Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Inadmissible representations of the tree automorphism group

Nicolas Monod

Abstract

The automorphism group of a regular locally finite tree is shown to admit irreducible Banach representations that are not admissible. The dense subspace of smooth vectors contains no algebraically irreducible component.

Inadmissible representations of the tree automorphism group

Abstract

The automorphism group of a regular locally finite tree is shown to admit irreducible Banach representations that are not admissible. The dense subspace of smooth vectors contains no algebraically irreducible component.
Paper Structure (8 sections, 5 theorems, 23 equations, 3 figures)

This paper contains 8 sections, 5 theorems, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Tree groups
  4. Proofs
  5. The tree
  6. The representations
  7. Fixed vectors
  8. Invariant subspaces

Key Result

Theorem 1

Let $G$ be the automorphism group of a regular tree of finite valency ${\geq 3}$. There exists topologically irreducible continuous $G$-representations on a Banach space $V$ that are not admissible. Moreover, the subspace ${V}_{\mathrm{sm}}$ of smooth vector does not contain any non-zero algebraical

Figures (3)

  • Figure 1: The values of the Busemann kernel depending on $\xi$, in the case of adjacent vertices $x,y$. Pictured for $q=2$.
  • Figure 2: The complete trees $S$ and $S'$, and the boundary orbits $A_1, \ldots, A_n$, with $q=3$. Only one $A_i$ is pictured for $q<i<n$.
  • Figure 3: The domain of $\varphi$ has two horizontal cuts; it contains $\mathbf{D}_{2q^{1/2}}$ but not $\pm (q+1)$.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • proof : Proof of \ref{['prop:special-or-spherical']}
  • Remark 2
  • Proposition 3
  • proof
  • Theorem 4
  • proof : Proof of \ref{['thm:irred']}
  • proof : Proof of \ref{['ithm:inad']}
  • ...and 1 more