On the transitivity of the group of orbifold diffeomorphisms

TL;DR

This paper explores the transitivity properties of orbifold diffeomorphism groups, revealing that unlike manifold diffeomorphisms, orbifold groups generally lack full transitivity due to isotropy group obstructions.

Contribution

It investigates the transitivity limitations of orbifold diffeomorphism groups and provides insights into their structure compared to manifold diffeomorphisms.

Findings

Manifold diffeomorphism groups are n-transitive for n points.

Orbifold diffeomorphism groups are generally not n-transitive due to isotropy group obstructions.

The paper includes an example involving area-preserving mappings.

Abstract

Consider a connected manifold of dimension at least two and the group of compactly supported diffeomorphisms that are compactly supported isotopic to the identity. This group acts $n$-transitive: Any tuple of $n$ points can be moved to any other tuple of $n$ points by a compactly supported diffeomorphism that is compactly supported isotopic to the identity. An orbifold diffeomorphism group is typically not $n$-transitive: simple obstructions are given by isomorphism classes of isotropy groups of points. In this paper we investigate the transitivity properties of the group of compactly supported diffeomorphisms of orbifolds that are compactly supported isotopic to the identity. We also study an example in the category of area preserving mappings.