Quasimorphisms on the group of density preserving diffeomorphisms of the Möbius band

TL;DR

The paper advances the study of quasimorphisms on groups of density-preserving diffeomorphisms of a non-orientable surface, focusing on the Möbius band $M$. It develops a Gambaudo-Ghys type cocycle from braid groups $B_2(M)$ to $\operatorname{Diff}_\omega(M,\partial M)_0$, proves the cocycle is well-defined and injective, and uses it to show that $Q(\operatorname{Diff}_\omega(M,\partial M)_0)$ is infinite-dimensional by transferring infinite-dimensionality from $Q(B_2(M))$. A foundational contractibility result for the identity component on non-orientable surfaces with boundary supports the cocycle construction, while a careful analysis of twisted differential forms, density forms, and a blowing-up compactification of $X_2(M)$ yields uniform word-length bounds essential for the technical estimates. The results extend known phenomena for orientable and low-genus cases to the non-orientable setting, highlighting a rich structure of unbounded homogeneous quasimorphisms in this context and establishing weak contractibility as a key structural property. The approach integrates twisted de Rham theory, mapping class group techniques, and geometric group theory methods to achieve the infinite-dimensional quasimorphism space claim.

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 Möbius 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.

Figures (5)

• Figure 6.1: A decomposition into foliated strips and junctions. The red rectangles are strips, foliated by the red geodesic arcs. The white rectangles are junctions which are adjacent to exactly three strips.
• Figure 6.2: Possible subarcs in a junction.
• Figure 6.3: A finite generating set of $P_2(M)$. In each figure, one of $\bar{z}_i$ does not move and the other moves along the indicated path. In particular, the braids of (A) and (B) represent the same braid, denoted by $A_{2,3}$. The braids of (C) and (D) are denoted by $\rho_2$ and $\rho_3$, respectively. See GoncalvesGuaschi17 and compare this with GoncalvesGuaschi17.
• Figure 6.4: The braid $B_{2,3}$. It exchanges the positions of $\bar{z}_i$, twisting the strands in the counter-clockwise direction.
• Figure 6.5: Each braid $\eta_{i,j}$ is represented as a pair of two trajectories: one constant path at $\bar{z}_{3-i}$; one non-trivial trajectory of $\bar{z}_i$.

Theorems & Definitions (60)

• Remark 1.1
• Proposition 2.1: See Calegari09
• Proposition 2.2
• Proposition 3.1: Homotopy invariance of twisted de Rham cohomologies
• Proposition 3.2
• proof
• Lemma 3.3
• proof
• Proposition 4.1
• proof