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We study the case of $G = H \\times K$ for infinite and finitely generated groups $H$ and $K$: on the one hand we show that if $K$ is nonamenable and $H$ has decidable word problem, then the free extension to $G$ of any $H$-subshift which is effectively closed is a sofic $G$-subshift. On the other hand we prove that if both $H$ and $K$ are amenable, there "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2309.02620","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2023-09-05T23:40:05Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"f5e982a2107dc6b34028f454f67a29173e7b5b9ed833ec8e1e0844cde9edfc95","abstract_canon_sha256":"4d4f5f7706ddc261951ea965ea178997808330ab9129cea456dbe440057e5e12"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:18:06.646938Z","signature_b64":"VNEA9GOr/9QytQNw7ri0N7u7wX+RyIcyt3bcns8b4WM6bYmz/XWZVApNgtPtNNNjPiw/zeEvXRbias1Q+jO2Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"72ed7210d2ff36a9a0cdbb3952610e60455d69f9f908280cbb90213a9bd58334","last_reissued_at":"2026-07-05T10:18:06.646524Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:18:06.646524Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Soficity of free extensions of effective subshifts","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.DS","authors_text":"Mathieu Sablik, Sebasti\\'an Barbieri, Ville Salo","submitted_at":"2023-09-05T23:40:05Z","abstract_excerpt":"Let $G$ be a group and $H\\leqslant G$ a subgroup. The free extension of an $H$-subshift $X$ to $G$ is the $G$-subshift $\\widetilde{X}$ whose configurations are those for which the restriction to every coset of $H$ is a configuration from $X$. We study the case of $G = H \\times K$ for infinite and finitely generated groups $H$ and $K$: on the one hand we show that if $K$ is nonamenable and $H$ has decidable word problem, then the free extension to $G$ of any $H$-subshift which is effectively closed is a sofic $G$-subshift. 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