{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:HSJDWDYEVEHML7TKYNF5RU3MLT","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"7e2fa3292644dbb7479b2274400f28470e3b816de89a7094a077c4ea5efc962b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-02-20T00:38:55Z","title_canon_sha256":"43091026e66b4e9ce11cc5dc52c8bea12d0a377dae555cf95417b93c93605150"},"schema_version":"1.0","source":{"id":"1202.4199","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.4199","created_at":"2026-05-18T04:01:25Z"},{"alias_kind":"arxiv_version","alias_value":"1202.4199v2","created_at":"2026-05-18T04:01:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.4199","created_at":"2026-05-18T04:01:25Z"},{"alias_kind":"pith_short_12","alias_value":"HSJDWDYEVEHM","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"HSJDWDYEVEHML7TK","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"HSJDWDYE","created_at":"2026-05-18T12:27:09Z"}],"graph_snapshots":[{"event_id":"sha256:3d95faaf59bade481b66cde77f2fb24515acd46f3e81760d4c9213db21f63fef","target":"graph","created_at":"2026-05-18T04:01:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper we investigate metric properties of the groups $\\Gamma_d(q)$ whose Cayley graphs are the Diestel-Leader graphs $DL_d(q)$ with respect to a given generating set $S_{d,q}$. These groups provide a geometric generalization of the family of lamplighter groups, whose Cayley graphs with respect to a certain generating set are the Diestel-Leader graphs $DL_2(q)$. Bartholdi, Neuhauser and Woess in \\cite{BNW} show that for $d \\geq 3$, $\\Gamma_d(q)$ is of type $F_{d-1}$ but not $F_d$. We show below that these groups have dead end elements of arbitrary depth with respect to the generating se","authors_text":"Jennifer Taback, Melanie Stein","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-02-20T00:38:55Z","title":"Metric Properties of Diestel-Leader Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.4199","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c6271858f1277c65eb5df163c5b1ab8092dbd1b659dd54fb392cb18fee57ef1e","target":"record","created_at":"2026-05-18T04:01:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7e2fa3292644dbb7479b2274400f28470e3b816de89a7094a077c4ea5efc962b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-02-20T00:38:55Z","title_canon_sha256":"43091026e66b4e9ce11cc5dc52c8bea12d0a377dae555cf95417b93c93605150"},"schema_version":"1.0","source":{"id":"1202.4199","kind":"arxiv","version":2}},"canonical_sha256":"3c923b0f04a90ec5fe6ac34bd8d36c5cf3818411aa638bfb9289651a2e683a4f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3c923b0f04a90ec5fe6ac34bd8d36c5cf3818411aa638bfb9289651a2e683a4f","first_computed_at":"2026-05-18T04:01:25.486387Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:01:25.486387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OfZPqcUZH5rIAM6a+qmxM/2Arpo/GZ+Vu5EY//kNZaLnpCyfiETMdGFdKs/aSgOLQa6GZChDtIzD3KqepxxnDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:01:25.487102Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.4199","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c6271858f1277c65eb5df163c5b1ab8092dbd1b659dd54fb392cb18fee57ef1e","sha256:3d95faaf59bade481b66cde77f2fb24515acd46f3e81760d4c9213db21f63fef"],"state_sha256":"59e5f3650c807ff457e86fa31d8c7d18e3cc9343187c0a5b5a29a52a99531b7c"}