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Some explicit cocycles on the Furstenberg boundary for products of isometries of hyperbolic spaces and $\mathrm{SL}(3,\mathbb{K})$

Michelle Bucher, Alessio Savini

TL;DR

The work advances explicit cocycle representatives for the kernel of the boundary evaluation map in measurable cohomology, focusing on products of real hyperbolic isometries and on SL(3, K). It uses Monod’s boundary spectral sequence to relate NH^k to invariants in $(\wedge^*\mathfrak{a})^{w_0}$ and constructs explicit injections and sections at the cocycle level, employing π_A-projections and contracting homotopies. For products of hyperbolic isometries, it provides concrete non-alternating cocycles in degrees 3 and 4 expressed via cross ratios, proving injectivity of the bounded-to-measurable map in degree 3 and yielding a parallel result for SL(3, K) in degrees 2 and 3. The results give explicit G-invariant cocycles on the Furstenberg boundary and illuminate the structure of bounded cohomology in higher-rank contexts.

Abstract

Nicolas Monod showed that the evaluation map $H^*_m(G\curvearrowright G/P)\longrightarrow H^*_m(G)$ between the measurable cohomology of the action of a connected semisimple Lie group $G$ on its Furstenberg boundary $G/P$ and the measurable cohomology of $G$ is surjective with a kernel that can be entirely described in terms of invariants in the cohomology of a maximal split torus $A<G$. In a recent paper we refine Monod's result and show in particular that the cohomology of non-alternating cocycles on $G/P$ is in general not trivial and lies in the kernel of the evaluation. In this paper we describe explicitly such non-alternating and alternating cocycles on $G/P$ in low degrees when $G$ is either a product of isometries of real hyperbolic spaces or $G=\mathrm{SL}(3,\mathbb{K})$, where $\mathbb{K}$ is either the real or the complex field. As a consequence, we deduce that the comparison map $H^*_{m,b}(G)\rightarrow H^*_m(G)$ from the measurable bounded cohomology is injective in degree $3$ for nontrivial products of isometries of hyperbolic spaces. We get also another proof of the injectivity for $G=\mathrm{SL}(3,\mathbb{K})$, when $\mathbb{K}$ is either the real field or the complex one.

Some explicit cocycles on the Furstenberg boundary for products of isometries of hyperbolic spaces and $\mathrm{SL}(3,\mathbb{K})$

TL;DR

The work advances explicit cocycle representatives for the kernel of the boundary evaluation map in measurable cohomology, focusing on products of real hyperbolic isometries and on SL(3, K). It uses Monod’s boundary spectral sequence to relate NH^k to invariants in and constructs explicit injections and sections at the cocycle level, employing π_A-projections and contracting homotopies. For products of hyperbolic isometries, it provides concrete non-alternating cocycles in degrees 3 and 4 expressed via cross ratios, proving injectivity of the bounded-to-measurable map in degree 3 and yielding a parallel result for SL(3, K) in degrees 2 and 3. The results give explicit G-invariant cocycles on the Furstenberg boundary and illuminate the structure of bounded cohomology in higher-rank contexts.

Abstract

Nicolas Monod showed that the evaluation map between the measurable cohomology of the action of a connected semisimple Lie group on its Furstenberg boundary and the measurable cohomology of is surjective with a kernel that can be entirely described in terms of invariants in the cohomology of a maximal split torus . In a recent paper we refine Monod's result and show in particular that the cohomology of non-alternating cocycles on is in general not trivial and lies in the kernel of the evaluation. In this paper we describe explicitly such non-alternating and alternating cocycles on in low degrees when is either a product of isometries of real hyperbolic spaces or , where is either the real or the complex field. As a consequence, we deduce that the comparison map from the measurable bounded cohomology is injective in degree for nontrivial products of isometries of hyperbolic spaces. We get also another proof of the injectivity for , when is either the real field or the complex one.
Paper Structure (12 sections, 17 theorems, 112 equations, 2 figures)

This paper contains 12 sections, 17 theorems, 112 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Products of isometries of real hyperbolic spaces
  3. The special linear group $\mathrm{SL}(3,\mathbb{K})$ for $\mathbb{K}=\mathbb{R},\mathbb{C}$
  4. (Un)bounded cohomology
  5. Monod's spectral sequence
  6. Explicit differentials
  7. Representing $H^p_m(A)^{w_0}$ in $C^{p,1}$ and $C^{p,2}$
  8. Homotopies for $(C_K^{*,q},d^\uparrow)$
  9. Explicit expression for the injection $i$ of the short exact sequence \ref{['SES']}
  10. Explicit expression for a section $s$ of the short exact sequence (\ref{['SES']})
  11. The spectral sequence for products of isometries of real hyperbolic spaces
  12. Isometries of $\mathbb{H}^n$

Key Result

Theorem 1

Monod Let $G$ be a connected semisimple Lie group with finite center. The evaluation map is surjective and its kernel fits into an exact sequence when $k\geqslant 3$, and for $k =2$, we have an isomorphism

Figures (2)

  • Figure 1: The first page $E_1$
  • Figure 2: The second page $E_2$

Theorems & Definitions (34)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark
  • Theorem 4
  • Remark
  • Remark
  • Theorem 5
  • Remark
  • Conjecture 6
  • ...and 24 more