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We show that for any finite collection of simultaneously inessential subgroups $\\{H_1, \\ldots, H_k\\}$ in $G$, there exists an element $g$ of infinite order such that $\\forall i$, $\\langle H_i, g\\rangle \\cong H_i * \\langle g\\rangle$. We apply this to show that any group, acting faithfully and geometrically on a non-Euclidean possibly reducible CAT(0) cube complex, has property $P_{naive}$ i.e. given any finite list $\\{g_1, \\ldots, g_k\\}"},"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":"1507.05744","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-07-21T08:28:55Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"e1fd1c3e514e9d11baa885be1bc430cc4da5c8cf481b1c3e0d2168cc0510cec5","abstract_canon_sha256":"f325ce02acbc7ffda0775444335cad54635ab2e7ccd6e98f0db763d37ddf4dca"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:14:53.055286Z","signature_b64":"O3xnbe3S1yy7Mzbx/uhlSWzyZOZPKHQ3imD5rdoAsz3gAsQaGVviKTtfV1/Gd5V62MwTuvIG2yk8enK0Ky4eAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b9311dfe9ca6c658ec67ef615d61425cf5cb0a2e27aeb88f398c9646dc2c5500","last_reissued_at":"2026-05-18T01:14:53.054691Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:14:53.054691Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ping pong on CAT(0) cube complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.GR","authors_text":"Aditi Kar, Michah Sageev","submitted_at":"2015-07-21T08:28:55Z","abstract_excerpt":"Let $G$ be a group acting properly and essentially on an irreducible, non-Euclidean finite dimensional CAT(0) cube complex $X$ without fixed points at infinity. 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