TY - JOUR
T1 - Non-commutative Barge-Ghys Quasimorphisms
AU - Brandenbursky, Michael
AU - Verbitsky, Misha
N1 - Publisher Copyright:
© The Author(s) 2024.
PY - 2024/8/1
Y1 - 2024/8/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85200924992&partnerID=8YFLogxK
U2 - 10.1093/imrn/rnae119
DO - 10.1093/imrn/rnae119
M3 - Article
AN - SCOPUS:85200924992
SN - 1073-7928
VL - 2024
SP - 11135
EP - 11158
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 15
ER -