The Diophantine problem in the Baumslag-Gersten group

Abstract

We show that, for every positive prime number $p$, the Baumslag-Gersten group $G_p=⟨a,t\mid (ta{t}^{-1})a(ta{t}^{-1}{)}^{-1}=a^p⟩$ has an undecidable Diophantine problem. This gives the first known examples of one-relator groups with undecidable Diophantine problems. Furthermore we apply this to show that the one-relator group $G_p\ast \mathbb{Z}$ has an undecidable single equation problem.

16 pages