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We prove that the image of $\\rho$ is finite. A similar result holds if $\\pi_1(S)$ is replaced by the free group $F_n$ on $n\\geq 2$ generators and where $Mod(S,\\, \\ast)$ is replaced by $Aut(F_n)$. We thus resolve a well-known question of M. Kisin. We show that if $G$ is a linear algebraic group and if the representation variety of $\\pi_"},"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":"1702.03622","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2017-02-13T04:15:13Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"bf3393640ef912167ad6656115f127e5c739fd01ec94eaf38a338fd4ce1ea0e0","abstract_canon_sha256":"1d287899ced64cc4ba580ebfb184e0f36b0a24ca6e35d86b31c803f1623df492"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:53.367127Z","signature_b64":"FjJnyCQSK7mIkIEgypz2Vu9ZXxI/qoET1Cw5+atLsgFYadwDkWzOY/AlzUsE4RiNZadURw4BDusejtYHBxeJDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a491f28d59976ea4237cd8ff5e9390da9eef6cca3c75e2ad0b00782fa113295","last_reissued_at":"2026-05-18T00:50:53.366640Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:53.366640Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Representations of surface groups with finite mapping class group orbits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Indranil Biswas, Mahan Mj, Ramanujan Santharoubane, Thomas Koberda","submitted_at":"2017-02-13T04:15:13Z","abstract_excerpt":"Let $(S,\\, \\ast)$ be a closed oriented surface with a marked point, let $G$ be a fixed group, and let $\\rho\\colon\\pi_1(S) \\longrightarrow G$ be a representation such that the orbit of $\\rho$ under the action of the mapping class group $Mod(S,\\, \\ast)$ is finite. We prove that the image of $\\rho$ is finite. A similar result holds if $\\pi_1(S)$ is replaced by the free group $F_n$ on $n\\geq 2$ generators and where $Mod(S,\\, \\ast)$ is replaced by $Aut(F_n)$. We thus resolve a well-known question of M. Kisin. 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