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PreprintCC BY-NC-ND 4.0arXiv

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

  • Jintao Luo
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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≤ 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 NSub(Fω) of countable groups, the STRP locus belongs to Π04, is Σ03-hard; the locus of finitely generated virtually free groups is Σ03-complete. For quotients Fω/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 SLn(ℚ) does not have \STRP for any n≥2.

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Cite

BibTeXRIS
BibTeX
@misc{luo2026strong,
  title = {{Strong Topological Rokhlin Property: Finite-Index Ascent, Descriptive Complexity, and Effective Obstructions}},
  author = {Jintao Luo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.18485},
  eprint = {2608.18485},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.18485},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Strong Topological Rokhlin Property: Finite-Index Ascent, Descriptive Complexity, and Effective Obstructions
AU  - Jintao Luo
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2608.18485
UR  - https://arxiv.org/abs/2608.18485
AB  - 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≤ 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 NSub(F_ω) of countable groups, the STRP locus belongs to Π^0_4, is Σ^0_3-hard; the locus of finitely generated virtually free groups is 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 SL_n(ℚ) does not have \STRP for any n≥2.
ER  -