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On growth of cocycles of isometric representations on $L^p$-spaces

Antonio López Neumann, Juan Paucar

Abstract

We study different notions of asymptotic growth for 1-cocycles of isometric representations on Banach spaces. One can see this as a way of quantifying the absence of fixed point properties on such spaces. Inspired by the work of Lafforgue, we show the following dichotomy: for a compactly generated group $G$, either all 1-cocycles of $G$ taking values in $L^p$-spaces are bounded (this is Property $FL^p$) or there exists a 1-cocycle of $G$ taking values in an $L^p$-space with relatively fast growth. We also obtain upper and lower bounds on the average growth of harmonic 1-cocycles with values in Banach spaces with convexity properties. As a consequence, we obtain bounds on the average growth of all 1-cocycles with values in $L^p$-spaces for groups with property $(T)$. Lastly, we show that for a unimodular compactly generated group $G$, the existence of a 1-cocycle with compression larger than $\sqrt{n}$ implies the Liouville property for a large family of probability measures on $G$.

On growth of cocycles of isometric representations on $L^p$-spaces

Abstract

We study different notions of asymptotic growth for 1-cocycles of isometric representations on Banach spaces. One can see this as a way of quantifying the absence of fixed point properties on such spaces. Inspired by the work of Lafforgue, we show the following dichotomy: for a compactly generated group , either all 1-cocycles of taking values in -spaces are bounded (this is Property ) or there exists a 1-cocycle of taking values in an -space with relatively fast growth. We also obtain upper and lower bounds on the average growth of harmonic 1-cocycles with values in Banach spaces with convexity properties. As a consequence, we obtain bounds on the average growth of all 1-cocycles with values in -spaces for groups with property . Lastly, we show that for a unimodular compactly generated group , the existence of a 1-cocycle with compression larger than implies the Liouville property for a large family of probability measures on .
Paper Structure (21 sections, 25 theorems, 97 equations, 1 figure)

This paper contains 21 sections, 25 theorems, 97 equations, 1 figure.

Table of Contents

  1. An $L^p$-analogue of a theorem by Lafforgue
  2. Harmonic cocycles and average growth
  3. Drift and equivariant compression
  4. Outline of the paper
  5. Acknowledgements
  6. Preliminaries
  7. Uniformly convex and uniformly smooth metric spaces
  8. Duality mappings in smooth Banach spaces
  9. Isometric representations on Banach spaces and 1-cohomology
  10. Cohomologically adapted probability measures
  11. Growth of cocycles in $L^\infty$-norm
  12. Ultrafilters and ultraproducts of representations
  13. Ultraproducts of isometric actions
  14. Growth of harmonic cocycles in $L^p$-norm
  15. Harmonicity and spectral gap
  16. ...and 6 more sections

Key Result

Theorem 1

lafforgue-typeneg Let $G$ be a locally compact, compactly generated group with compact generating set $S$ and without property $(T)$. Then there exists a Hilbert space $\mathcal{H}$, a unitary representation $\pi : G \to \mathcal{U} (\mathcal{H})$ and a 1-cocycle $b \in Z^1(G, \pi)$ such that for al

Figures (1)

  • Figure :

Theorems & Definitions (55)

  • Theorem
  • Definition 1
  • Theorem A
  • Remark
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • ...and 45 more