Equivariant maps for measurable cocycles with values into higher rank Lie groups

Abstract

Let G be a semisimple Lie group of non-compact type and let X G be the Riemannian symmetric space associated to it.Suppose X G has dimension n and it does not contain any factor isometric to either H 2 or SL(3, R)/SO(3).Given a closed n-dimensional complete Riemannian manifold N , let Γ = π1(N ) be its fundamental group and Y its universal cover.Consider a representation ρ : Γ → G with a measurable ρ-equivariant map ψ : Y → X G. Connell-Farb described a way to construct a map F : Y → X G which is smooth, ρ-equivariant and with uniformly bounded Jacobian.In this paper we extend the construction of Connell-Farb to the context of measurable cocycles.More precisely, if (Ω, µΩ) is a standard Borel probability Γ-space, let σ : Γ × Ω → G be measurable cocycle.We construct a measurable map F : Y × Ω → X G which is σ-equivariant, whose slices are smooth and they have uniformly bounded Jacobian.For such equivariant maps we define also the notion of volume and we prove a sort of mapping degree theorem in this particular context.

Locations

• Pacific Journal of Mathematics - View
• arXiv (Cornell University) - View - PDF
• Virtual Community of Pathological Anatomy (University of Castilla La Mancha) - View - PDF
• DataCite API - View