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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 aBaumslag-Solitar group $BS(m,n)$ with $m \neq \pm n$, but is not automatic, notCAT(0), and cannot act freely on a CAT(0) cube complex. This answers along-standing question on the automaticity of one-relator groups and givescounterexamples to a conjecture of Wise.

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