Quasihomomorphisms to real algebraic groups

TL;DR

The paper proves a rigidity theorem for quasihomomorphisms from a group $\Gamma$ into real algebraic groups $G$: after passing to a finite-index subgroup and applying a bounded modification, the defect is controlled by a bounded normal algebraic subgroup, and in a stronger form by factoring through a compact quotient and a rigid abelian normal subgroup. The authors develop a framework combining Zariski closures, parabolic contraction, and the Adjoint action to show that the defect cannot wander outside a bounded normal region; they also derive corollaries linking to bounded cohomology and almost-homomorphism behavior. A key advance is extending the Fujiwara–Kapovich-type rigidity from discrete targets to real algebraic groups, while clarifying the role of normal versus central defect and providing a counterexample illustrating the limits of central-defect rigidity. The results illuminate how real-algebraic structure and Zimmer-type dynamics constrain approximate homomorphisms, with potential implications for higher-rank lattices and model-theoretic perspectives on approximate subgroups.

Abstract

A quasihomomorphism is a map that satisfies the homomorphism relation up to bounded error. Fujiwara and Kapovich proved a rigidity result for quasihomomorphisms taking values in discrete groups, showing that all quasihomomorphisms can be built from homomorphisms and sections of bounded central extensions. We study quasihomomorphisms with values in real linear algebraic groups, and prove an analogous rigidity theorem.

Theorems & Definitions (54)

• Theorem A: Main theorem, succinct form
• Remark
• Theorem B: Main theorem, stronger form
• Corollary C
• Theorem D
• Theorem E
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• proof