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A group $G$ has property (PH) if it virtually has property (PH'). This notion is a generalization of property (QT) introduced by Bestvina-Bromberg-Fujiwara \\cite{BBF21}. In this paper, we initiate the study of property (PH) of groups and give a complete characterization of groups with property (PH') or (PH) from lineal actions.\n  In addition, by con"},"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":"2506.04856","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2025-06-05T10:24:42Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"13075f032580541e6ea05eab585bd0b808b9a1dd51a544945ddcb3b373c0e4cc","abstract_canon_sha256":"4ba7865586138589c81695bed499d2479b8c6d7975afbeacf788d0ffb3c90fef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:17:00.602916Z","signature_b64":"6w0AbL26JU0pZWatZkdaxmI/uSG/J/qHfeTPwzupyxnDTKM70kDRnQpB9PcmJeIILOMBd9i9wrzyUJlYl0vYAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"844ac5aed7e77f07218d3fd6805521f54f8d5489d2d439b04e1959346453a82c","last_reissued_at":"2026-07-05T11:17:00.602367Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:17:00.602367Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proper actions on finite products of hyperbolic spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Bingxue Tao, Renxing Wan","submitted_at":"2025-06-05T10:24:42Z","abstract_excerpt":"A group $G$ is said to have property (PH') if there exist finitely many hyperbolic spaces $X_1,\\cdots,X_n$ on which $G$ acts coboundedly such that the diagonal action of $G$ on the product $\\prod_{i=1}^nX_i$ equipped with $\\ell^1$-metric is proper. 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