Quasimorphisms on the Group of Density Preserving Diffeomorphisms of the Möbius Band

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, groups of “area”-preserving diffeomorphisms on non-orientable manifolds have not been studied. 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 Möbius band admits countably many homogeneous quasimorphisms that are linearly independent. Also, we discuss the higher-genus cases implicitly suggested by the literature. Along the proof, we show that groups of density preserving diffeomorphisms on compact, connected, non-orientable surfaces with non-empty boundary are weakly contractible.