Vol. 312, No. 2, 2021

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

Alessio Savini

Vol. 312 (2021), No. 2, 505–525

Let G be a semisimple Lie group of noncompact type and let 𝒳G be the Riemannian symmetric space associated to it. Suppose 𝒳G has dimension n and does not contain any factor isometric to either 2 or SL(3, )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 𝒳G. Connell and Farb described a way to construct a map F : Y 𝒳G which is smooth, ρ-equivariant and with uniformly bounded Jacobian.

We extend the construction of Connell and 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 𝒳G 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.

uniform lattice, Zimmer cocycle, Patterson–Sullivan measure, natural map, Jacobian, mapping degree
Mathematical Subject Classification
Primary: 22E40, 57M50
Received: 10 February 2020
Revised: 28 April 2021
Accepted: 30 April 2021
Published: 31 August 2021
Alessio Savini
University of Geneva