Invertible Sharplyn-Transitive Sets

We prove various results about sharplyn-transitive sets of homeomorphisms of “nice” topological spaces like the real line and the circle. Our main results concern sharply 3-transitive setsGof homeomorphisms of the circle to itself. IfG=G−1, thenGcontains a real hyperbolic part (a set of involutions of the circle having special properties). We show that every real hyperbolic part is the hyperbolic part of a real abstract oval in the sense of Buekenhout and that every real abstract oval arises from a topological oval in a flat projective plane. This establishes a new relationship between flat Minkowski planes that admit automorphisms which are circle-symmetries and flat projective planes containing topological ovals. We also consider sharplyn-transitive sets of permutations acting on finite sets. We find that our results about flat geometries and sharplyn-transitive sets of homeomorphisms have counterparts in the finite case.