Non-commutative Barge-Ghys Quasimorphisms

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.