Quasimorphisms and bounded cohomology of transformation groups of nonorientable surfaces
Abstract
Let Ng be a closed connected nonorientable surface of genus g≥1, let ω be a smooth positive density, and let μ be a good measure. In this paper we generalize the Gambaudo--Ghys construction using homogeneous quasimorphisms on the central quotients of the braid groups of Ng. We prove injectivity of this map for density-preserving diffeomorphisms. For g≥2 and quasimorphisms vanishing on the disc braid group, we obtain an injective map to C0-continuous homogeneous quasimorphisms on Homeo0(Ng,μ). We also obtain undistortion criteria in Homeo0(Ng) and Diff0(Ng). For g=1,2 and every d≥2, we prove dimH̅bd(Homeo0(Ng,μ))=∞, dimEH̅bd(Diff0(Ng,ω))=∞. Moreover, EH̅bd(Homeo0(N2,μ)) is infinite dimensional in every positive even degree d. Here H̅bd and EH̅bd denote reduced bounded cohomology and reduced exact bounded cohomology, respectively.
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BibTeX
@misc{brandenbursky2026quasimorphisms,
title = {{Quasimorphisms and bounded cohomology of transformation groups of nonorientable surfaces}},
author = {Michael Brandenbursky},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02847},
eprint = {2611.02847},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02847},
}
RIS
TY - UNPB
TI - Quasimorphisms and bounded cohomology of transformation groups of nonorientable surfaces
AU - Michael Brandenbursky
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02847
UR - https://arxiv.org/abs/2611.02847
AB - Let N_g be a closed connected nonorientable surface of genus g≥1, let ω be a smooth positive density, and let μ be a good measure. In this paper we generalize the Gambaudo--Ghys construction using homogeneous quasimorphisms on the central quotients of the braid groups of N_g. We prove injectivity of this map for density-preserving diffeomorphisms. For g≥2 and quasimorphisms vanishing on the disc braid group, we obtain an injective map to C^0-continuous homogeneous quasimorphisms on Homeo_0(N_g,μ). We also obtain undistortion criteria in Homeo_0(N_g) and Diff_0(N_g). For g=1,2 and every d≥2, we prove dimH̅_b^d(Homeo_0(N_g,μ))=∞, dimEH̅_b^d(Diff_0(N_g,ω))=∞. Moreover, EH̅_b^d(Homeo_0(N_2,μ)) is infinite dimensional in every positive even degree d. Here H̅_b^d and EH̅_b^d denote reduced bounded cohomology and reduced exact bounded cohomology, respectively.
ER -