Louder, L; Wilton, H; (2022) Negative immersions for one-relator groups. Duke Mathematical Journal , 171 (3) pp. 547-594. 10.1215/00127094-2021-0024.

Preview

Text Louder_Negative immersions for one-relator groups_AAM.pdf Download (746kB) | Preview

Abstract

We prove a freeness theorem for low-rank subgroups of one-relator groups. Let F be a free group, and let w ∈ F be a nonprimitive element. The primitivity rank of w, π(w), is the smallest rank of a subgroup of F containing w as an imprimitive element. Then any subgroup of the one-relator group G = F/⟨⟨w⟩⟩ generated by fewer than π(w) elements is free. In particular, if π(w) > 2, then G does not contain any Baumslag–Solitar groups. The hypothesis that π(w) > 2 implies that the presentation complex X of the one-relator group G has negative immersions: if a compact, connected complex Y immerses into X and X(Y) ≥ 0, then Y Nielsen reduces to a graph. The freeness theorem is a consequence of a dependence theorem for free groups, which implies several classical facts about free and one-relator groups, including Magnus’ Freiheitssatz and theorems of Lyndon, Baumslag, Stallings, and Duncan–Howie. The dependence theorem strengthens Wise’s w-cycles conjecture, proved independently by the authors and Helfer–Wise, which implies that the one-relator complex X has nonpositive immersions when π(w) > 1.

Download activity - last month

Export as XMLCSVJSON

Download activity - last 12 months

Export as XMLCSVJSON

Downloads by country - last 12 months

Export as JSONXMLCSV

Archive Staff Only

View Item