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Invariance of non-vanishing of first $l^p$-cohomology under $L^q$-Measured Equivalence

Kajal Das

TL;DR

The paper proves that for non-amenable groups, the non-vanishing of the first $l^p$-cohomology $l^pH^1$ is preserved under $L^q$-Measured Equivalence when $q\ge p>1$, with a weaker statement for $p=1$. The approach relies on inducing representations and cocycles across the $L^q$-ME coupling, producing a preserved affine-action framework, and transferring nontrivial cohomology from one group to another via induced representations and a fixed-point/scalar-argument. As applications, the authors derive conformal-dimension invariance for hyperbolic groups with CLP under suitable $L^q$-ME, reprove non-$L^1$-ME for free vs. surface groups, and obtain rigidity results for 3-manifold groups across Thurston geometries. The work also establishes a foundational link between direct-integral Banach-space cohomology and fiberwise cohomology, enabling the main transfer arguments and suggesting avenues for higher-degree cohomology questions.

Abstract

The first $l^p$-cohomology is an algebro-analytical object attached to a finitely generated discrete group and introduced by M. Gromov. It is well known that it is invariant under quasi-isometry. In this article, we prove that the non-vanishing of the first $l^p$-cohomology of a non-amenable group is invariant under $L^q$-Measured Equiavalence (an equivalence relation introduced by Gromov), where $q\geq p$. We also discuss many applications of this result. We prove that for hyperbolic (in the sense of Gromov) Coxeter groups with boundaries having Combinatorial Loewner Property, conformal dimension (of the canonical conformal gauge) of the Gromov boundary is invariant under $L^q$-Measured Equivalence for some large $q$. We prove that the finitely generated free groups and surface groups are not $L^1$-Measured Equivalent. We also give a lower bound of the critical exponent for the first $l^p$-cohomology of any lattice in $SO(n,1)$. Finally, we discuss $L^q$-Measured Equivalence between non-amenable 3-manifold groups corresponding to Thurston's three geometries $\mathbb{H}^3$, $\mathbb{H}^2\times\mathbb{R}$ and $\widetilde{SL_2(\mathbb{R})}$.

Invariance of non-vanishing of first $l^p$-cohomology under $L^q$-Measured Equivalence

TL;DR

The paper proves that for non-amenable groups, the non-vanishing of the first -cohomology is preserved under -Measured Equivalence when , with a weaker statement for . The approach relies on inducing representations and cocycles across the -ME coupling, producing a preserved affine-action framework, and transferring nontrivial cohomology from one group to another via induced representations and a fixed-point/scalar-argument. As applications, the authors derive conformal-dimension invariance for hyperbolic groups with CLP under suitable -ME, reprove non--ME for free vs. surface groups, and obtain rigidity results for 3-manifold groups across Thurston geometries. The work also establishes a foundational link between direct-integral Banach-space cohomology and fiberwise cohomology, enabling the main transfer arguments and suggesting avenues for higher-degree cohomology questions.

Abstract

The first -cohomology is an algebro-analytical object attached to a finitely generated discrete group and introduced by M. Gromov. It is well known that it is invariant under quasi-isometry. In this article, we prove that the non-vanishing of the first -cohomology of a non-amenable group is invariant under -Measured Equiavalence (an equivalence relation introduced by Gromov), where . We also discuss many applications of this result. We prove that for hyperbolic (in the sense of Gromov) Coxeter groups with boundaries having Combinatorial Loewner Property, conformal dimension (of the canonical conformal gauge) of the Gromov boundary is invariant under -Measured Equivalence for some large . We prove that the finitely generated free groups and surface groups are not -Measured Equivalent. We also give a lower bound of the critical exponent for the first -cohomology of any lattice in . Finally, we discuss -Measured Equivalence between non-amenable 3-manifold groups corresponding to Thurston's three geometries , and .

Paper Structure

This paper contains 20 sections, 17 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Statement of the main theorem and the corollaries
  3. Acknowledgements
  4. $L^p$-Measure Equivalence
  5. The first $l^p$-cohomology of groups
  6. Definition of $l^p$-cohomology
  7. List of some known results of first $l^p$-cohomology of discrete groups:
  8. 1-cohomology with coeffcients in a Banach Space
  9. Induced representation and Induced affine action
  10. Induced representation
  11. Induced cocycle in the induced representation
  12. Induced representation from right-regular action
  13. Induced affine action
  14. Proof of the main theorem
  15. Applications
  16. ...and 5 more sections

Key Result

Theorem 1.1

(Main Theorem) Suppose two non-amenable groups $\Gamma$ and $\Lambda$ are $L^q$-ME for some $q\geq 1$. Then $l^pH^1(\Gamma)\neq 0$ if and only if $l^pH^1(\Lambda)\neq 0$, when $1< p\leq q$. If $p=1$, then $l^pH^1(\Lambda)=0$ implies that all affine actions associated to $l^pH^1(\Gamma)$ have bounded

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Remark 2.1
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • ...and 18 more