{"paper":{"title":"Free products and the algebraic structure of diffeomorphism groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.GT"],"primary_cat":"math.GR","authors_text":"Sang-hyun Kim, Thomas Koberda","submitted_at":"2017-07-19T14:29:50Z","abstract_excerpt":"Let $M$ be a compact one--manifold, and let $\\mathrm{Diff}^{1+\\mathrm{bv}}(M)$ denote the group of $C^1$ orientation preserving diffeomorphisms of $M$ whose first derivatives have bounded variation. We prove that if $G$ is a group which is not virtually metabelian, then $(G\\times\\mathbb{Z})*\\mathbb{Z}$ is not realized as a subgroup of $\\mathrm{Diff}^{1+\\mathrm{bv}}(M)$. This gives the first examples of finitely generated groups $G,H\\le \\mathrm{Diff}_+^\\infty(M)$ such that $G\\ast H$ does not embed into $\\mathrm{Diff}^{1+\\mathrm{bv}}(M)$. By contrast, for all countable groups $G,H\\le\\mathrm{Home"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.06115","kind":"arxiv","version":3},"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"}