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

© 2018, American Mathematical Society. For every pair of positive integers p > q we construct a one-relator group R p,q whose Dehn function is ≃ n 2α where α = log 2 (2p/q). The group R p,q has no subgroup isomorphic to a Baumslag–Solitar group BS(m, n) with m ≠ ±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.