The geometry of one-relator groups satisfying a polynomial isoperimetric inequality

For every pair of positive integers $p > q$ we construct a one-relator group $R_{p,q}$ whose Dehn function is $\simeq n^{2 \alpha }$ where $\alpha = \log _2(2p / q)$. The group $R_{p, q}$ has no subgroup isomorphic to a Baumslag-Solitar group $BS(m, n)$ with $m \neq \pm n$, but it is not automatic, not CAT(0), and cannot act freely on a CAT(0) cube complex. This answers a long-standing question on the automaticity of one-relator groups and gives counterexamples to a conjecture of Wise.