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In the case $G=\\mathrm{SL}(2,\\mathbb C)$ this space has been thoroughly studied.\n  Following work of Thurston, as presented by Culler-Shalen, we give a lower bound for the dimension of irreducible components of $X(\\Gamma,G)$ in terms of the Euler characteristic $\\chi(M)$ of $M$, the number $t$ of "},"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":"1510.00567","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-10-02T11:42:12Z","cross_cats_sorted":[],"title_canon_sha256":"e87d63d0eec464875317727d916da71419e5a43242f85119bfc49125e51eb0a8","abstract_canon_sha256":"0461c15a9d391fb20d3a672f9ce528b15f3c914d85c19d1a8e4bbd6caff2edaa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:12.636730Z","signature_b64":"GzgQtWeOKdWmtwjFHnfiLJbGLBX4HqW4gp9xDAT2orDEUFVdUhwHWtLoLUmyzu/C5fAn8pQEuLleGlEqVn8LDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"92e1b5fe894bc2c8eb05e4adf981a3d746b1c2400267ed177119ee094061e016","last_reissued_at":"2026-05-18T01:31:12.636069Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:12.636069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dimension of character varieties for $3$-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Antonin Guilloux, Elisha Falbel","submitted_at":"2015-10-02T11:42:12Z","abstract_excerpt":"Let $M$ be a $3$-manifold, compact with boundary and $\\Gamma$ its fundamental group. Consider a complex reductive algebraic group G. The character variety $X(\\Gamma,G)$ is the GIT quotient $\\mathrm{Hom}(\\Gamma,G)//G$ of the space of morphisms $\\Gamma\\to G$ by the natural action by conjugation of $G$. In the case $G=\\mathrm{SL}(2,\\mathbb C)$ this space has been thoroughly studied.\n  Following work of Thurston, as presented by Culler-Shalen, we give a lower bound for the dimension of irreducible components of $X(\\Gamma,G)$ in terms of the Euler characteristic $\\chi(M)$ of $M$, the number $t$ of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.00567","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1510.00567","created_at":"2026-05-18T01:31:12.636176+00:00"},{"alias_kind":"arxiv_version","alias_value":"1510.00567v1","created_at":"2026-05-18T01:31:12.636176+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.00567","created_at":"2026-05-18T01:31:12.636176+00:00"},{"alias_kind":"pith_short_12","alias_value":"SLQ3L7UJJPBM","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_16","alias_value":"SLQ3L7UJJPBMR2YF","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_8","alias_value":"SLQ3L7UJ","created_at":"2026-05-18T12:29:42.218222+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SLQ3L7UJJPBMR2YF4SW7TAND25","json":"https://pith.science/pith/SLQ3L7UJJPBMR2YF4SW7TAND25.json","graph_json":"https://pith.science/api/pith-number/SLQ3L7UJJPBMR2YF4SW7TAND25/graph.json","events_json":"https://pith.science/api/pith-number/SLQ3L7UJJPBMR2YF4SW7TAND25/events.json","paper":"https://pith.science/paper/SLQ3L7UJ"},"agent_actions":{"view_html":"https://pith.science/pith/SLQ3L7UJJPBMR2YF4SW7TAND25","download_json":"https://pith.science/pith/SLQ3L7UJJPBMR2YF4SW7TAND25.json","view_paper":"https://pith.science/paper/SLQ3L7UJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1510.00567&json=true","fetch_graph":"https://pith.science/api/pith-number/SLQ3L7UJJPBMR2YF4SW7TAND25/graph.json","fetch_events":"https://pith.science/api/pith-number/SLQ3L7UJJPBMR2YF4SW7TAND25/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SLQ3L7UJJPBMR2YF4SW7TAND25/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SLQ3L7UJJPBMR2YF4SW7TAND25/action/storage_attestation","attest_author":"https://pith.science/pith/SLQ3L7UJJPBMR2YF4SW7TAND25/action/author_attestation","sign_citation":"https://pith.science/pith/SLQ3L7UJJPBMR2YF4SW7TAND25/action/citation_signature","submit_replication":"https://pith.science/pith/SLQ3L7UJJPBMR2YF4SW7TAND25/action/replication_record"}},"created_at":"2026-05-18T01:31:12.636176+00:00","updated_at":"2026-05-18T01:31:12.636176+00:00"}