Invertible Sharply n-Transitive Sets

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Abstract

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.

Original languageEnglish
Pages (from-to)231-254
Number of pages24
JournalJournal of Combinatorial Theory. Series A
Volume81
Issue number2
DOIs
Publication statusPublished - Feb 1998
Externally publishedYes

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