The deep-fall property and the Atiyah Conjecture

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Find many examples of left-orderable groups and their free actions on complexes that have the deep-fall property in some dimension $i$. Find free actions by left orderable groups that do not have the deep-fall property.  See what can be said about non-free actions on ordered complexes. The same will formally apply to matrices with entries in the group ring over $\mathbb{C}$.

## Workshop introduction

This problem list concerns the study of the asymptotic behavior of some natural invariants of finitely generated groups.

## Question context

This notion was used in the paper ``Sumultiplicativity and the Hanna Neumann Conjecture". 
Let $\hat{Y}$ be a complex with a free action by a left-orderable group~$\Gamma$ and $i\ge 0$.
This induces a $\Gamma$-invariant total order on the set of $i$-cells in $\hat{Y}$, $\Sigma_i^{\hat{Y}}$.
For  $\sigma\in \Sigma_i^{\hat{Y}}$ and $E\subseteq \Sigma_i^{\hat{Y}}\setminus \{\sigma\}$,
let 
$$
[E<\sigma] := \{ \tau\in E |\tau<\sigma \} .
$$
We say that $\sigma$ {\em falls into} $E$ if 
$\partial\sigma\in \overline{\partial(\ell^2(E))}$.
A cell $\sigma\in \Sigma_i^{\hat{Y}}$ is called {\bf order-essential} if it falls into 
$[\Sigma^{\hat{Y}}_i< \sigma]$, i.e. 
$$\partial\sigma\in 
\overline{\partial(\ell^2[\Sigma^{\hat{Y}}_i< \sigma])}.$$ Call it {\bf order-inessential} otherwise.
Let $\mathbb{E}^{\hat{Y}}_i$ and $\mathbb{I}^{\hat{Y}}_i$ denote the sets of order-essential and order-inessential edges in $\hat{Y}$, respectively. 
We say that the $\Gamma$-action on $\hat{Y}$ has the {\bf deep-fall property}, or more precisely  $i$-deep-fall property, if for any 
$\sigma\in \mathbb{E}^{\hat{Y}}_i$ we have $\partial\sigma\in 
\overline{\partial(\ell^2[\mathbb{I}^{\hat{Y}}_i< \sigma])}$.

The argument in ``Sumbmultiplicativity ..." implies that each free action of a left-orderable group $\Gamma$ on a complex with the deep-fall property for some $i$ provides an instance when the (integral) Atiyah Conjecture holds. (Represent the $i$th boundary map by a matrix with entries in $\mathbb{Z}\Gamma$ and see that the kernel has integral dimension.) So for future investigation we propose

Historical workshop record. Current open/solved status has not been independently reassessed.

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