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Bounded Cohomology and Unitary Representations of Automorphism Groups of Regular Trees

Cunyuan Zhao

TL;DR

This work determines the full spectrum of continuous bounded cohomology $ ext{H}_ ext{cb}^n( ext{Aut}(T),\, rak H_ ho)$ for all irreducible unitary representations $ ho$ of the automorphism group of a regular tree $T$. The authors classify representations into spherical, special, and cuspidal, and show a precise vanishing pattern: $ ext{H}_ ext{cb}^n(G, rak H_ ho)=0$ for all $n eq 2$; for $n=2$, nontrivial cohomology occurs only for a countable family of cuspidal representations associated to centipede or spider subtrees, each giving a one-dimensional class. They construct explicit nontrivial 2-cocycles from centipede/spider data and connect these to Monod–Shalom cocycles, providing a complete and explicit account of the 2nd cohomology while establishing universal vanishing in higher degrees. The results deepen the understanding of bounded cohomology with nontrivial coefficients for groups acting on trees and yield concrete geometric and representation-theoretic criteria for nonvanishing, with potential applications to rigidity phenomena and orbit equivalence in tree-related contexts.

Abstract

We compute the continuous bounded cohomology of the full automorphism groups of regular trees in all positive degrees, with coefficients arising from any irreducible continuous unitary representations. To the author's knowledge, this seems to be the first instance where the continuous bounded cohomology is determined in all positive degrees with coefficients arising from any irreducible continuous unitary representations without being zero in all cases.

Bounded Cohomology and Unitary Representations of Automorphism Groups of Regular Trees

TL;DR

This work determines the full spectrum of continuous bounded cohomology for all irreducible unitary representations of the automorphism group of a regular tree . The authors classify representations into spherical, special, and cuspidal, and show a precise vanishing pattern: for all ; for , nontrivial cohomology occurs only for a countable family of cuspidal representations associated to centipede or spider subtrees, each giving a one-dimensional class. They construct explicit nontrivial 2-cocycles from centipede/spider data and connect these to Monod–Shalom cocycles, providing a complete and explicit account of the 2nd cohomology while establishing universal vanishing in higher degrees. The results deepen the understanding of bounded cohomology with nontrivial coefficients for groups acting on trees and yield concrete geometric and representation-theoretic criteria for nonvanishing, with potential applications to rigidity phenomena and orbit equivalence in tree-related contexts.

Abstract

We compute the continuous bounded cohomology of the full automorphism groups of regular trees in all positive degrees, with coefficients arising from any irreducible continuous unitary representations. To the author's knowledge, this seems to be the first instance where the continuous bounded cohomology is determined in all positive degrees with coefficients arising from any irreducible continuous unitary representations without being zero in all cases.

Paper Structure

This paper contains 20 sections, 41 theorems, 121 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries on Trees
  3. Unitary Representations of $\mathrm{Aut}(T)$
  4. Unitary Spherical Representations and Smooth Vectors
  5. Unitary Spherical Representations
  6. An Interlude on Smooth Vectors
  7. Unitary Special Representations
  8. Induced Representations and Unitary Cuspidal Representations
  9. Continuous Bounded Cohomology
  10. Almost Everywhere Identities
  11. Proof of Theorem \ref{['thm: A']} and \ref{['thm: B']}
  12. The Flipping Lemma
  13. The Spherical Cases
  14. The Cuspidal Cases
  15. Vanishing of Bounded Cohomology in Higher Degrees
  16. ...and 5 more sections

Key Result

Theorem A

Let $(\pi,\mathcal{H}_\pi)\in \widehat{G}$ be a unitary irreducible representation of $G$. If $\mathcal{M}_\pi$ does not consist of centipedes or spiders, then $\mathrm{H}_\mathrm{cb}^n(G, \mathcal{H}_\pi)=0$ for all $n\geq 1$.

Figures (3)

  • Figure 1: A $4$-centipede in a $4$-regular tree (left) and a spider in an $8$-regular tree (right). The two heads of the $4$-centipede are indicated by thick red segments.
  • Figure 2: The element $h_{35}\in G(S)$ flips $\gamma_3$ and $\gamma_5$ and stabilizes all other $\gamma_k$'s.
  • Figure 3: A $4$-centipede (thick red) that lies on $L(\gamma_0,\gamma_1)$ in the $4$-regular tree.

Theorems & Definitions (90)

  • Definition 1.1
  • Definition 1.2
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 80 more