math.GR — Group Theory

Three-Relator Universal Group, Four-Relator Adian–Rabin Family, and Unrecognizability of $\#^2(S^2\times S^2)$

Contributed by Qiuyu Ren

We construct a universal group with 2 generators and 3 relators, and an Adian–Rabin family with 2 generators and 4 relators whose nontrivial groups are universal. The resulting presentations show that the standard smooth manifold $\#^n(S^2\times S^2)$ is unrecognizable up to homeomorphism or diffeomorphism for every fixed n ≥ 2.