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Equivariant maps for measurable cocycles with values into higher rank Lie groups
Geometric Topology · Submitted 13 Nov 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We construct a measurable map $F: Y \times \Omega \rightarrow \mathcal{X}_G$ which is $\sigma$-equivariant, whose slices are smooth and they have uniformly bounded Jacobian.
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The abstract, sentence by sentence
Let $G$ a semisimple Lie group of non-compact type and let $\mathcal{X}_G$ be the Riemannian symmetric space associated to it. Suppose $\mathcal{X}_G$ has dimension $n$ and it has no factor isometric to either $\mathbb{H}^2$ or $\text{SL}(3,\mathbb{R})/\text{SO}(3)$. Given a closed $n$-dimensional Riemannian manifold $N$, let $\Gamma=\pi_1(N)$ be its fundamental group and $Y$ its universal cover. Consider a representation $\rho:\Gamma \rightarrow G$ with a measurable $\rho$-equivariant map $\psi:Y \rightarrow \mathcal{X}_G$. Connell-Farb described a way to construct a map $F:Y\rightarrow \mathcal{X}_G$ which is smooth, $\rho$-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 $(\Omega,\mu_\Omega)$ is a standard Borel probability $\Gamma$-space, let $\sigma:\Gamma \times \Omega \rightarrow G$ be a measurable cocycle. We construct a measurable map $F: Y \times \Omega \rightarrow \mathcal{X}_G$ which is $\sigma$-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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $G$ a semisimple Lie group of non-compact type and let $\mathcal{X}_G$ be the Riemannian symmetric space associated to it. | background 1.00 | 0.05 | 0.01 | asserted 0.94 |
| s1 | Suppose $\mathcal{X}_G$ has dimension $n$ and it has no factor isometric to either $\mathbb{H}^2$ or $\text{SL}(3,\mathbb{R})/\text{SO}(3)$. | background 0.78 | 0.06 | 0.02 | asserted 0.83 |
| s2 | Given a closed $n$-dimensional Riemannian manifold $N$, let $\Gamma=\pi_1(N)$ be its fundamental group and $Y$ its universal cover. | background 1.00 | 0.07 | 0.01 | asserted 0.90 |
| s3 | Consider a representation $\rho:\Gamma \rightarrow G$ with a measurable $\rho$-equivariant map $\psi:Y \rightarrow \mathcal{X}_G$. | background 0.99 | 0.09 | 0.02 | asserted 0.66 |
| s4 | Connell-Farb described a way to construct a map $F:Y\rightarrow \mathcal{X}_G$ which is smooth, $\rho$-equivariant and with uniformly bounded Jacobian. | background 1.00 | 0.03 | 0.04 | asserted 1.00 |
| s5 | In this paper we extend the construction of Connell-Farb to the context of measurable cocycles. | contribution 1.00 | 0.84 | 0.21 | asserted 1.00 |
| s6 | More precisely, if $(\Omega,\mu_\Omega)$ is a standard Borel probability $\Gamma$-space, let $\sigma:\Gamma \times \Omega \rightarrow G$ be a measurable cocycle. | background 0.43 | 0.20 | 0.03 | asserted 0.87 |
| s7 | We construct a measurable map $F: Y \times \Omega \rightarrow \mathcal{X}_G$ which is $\sigma$-equivariant, whose slices are smooth and they have uniformly bounded Jacobian. | contribution 0.87 | 0.97 | 0.71 | asserted 1.00 |
| s8 | For such equivariant maps we define also the notion of volume and we prove a sort of mapping degree theorem in this particular context. | contribution 0.62 | 0.95 | 0.30 | asserted 0.99 |
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The paper
Original abstract
Let $G$ a semisimple Lie group of non-compact type and let $\mathcal{X}_G$ be the Riemannian symmetric space associated to it. Suppose $\mathcal{X}_G$ has dimension $n$ and it has no factor isometric to either $\mathbb{H}^2$ or $\text{SL}(3,\mathbb{R})/\text{SO}(3)$. Given a closed $n$-dimensional Riemannian manifold $N$, let $\Gamma=\pi_1(N)$ be its fundamental group and $Y$ its universal cover. Consider a representation $\rho:\Gamma \rightarrow G$ with a measurable $\rho$-equivariant map $\psi:Y \rightarrow \mathcal{X}_G$. Connell-Farb described a way to construct a map $F:Y\rightarrow \mathcal{X}_G$ which is smooth, $\rho$-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 $(\Omega,\mu_\Omega)$ is a standard Borel probability $\Gamma$-space, let $\sigma:\Gamma \times \Omega \rightarrow G$ be a measurable cocycle. We construct a measurable map $F: Y \times \Omega \rightarrow \mathcal{X}_G$ which is $\sigma$-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.
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What it cites · 2 sentences without a confident label
Builds on 2 papers
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As already done by the author for measurable cocycles of rank-one lattices [ref] , in this paper we would like to apply the barycenter to build natural maps for measurable cocycles taking values into higher rank Lie groups.
Natural maps for measurable cocycles of compact hyperbolic manifolds · Introduction · 0.96
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As already done by the author for measurable cocycles of rank-one lattices [ref] , in this paper we would like to apply the barycenter to build natural maps for measurable cocycles taking values into higher rank Lie groups.
Integrable tautness of isometries of complex hyperbolic spaces · Introduction · 0.96
Uses the method or tool of 1 paper
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The proof follows the line of [ref] .
Natural maps for measurable cocycles of compact hyperbolic manifolds · Volume of equivariant maps · 1.00
Compares with 1 paper
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It is worth noticing that for constructing the $σ$-equivariant family ${ μ^s_a,x }_a ∈Y, x ∈Ω$ and hence for defining the map $F^s:Y ×Ω→_G$ we exploited the existence of a measurable map $ψ:Y ×Ω→_G$ and not of a boundary map as in [ref] .
Natural maps for measurable cocycles of compact hyperbolic manifolds · Natural maps associated to measurable cocycles of higher rank lattices · 0.91
Cites as background 1 paper
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Among the other possible applications of the barycenter construction and natural maps, it is worth mentioning the rigidity result obtained by Boland-Connell [ref] for foliations of Riemannian manifolds with negatively curved leaves and the rigidity phenomena proved by Boland-Newberger [ref] and by Adeboye-Bray-Constantine [ref] and the author [ref] for Finsler/Benoist manifolds.
Entropy rigidity for foliations by strictly convex projective manifolds · Introduction · 0.95
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