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If ${\\mathcal B}$ is the group $C^*$-algebra for a discrete group $G$ generated by $A_1,\\ A_2,\\ ...,\\ A_n$ with respect to a representation $\\rho$, then $P(A)$ is an invariant of (weak) equivalence for $\\rho$. This paper computes the joint spectrum of $(1,\\ a,\\ t)$ for the infinite dihedral group $D_{\\infty}=<a,\\ t\\ |\\ a^2=t^2="},"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":"1605.01547","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-05-05T10:43:09Z","cross_cats_sorted":[],"title_canon_sha256":"78bc879bbc3387c452f3a8664000c4b42ab6e330adfcdd55584f0a44797c2333","abstract_canon_sha256":"92aec9ba391cd4329ef52e922ef2aa2aae936781de3e605047c54dd5589e28bb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:13.084640Z","signature_b64":"g1p6dUY4s4PAmd2ep6x2rv+Xn/hO/M161+C3RRxt+baJQ3B8oOzG11o49tKf7PXq3jwH+6LSqBNIE4vQ+HguAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d43b7f8c7d85fbfda6db75e1ce3744bf45479925e2b7ae5e0a0650cd3bae71f0","last_reissued_at":"2026-05-18T00:42:13.083930Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:13.083930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Joint spectrum and infinite dihedral group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Rongwei Yang, Rostilav Grigorchuk","submitted_at":"2016-05-05T10:43:09Z","abstract_excerpt":"For a tuple $A=(A_1,\\ A_2,\\ ...,\\ A_n)$ of elements in a unital Banach algebra ${\\mathcal B}$, its {\\em projective joint spectrum} $P(A)$ is the collection of $z\\in {\\mathbb C}^n$ such that the multiparameter pencil $A(z)=z_1A_1+z_2A_2+\\cdots +z_nA_n$ is not invertible. If ${\\mathcal B}$ is the group $C^*$-algebra for a discrete group $G$ generated by $A_1,\\ A_2,\\ ...,\\ A_n$ with respect to a representation $\\rho$, then $P(A)$ is an invariant of (weak) equivalence for $\\rho$. This paper computes the joint spectrum of $(1,\\ a,\\ t)$ for the infinite dihedral group $D_{\\infty}=<a,\\ t\\ |\\ a^2=t^2="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.01547","kind":"arxiv","version":3},"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":"1605.01547","created_at":"2026-05-18T00:42:13.084048+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.01547v3","created_at":"2026-05-18T00:42:13.084048+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.01547","created_at":"2026-05-18T00:42:13.084048+00:00"},{"alias_kind":"pith_short_12","alias_value":"2Q5X7DD5QX57","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_16","alias_value":"2Q5X7DD5QX573JW3","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_8","alias_value":"2Q5X7DD5","created_at":"2026-05-18T12:29:55.572404+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/2Q5X7DD5QX573JW3OXQ44N2EX5","json":"https://pith.science/pith/2Q5X7DD5QX573JW3OXQ44N2EX5.json","graph_json":"https://pith.science/api/pith-number/2Q5X7DD5QX573JW3OXQ44N2EX5/graph.json","events_json":"https://pith.science/api/pith-number/2Q5X7DD5QX573JW3OXQ44N2EX5/events.json","paper":"https://pith.science/paper/2Q5X7DD5"},"agent_actions":{"view_html":"https://pith.science/pith/2Q5X7DD5QX573JW3OXQ44N2EX5","download_json":"https://pith.science/pith/2Q5X7DD5QX573JW3OXQ44N2EX5.json","view_paper":"https://pith.science/paper/2Q5X7DD5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.01547&json=true","fetch_graph":"https://pith.science/api/pith-number/2Q5X7DD5QX573JW3OXQ44N2EX5/graph.json","fetch_events":"https://pith.science/api/pith-number/2Q5X7DD5QX573JW3OXQ44N2EX5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2Q5X7DD5QX573JW3OXQ44N2EX5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2Q5X7DD5QX573JW3OXQ44N2EX5/action/storage_attestation","attest_author":"https://pith.science/pith/2Q5X7DD5QX573JW3OXQ44N2EX5/action/author_attestation","sign_citation":"https://pith.science/pith/2Q5X7DD5QX573JW3OXQ44N2EX5/action/citation_signature","submit_replication":"https://pith.science/pith/2Q5X7DD5QX573JW3OXQ44N2EX5/action/replication_record"}},"created_at":"2026-05-18T00:42:13.084048+00:00","updated_at":"2026-05-18T00:42:13.084048+00:00"}