Uniform negative immersions and the coherence of one-relator groups

Abstract

Previously, the authors proved that the presentation complex of a onerelator group G satisfies a geometric condition called negative immersions if every two-generator, one-relator subgroup of G is free. Here, we prove that one-relator groups with negative immersions are coherent, answering a question of Baumslag in this case. Other strong constraints on the finitely generated subgroups also follow such as, for example, the co-Hopf property. The main new theorem strengthens negative immersions to uniform negative immersions, using a rationality theorem proved with linear-programming techniques.

Primitivity rank for random elements in free groups

“…Since generic elements w ∈ F r have π(w) = r, it follows that for r ≥ 3 and a generic w ∈ F r the group G w is coherent. This fact was first proved by Sapir and Spukalova [23], and also observed by Louder and Wilton [16], with a reference to Puder's result on the primitivity rank of generic elements [21].…”

“…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 , . .…”

One-relator hierarchies

“…Another related property is that of uniform negative immersions, introduced in [LW21b]. Results analogous to those proved by Louder and Wilton [LW21a,LW21b] for one-relator groups with negative immersions have also been shown for fundamental groups of two-complexes with a stronger form of uniform negative immersions by Wise [Wis20]. However, having uniform negative immersions turns out to be equivalent to having negative immersions in the case of one-relator complexes [LW21b, Theorem C].…”