{"paper":{"title":"On the generic triangle group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.GR"],"primary_cat":"math.MG","authors_text":"Riccardo Piergallini, Stefano Isola","submitted_at":"2014-05-08T11:11:45Z","abstract_excerpt":"We introduce the concept of a generic Euclidean triangle $\\tau$ and study the group $G_\\tau$ generated by the reflection across the edges of $\\tau$. In particular, we prove that the subgroup $T_\\tau$ of all translations in $G_\\tau$ is free abelian of infinite rank, while the index 2 subgroup $H_\\tau$ of all orientation preserving transformations in $G_\\tau$ is free metabelian of rank 2, with $T_\\tau$ as the commutator subgroup. As a consequence, the group $G_\\tau$ cannot be finitely presented and we provide explicit minimal infinite presentations of both $H_\\tau$ and $G_\\tau$. This answers in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.1881","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":""},"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"}