We prove a freeness theorem for low-rank subgroups of one-relator groups. Let F be free group, and let w∈F nonprimitive element. The primitivity rank w, π(w), is the smallest subgroup containing w as an imprimitive Then any group G=F∕⟨⟨w⟩⟩ generated by fewer than π(w) elements free. In particular, if π(w)>2, then G does not contain Baumslag–Solitar hypothesis that π(w)>2 implies presentation co...

We prove that the natural map G → Ĝ, where G is a torsionfree group and Ĝ is obtained by adding a new generator t and a new relator w , is surjective only if w is conjugate to gt where g ∈ G . This solves a special case of the surjectivity problem for group extensions, raised by Cohen [2]. AMS Classification 20E22, 20F05; 57M20, 57Q10

We show that the free-by-cyclic groups of the form F2 o Z act properly cocompactly on CAT(0) square complexes. We also show using generalized Baumslag–Solitar groups that all known groups defined by a 2-generator 1-relator presentation are either SQ-universal or are cyclic or isomorphic to a soluble Baumslag–Solitar group.

the freiheitssatz of magnus for one-relator groups is one of the cornerstones of combinatorial group theory. in this short note which is mostly expository we discuss the relationship between the freiheitssatz and corre-sponding results in free power series rings over fields. these are related to results of schneerson not readily available in english. this relationship uses a faithful representa...

Pride groups, or “groups given by presentations in which each defining relator involves at most two types of generators”, include Coxeter groups, Artin groups, triangles of groups, and Vinberg’s groups defined by periodic paired relations. We show that every non-spherical Pride group that is not a triangle of groups satisfies the Tits alternative.