$n$-transitivity of bisection groups of a Lie groupoid

TL;DR

This paper extends the concept of $n$-transitivity from diffeomorphism groups to bisection groups of Lie groupoids, proving that such groups are $n$-transitive under mild conditions, with applications to symplectic groupoids.

Contribution

It establishes that groups of $C^r$-bisections of Lie groupoids are $n$-transitive for all $n$, generalizing known results for diffeomorphism groups.

Findings

Groups of all bisections of any Lie groupoid are $n$-transitive.

Lagrangian bisections of symplectic groupoids are $n$-transitive.

If the groupoid is source connected, there exists a bisection passing through any given arrow.

Abstract

The notion of $n$-transitivity can be carried over from groups of diffeomorphisms on a manifold $M$ to groups of bisections of a Lie groupoid over $M$. The main theorem states that the $n$-transitivity is fulfilled for all $n\in\mathbb N$ by an arbitrary group of $C^r$-bisections of a Lie groupoid $\Gamma$ of class $C^r$, where $1\leq r\leq\omega$, under mild conditions. For instance, the group of all bisections of any Lie groupoid and the group of all Lagrangian bisections of any symplectic groupoid are $n$-transitive in the sense of this theorem. In particular, if $\Gamma$ is source connected for any arrow $\gamma\in\Gamma$ there is a bisection passing through $\gamma$.