{"paper":{"title":"Cross-ratio dynamics on ideal polygons","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DS","authors_text":"Dmitry Fuchs, Ivan Izmestiev, Maxim Arnold, Serge Tabachnikov","submitted_at":"2018-12-13T09:41:49Z","abstract_excerpt":"Two ideal polygons, $(p_1,\\ldots,p_n)$ and $(q_1,\\ldots,q_n)$, in the hyperbolic plane or in hyperbolic space are said to be $\\alpha$-related if the cross-ratio $[p_i,p_{i+1},q_i,q_{i+1}] = \\alpha$ for all $i$ (the vertices lie on the projective line, real or complex, respectively). For example, if $\\alpha = -1$, the respective sides of the two polygons are orthogonal. This relation extends to twisted ideal polygons, that is, polygons with monodromy, and it descends to the moduli space of M\\\"obius-equivalent polygons. We prove that this relation, which is, generically, a 2-2 map, is completely"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.05337","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}