Example of an unbounded diffeomorphism group

TL;DR

This paper demonstrates that the identity component of the diffeomorphism group of a punctured torus is unbounded, leading to the conclusion that its fragmentation norm is also unbounded, revealing new geometric properties.

Contribution

It establishes the unboundedness of the identity component of the diffeomorphism group of a punctured torus, a novel result in geometric topology.

Findings

The identity component of the diffeomorphism group of the punctured torus is unbounded.

The fragmentation norm of the punctured torus is unbounded.

Provides new insights into the structure of diffeomorphism groups on surfaces.

Abstract

It is shown that the compactly supported identity component of the diffeomorphism group of the 2-dimensional punctured torus $\mathbb T^2_p$ is an unbounded group. It follows that the fragmentation norm of $\mathbb T^2_p$ is unbounded.