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Given such a pair (called a left-right pair), we can construct a directed graph $\\mathcal{F}(L, R)$ with vertices $\\mathcal{D}_0$ and edges $\\{(z, R(z))\\}_{z \\in \\mathcal{D}_0} \\cup \\{(z, L(z))\\}_{z \\in \\mathcal{D}_0}$, which is a collection of infinite binary trees. We answer two questions of Nathanson by classifying all the pairs of elements of $\\mathrm{SL}_2(\\mathbb"},"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":"1810.04349","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-10T03:15:17Z","cross_cats_sorted":[],"title_canon_sha256":"f325fe17b26095707cfb3d70bb4e0b6f961c5277fa3c6d7373432bf73ff3f928","abstract_canon_sha256":"467d36a5baf8753981e26c20bf39812351f10092be74abc83942470efbcbe5b4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:40.349637Z","signature_b64":"9q/jatp7ui2aaMN80nFyTuc8aVU/L2LvJkoOEnbXgoQuXTkLWch7aTBG3eDuqXbbzROQAg7ynraJZqXPoXRkBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"acd4fb6538faa82de8dafa7af0ac188f0f9d60f370901af6800c5d6b8a469473","last_reissued_at":"2026-05-18T00:03:40.349208Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:40.349208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Left-Right Pairs and Complex Forests of Infinite Rooted Binary Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Nina Zubrilina","submitted_at":"2018-10-10T03:15:17Z","abstract_excerpt":"Let $\\mathcal{D}_0:= \\{x + iy \\ \\vert x, y >0\\}$, and let $(L, R)$ be a pair of M\\\"{o}bius transformations corresponding to $\\mathrm{SL}_2(\\mathbb{N}_0)$ matrices such that $R(\\mathcal{D}_0)$ and $L(\\mathcal{D}_0)$ are disjoint. 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