Non-commutative Barge-Ghys quasimorphisms
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866915096283840512 |
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| author | Brandenbursky, Michael Verbitsky, Misha |
| author_facet | Brandenbursky, Michael Verbitsky, Misha |
| contents | A (non-commutative) Ulam quasimorphism is a map $q$ from a group $Γ$ to a topological group $G$ such that $q(xy)q(y)^{-1}q(x)^{-1}$ belongs to a fixed compact subset of $G$. Generalizing the construction of Barge and Ghys, we build a family of quasimorphisms on a fundamental group of a closed manifold $M$ of negative sectional curvature, taking values in an arbitrary Lie group. This construction, which generalizes the Barge-Ghys quasimorphisms, associates a quasimorphism to any principal $G$-bundle with connection on $M$. Kapovich and Fujiwara have shown that all quasimorphisms taking values in a discrete group can be constructed from group homomorphisms and quasimorphisms taking values in a commutative group. We construct Barge-Ghys type quasimorphisms taking prescribed values on a given subset in $Γ$, producing counterexamples to the Kapovich and Fujiwara theorem for quasimorphisms taking values in a Lie group. Our construction also generalizes a result proven by D. Kazhdan in his paper ``On $ε$-representations''. Kazhdan has proved that for any $ε>0$, there exists an $ε$-representation of the fundamental group of a Riemann surface of genus 2 which cannot be $1/10$-approximated by a representation. We generalize his result by constructing an $ε$-representation of the fundamental group of a closed manifold of negative sectional curvature taking values in an arbitrary Lie group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_12958 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Non-commutative Barge-Ghys quasimorphisms Brandenbursky, Michael Verbitsky, Misha Differential Geometry Group Theory Geometric Topology 53C05, 20F67, 22E25 A (non-commutative) Ulam quasimorphism is a map $q$ from a group $Γ$ to a topological group $G$ such that $q(xy)q(y)^{-1}q(x)^{-1}$ belongs to a fixed compact subset of $G$. Generalizing the construction of Barge and Ghys, we build a family of quasimorphisms on a fundamental group of a closed manifold $M$ of negative sectional curvature, taking values in an arbitrary Lie group. This construction, which generalizes the Barge-Ghys quasimorphisms, associates a quasimorphism to any principal $G$-bundle with connection on $M$. Kapovich and Fujiwara have shown that all quasimorphisms taking values in a discrete group can be constructed from group homomorphisms and quasimorphisms taking values in a commutative group. We construct Barge-Ghys type quasimorphisms taking prescribed values on a given subset in $Γ$, producing counterexamples to the Kapovich and Fujiwara theorem for quasimorphisms taking values in a Lie group. Our construction also generalizes a result proven by D. Kazhdan in his paper ``On $ε$-representations''. Kazhdan has proved that for any $ε>0$, there exists an $ε$-representation of the fundamental group of a Riemann surface of genus 2 which cannot be $1/10$-approximated by a representation. We generalize his result by constructing an $ε$-representation of the fundamental group of a closed manifold of negative sectional curvature taking values in an arbitrary Lie group. |
| title | Non-commutative Barge-Ghys quasimorphisms |
| topic | Differential Geometry Group Theory Geometric Topology 53C05, 20F67, 22E25 |
| url | https://arxiv.org/abs/2212.12958 |