4. The Atiyah Conjecture

The deepfall property and the Atiyah Conjecture
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 leftorderable 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 $\sigma$ falls into $E$ if $\partial\sigma\in \overline{\partial(\ell^2(E))}$. A cell $\sigma\in \Sigma_i^{\hat{Y}}$ is called orderessential 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 orderinessential otherwise. Let $\mathbb{E}^{\hat{Y}}_i$ and $\mathbb{I}^{\hat{Y}}_i$ denote the sets of orderessential and orderinessential edges in $\hat{Y}$, respectively. We say that the $\Gamma$action on $\hat{Y}$ has the deepfall property, or more precisely $i$deepfall 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 leftorderable group $\Gamma$ on a complex with the deepfall 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 proposeProblem 4.1.
[Igor Mineyev] Find many examples of leftorderable groups and their free actions on complexes that have the deepfall property in some dimension $i$. Find free actions by left orderable groups that do not have the deepfall property. See what can be said about nonfree actions on ordered complexes. The same will formally apply to matrices with entries in the group ring over $\mathbb{C}$.
Cite this as: AimPL: $L^2$ invariants for finitely generated groups, available at http://aimpl.org/l2invariantsgroups.