An obstruction to $\ell^p$-dimension

TL;DR

The paper demonstrates an obstruction to defining a consistent $ ext{ell}^p$-dimension in certain group settings, providing a negative answer to a question posed by Gaboriau.

Contribution

It constructs specific invariant subspaces in $ ext{ell}^p$ spaces over groups with elementary amenable subgroups, showing limitations of $ ext{ell}^p$-dimension theory.

Findings

Existence of invariant subspaces with trivial intersections

Obstruction to $ ext{ell}^p$-dimension in certain groups

Negative answer to Gaboriau's question

Abstract

For any group $G$ containing an infinite elementary amenable subgroup, and any $2<p<\infty$, there exists closed invariant subspaces $E_i\nearrow \ell^pG$ and $F\neq 0$ such that $E_i\cap F = 0$ for all $i$. This is an obstacle to $\ell^p$-dimension and gives a negative answer to a question of Gaboriau.