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Abstract Let $$\Delta $$ Δ be a hyperbolic triangle with a fixed area $$\varphi $$ φ . We prove that for all but countably many $$\varphi $$ φ , generic choices of $$\Delta $$ Δ have the property that the group generated by the $$\pi $$ π -rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $$\varphi \in (0,\pi ){\setminus }\mathbb {Q}\pi $$ φ ∈ ( 0 , π ) \ Q π , a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space $$\mathfrak {C}_\theta $$ C θ of singular hyperbolic metrics on a torus with a single cone point of angle $$\theta =2(\pi -\varphi )$$ θ = 2 ( π - φ ) , and answer an analogous question for the holonomy map $$\rho _\xi $$ ρ ξ of such a hyperbolic structure $$\xi $$ ξ . In an appendix by Gao, concrete examples of $$\theta $$ θ and $$\xi \in \mathfrak {C}_\theta $$ ξ ∈ C θ are given where the image of each $$\rho _\xi $$ ρ ξ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3-manifolds. 

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 $$\operatorname{Homeo}^{+}(I)$$ . 

In this paper, we develop the theory of residually finite rationally [Formula: see text] (RFR[Formula: see text]) groups, where [Formula: see text] is a prime. We first prove a series of results about the structure of finitely generated RFR[Formula: see text] groups (either for a single prime [Formula: see text], or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFR[Formula: see text] groups, which we use to study the boundary manifolds of algebraic curves [Formula: see text] and in [Formula: see text]. We show that boundary manifolds of a large class of curves in [Formula: see text] (which includes all line arrangements) have RFR[Formula: see text] fundamental groups, whereas boundary manifolds of curves in [Formula: see text] may fail to do so.