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Abstract Several algebraic criteria, reflecting displacement properties of transformation groups, have been used in the past years to prove vanishing of bounded cohomology and stable commutator length. Recently, the authors introduced the property ofcommuting cyclic conjugates, a new displacement technique that is widely applicable and provides vanishing of the bounded cohomology in all positive degrees and all dual separable coefficients. In this note we consider the most recent along with the by now classical displacement techniques and we study implications among them as well as counterexamples. 

We introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using elementary tools and well-known examples from combinatorial group theory, combined with the Baire category theorem, we obtain a plethora of results demonstrating that several phenomena in group theory are generic. In effect, we provide a new topological framework for the analysis of various well known problems in group theory. We also provide a connection between genericity in these spaces, the word problem for finitely generated groups and model-theoretic forcing. Using these connections, we investigate a natural question raised by Osin: when does a certain space of enumerated groups contain a comeager isomorphism class? We obtain a sufficient condition that allows us to answer Osin’s question in the negative for the space of all enumerated groups and the space of left orderable enumerated groups. We document several open questions in connection with these considerations. 

We study 2-generated subgroups $$\langle f,g\rangle <\operatorname{Homeo}^{+}(I)$$ such that $$\langle f^{2},g^{2}\rangle$$ is isomorphic to Thompson’s group $$F$$ , and such that the supports of $$f$$ and $$g$$ form a chain of two intervals. We show that this class contains uncountably many isomorphism types. These include examples with non-abelian free subgroups, examples which do not admit faithful actions by $$C^{2}$$ diffeomorphisms on 1-manifolds, examples which do not admit faithful actions by $PL$ homeomorphisms on an interval, and examples which are not finitely presented. We thus answer questions due to Brin. We also show that many relatively uncomplicated groups of homeomorphisms can have very complicated square roots, thus establishing the behavior of square roots of $$F$$ as part of a general phenomenon among subgroups of $$\operatorname{Homeo}^{+}(I)$$ .