{"paper":{"title":"Transverse properties of parabolic subgroups of Garside groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Luis Paris, Yago Antol\\'in","submitted_at":"2019-02-26T20:35:16Z","abstract_excerpt":"Let $G$ be a Garside group endowed with the generating set $\\mathcal{S}$ of non-trivial simple elements, and let $H$ be a parabolic subgroup of $G$. We determine a transversal $T$ of $H$ in $G$ such that each $\\theta \\in T$ is of minimal length in its right-coset, $H \\theta$, for the word length with respect to $\\mathcal{S}$. We show that there exists a regular language $L$ on $\\mathcal{S} \\cup \\mathcal{S}^{-1}$ and a bijection $\\mathrm{ev} : L \\to T$ satisfying $\\mathrm{lg} (U) = \\mathrm{lg}_\\mathcal{S}( \\mathrm{ev}(U))$ for all $U \\in L$. From this we deduce that the coset growth series of $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.10207","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"}