Negative immersions for one-relator groups

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 non-primitive 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 onerelator group G = F/ w generated by fewer than π(w) elements is free. In particular, if π(w) > 2 then G doesn't 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 χ(Y ) ≥ 0 then Y is Nielsen equivalent 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 non-positive immersions when π(w) > 1.

Primitivity rank for random elements in free groups

“…As noted above, the results of Louder and Wilton [15,16] imply that if r ≥ 3 and w ∈ F r has π(w) ≥ 3 then the group G w = a 1 , . .…”

“…We are grateful to Doron Puder for bringing to our attention his result from [21] that generic elements of F r have primitivity rank r and for pointing out that the primitivity rank plays a key role in the work of Louder and Wilton [15,16]. We also thank Henry Wilton for pointing us to the work of Cashen and Hoffmann [4].…”

“…Indeed, Klimakov [14] extended the notion of primitivity rank to the context of free algebras of Schreier varieties and obtained some structural results there. Also, a recent series of papers of Louder and Wilton [15,16] shows that for w ∈ F r there is a connection between the primitivity rank π(w) and subgroup properties of the one-relator group G w = a 1 , . .…”

“…Louder and Wilton conjectured [15] that if 1 = w ∈ F r is not a proper power and has π(w) > 2 then the corresponding one-relator group G w is word-hyperbolic (note that by [15,Theorem 1.4] all 2-generator subgroups in G w are free and hence G w has no Baumslag-Solitar subgroups). Cashen and Hoffmann [4] obtained some experimental evidence for this conjecture, verifying it for all w with |w| ≤ 17 and r ≤ 4.…”

“…For 1 = w ∈ F r which is not a proper power Louder and Wilton define w-subgroups of F r as maximal with respect to inclusion elements of Crit(w). They show in [15] that for w as above if π(w) = 2 then w has a unique w-subgroup. In [4] raise the question of whether the a w-subgroup is always unique.…”