{"paper":{"title":"Virtual braid groups, virtual twin groups and crystallographic groups","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Oscar Ocampo, Paulo Cesar Cerqueira dos Santos J\\'unior","submitted_at":"2021-10-05T22:45:41Z","abstract_excerpt":"Let $n\\ge 2$. Let $VB_n$ (resp. $VP_n$) be the virtual braid group (resp. the pure virtual braid group), and let $VT_n$ (resp. $PVT_n$) be the virtual twin group (resp. the pure virtual twin group). Let $\\Pi$ be one of the following quotients: $VB_n/\\Gamma_2(VP_n)$ or $VT_n/\\Gamma_2(PVT_n)$ where $\\Gamma_2(H)$ is the commutator subgroup of $H$. In this paper, we show that $\\Pi$ is a crystallographic group and we characterize the elements of finite order and the conjugacy classes of elements in $\\Pi$. Furthermore, we realize explicitly some Bieberbach groups and infinite virtually cyclic groups"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.02392","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2110.02392/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}