A property of the variety of 2-Engel groups
Let G be a group which acts by homeomorphisms on a metric space X. We say the action of G is locally moving on X if for every open U ⊆ X there is a g ∈ G such that g↾X ≠ Id while g↾(X∖U) = Id. We prove the following theorem: Theorem A. Let X,Y be completely metrizable spaces and let G be a group which acts on X and Y with locally moving actions. If the orbits of the action of G on X are of the second category in X and the orbits of the action of G on Y are of the second category...
A short proof, using graphs and groupoids, is given of Brodskii’s theorem that torsion-free one-relator groups are locally indicable.