT障碍，包括子群和直积障碍，并意味着对于任意$n\geq2$，$\mathrm{SL}_n(\mathbb Q)$不具有STRP。
英文摘要
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 $n\geq2$.
