Vol. 312, No. 2, 2021

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 314: 1  2
Vol. 313: 1  2
Vol. 312: 1  2
Vol. 311: 1  2
Vol. 310: 1  2
Vol. 309: 1  2
Vol. 308: 1  2
Vol. 307: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Contacts
 
Submission Guidelines
Submission Form
Policies for Authors
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
 
Other MSP Journals
Equivariant maps for measurable cocycles with values into higher rank Lie groups

Alessio Savini

Vol. 312 (2021), No. 2, 505–525
Abstract

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.

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