Quasimorphisms on nonorientable surface diffeomorphism groups

Abstract Bowden, Hensel and Webb constructed infinitely many quasimorphisms on the diffeomorphism groups of orientable surfaces. In this paper, we extend their result to nonorientable surfaces. Namely, we prove that the space of nontrivial quasimorphisms <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover> <m:mrow> <m:mi>Q</m:mi> <m:mi>H</m:mi> </m:mrow> <m:mo>˜</m:mo> </m:mover> <m:mfenced open="(" close=")"> <m:mrow> <m:mrow> <m:msub> <m:mo form="prefix">Diff</m:mo> <m:mn>0</m:mn> </m:msub> </m:mrow> <m:mfenced open="(" close=")"> <m:msub> <m:mi>N</m:mi> <m:mi>g</m:mi> </m:msub> </m:mfenced> </m:mrow> </m:mfenced> </m:math> $\widetilde{Q H}\left(\operatorname{Diff}_0\left(N_g\right)\right)$ on the identity component of the diffeomorphism group Diff 0 ( N g ) on a closed nonorientable surface N g of genus g ≥ 3 is infinite-dimensional. As a corollary, we obtain the unboundedness of the commutator length and the fragmentation length on Diff 0 ( N g ).