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 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.
We generalise the constructions in [2] and [11] to give infinite families of hyperbolic groups, each having a finitely presented subgroup that is not of type F3. By calculating the Euler characteristic of the hyperbolic groups constructed, we prove that infinitely many of them are pairwise non isomorphic. We further show that the first of these constructions cannot be generalised to dimensions higher than 3.
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