Quasimorphisms on the group of density preserving diffeomorphisms of the M\"{o}bius band

TL;DR

This paper explores the existence of quasimorphisms on the group of density-preserving diffeomorphisms of the Möbius band, revealing new structures and properties in non-orientable manifolds.

Contribution

It introduces the study of density-preserving diffeomorphism groups on non-orientable manifolds and constructs infinitely many linearly independent unbounded quasimorphisms.

Findings

Existence of countably many unbounded quasimorphisms on the Möbius band

Groups of density-preserving diffeomorphisms on certain non-orientable surfaces are weakly contractible

Generalization of volume-preserving concepts to non-orientable manifolds

Abstract

The existence of quasimorphisms on groups of homeomorphisms of manifolds has been extensively studied under various regularity conditions, such as smooth, volume-preserving, and symplectic. However, in this context, nothing is known about groups of `area'-preserving diffeomorphisms on non-orientable manifolds. In this paper, we initiate the study of groups of density-preserving diffeomorphisms on non-orientable manifolds. Here, the density is a natural concept that generalizes volume without concerning orientability. We show that the group of density-preserving diffeomorphisms on the M\"obius band admits countably many unbounded quasimorphisms which are linearly independent. Along the proof, we show that groups of density preserving diffeomorphisms on compact, connected, non-orientable surfaces with non-empty boundary are weakly contractible.