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In this paper, we generalise Magnus Freiheitssatz and solution to the word problem for one-relator groups by considering one relator quotients of certain classes of right-angled Artin groups and graph products of locally indicable polycyclic groups.
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We prove a freeness theorem for low-rank subgroups of one-relator groups. Let $F$ be a free group, and let $win F$ be a non-primitive element. The primitivity rank of $w$, $pi(w)$, is the smallest rank of a subgroup of $F$ containing $w$ as an imprimitive element. Then any subgroup of the one-relator group $G=F/langlelangle wranglerangle$ generated by fewer than $pi(w)$ elements is free. In particular, if $pi(w)>2$ then $G$ doesnt contain any Baumslag--Solitar groups. The hypothesis that $pi(w)>2$ 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 $chi(Y)geq 0$ 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 Wises $w$-cycles conjecture, proved independently by the authors and Helfer--Wise, which implies that the one-relator complex $X$ has non-positive immersions when $pi(w)>1$.
We show that any one-relator group $G=F/langlelangle wranglerangle$ with torsion is coherent -- i.e., that every finitely generated subgroup of $G$ is finitely presented -- answering a 1974 question of Baumslag in this case.
In this paper, we prove that two-generator one-relator groups with depth less than or equal to 3 can be effectively embedded into a tower of HNN-extensions in which each group has the effective standard normal form. We give an example to show how to deal with some general cases for one-relator groups. By using the Magnus method and Composition-Diamond Lemma, we reprove the G. Higman, B. H. Neumann and H. Neumanns embedding theorem.
We prove the statement in the title and exhibit examples of quotients of arbitrary nilpotency class. This answers a question by D. F. Holt.
Previously, the authors proved that the presentation complex of a one-relator 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.
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