Non-commutative Barge-Ghys quasimorphisms

TL;DR

The paper extends BG quasimorphisms to non-commutative targets by constructing Ulam quasimorphisms from $\pi_1(M)$ to Lie groups via holonomy of connections on bundles over negatively curved manifolds. It shows the existence of non-constructible, homogeneous BG-type quasimorphisms valued in Lie groups, providing counterexamples to Kapovich–Fujiwara for non-abelian targets and generalizing Kazhdan’s $\varepsilon$-representation results to arbitrary Lie groups. The authors develop a parallel framework using principal and vector bundles, geodesic polygon holonomy bounds, and Zariski-dense lattice arguments to contrast constructible vs non-constructible quasimorphisms. Their results reveal rich non-abelian phenomena in Ulam stability and bounded cohomology, with explicit constructions yielding $\varepsilon$-representations that resist approximation by honest representations, highlighting new structural insights into fundamental groups of negatively curved manifolds.

Abstract

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.

Figures (1)

• Figure 1: Lift of the geodesic polygon to the universal cover.