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Sublinear growth of 1-cocycles and uniform convexity

Andreas Thom

Abstract

Let G be a finitely generated group, let $π\colon G \to {\rm GL}(E)$ be a uniformly bounded $c_0$-representation on a superreflexive Banach space $E$, and let $b \colon G \to E$ be a $1$-cocycle for $π$. Then $b$ has sublinear growth with respect to the word length. As a corollary we obtain the corresponding Hilbert space statement for strongly mixing unitary representations.

Sublinear growth of 1-cocycles and uniform convexity

Abstract

Let G be a finitely generated group, let be a uniformly bounded -representation on a superreflexive Banach space , and let be a -cocycle for . Then has sublinear growth with respect to the word length. As a corollary we obtain the corresponding Hilbert space statement for strongly mixing unitary representations.
Paper Structure (4 sections, 4 theorems, 45 equations)

This paper contains 4 sections, 4 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Proof of the main theorem
  3. The setting of Hilbert spaces
  4. Vanishing of reduced cohomology for amenable groups

Key Result

Theorem 1

Let $G$ be a finitely generated group, $\mathcal{E}$ be a superreflexive Banach space, $\pi\colon G\to {\rm GL}(\mathcal{E})$ be a uniformly bounded $c_{0}$-representation, and $b\colon G\to \mathcal{E}$ be a $1$-cocycle. Then Equivalently, every such $1$-cocycle has sublinear growth.

Theorems & Definitions (9)

  • Theorem 1
  • proof : Proof of Theorem \ref{['thm:main']}
  • Corollary 2
  • proof
  • Proposition 3
  • proof
  • Remark 4
  • Corollary 5
  • Remark 6