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 $\alpha = \log_2(2p / q)$\mathrm{.}Thegroup$R_{p,q}$ has no subgroup isomorphic to a Baumslag-Solitar group $BS(m,n)$ with $m\mathrm{\ne }\pm n$, but 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.