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We call the space $Z$ (and action of $G$ on it) {\\it pullback} space (action). In such general setting a negative answer follows from a recent result of O. Baker and T. Riley [BR].\n  Suppose, in addition, that the initial actions are relatively hyperbolic that is they are non-parabolic and the induced action on the distinct pairs a"},"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":"1305.6649","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-05-28T22:40:23Z","cross_cats_sorted":[],"title_canon_sha256":"cee19343a362e4d59a42a5cb3d6ce43342a2e07cc4b7a0dafa96c8547a38e938","abstract_canon_sha256":"40943f7582712e067656c98b3d12ee729ef1a12c76709d8fd2cb4e272e2265f0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:59.008068Z","signature_b64":"I7LE7Y0KdtTvuxvkvnEeryzDFSz0HmGQkt+bcVnHLTuwN2VkxuoT1m5iqG75esqx08WDzDEByxRmYmO54xwwDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f90108e8ab2db564dc46c097ee40a8e74598f47b44caf1299c7ea4357e463ad3","last_reissued_at":"2026-05-18T02:55:59.007678Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:59.007678Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Similar relatively hyperbolic actions of a group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Leonid Potyagailo, Victor Gerasimov","submitted_at":"2013-05-28T22:40:23Z","abstract_excerpt":"Let a discrete group $G$ possess two convergence actions by homeomorphisms on compacta $X$ and $Y$. 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