Genericity, the Arzhantseva-Ol'shanskii method and the Isomorphism Problem for One-Relator Groups

Abstract

We apply the method of Arzhantseva-Ol'shanskii to prove that for an exponentially generic (in the sense of Ol'shanskii) class of one-relator groups the isomorphism problem is solvable in at most exponential time. This is obtained as a corollary of our more general result that for any fixed integers $m>1, n>0$ there is an exponentially generic class of $m$-generator $n$-relator groups where every group has only one Nielsen equivalence class of $m$-tuples generating non-free subgroups. This means that a group $G$ in this class has has only one non-free $m$-generated subgroup, namely $G$ itself. Hence for any homomorphism for an $m$-generated group to $G$ the image of this homomorphism is either free or is equal to $G$. Applied to injective homomorphisms from $G$ to itself this implies that $G$ is co-Hopfian. Moreover, every automorphism of $G$ is "freely induced", that is, it lifts to an automorphism of the free group $F_m$. All of these results are obtained by folding methods without using the theory of JSJ-decomposition or the R-tree techniques deployed by Zlil Sela in his famous solution of the isomorphism problem for torsion-free word-hyperbolic groups.