{"paper":{"title":"$\\ell^{2}$-torsion of free-by-cyclic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Matt Clay","submitted_at":"2015-09-30T17:14:41Z","abstract_excerpt":"We provide an upper bound on the $\\ell^{2}$-torsion of a free-by-cyclic group, $-\\rho^{(2)}(\\mathbb{F} \\rtimes_{\\Phi} \\mathbb{Z})$, in terms of a relative train-track representative for $\\Phi \\in \\mathrm{Aut}(\\mathbb{F})$. Our result shares features with a theorem of L\\\"uck-Schick computing the $\\ell^{2}$-torsion of the fundamental group of a 3-manifold that fibers over the circle in that it shows that the $\\ell^{2}$-torsion is determined by the exponential dynamics of the monodromy. In light of the result of L\\\"uck-Schick, a special case of our bound is analogous to the bound on the volume of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.09258","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"}