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4. The Atiyah Conjecture

    1. The deep-fall property and the Atiyah Conjecture

          This notion was used in the paper ‚ÄúSumultiplicativity 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 $\sigma$ falls into $E$ if $\partial\sigma\in \overline{\partial(\ell^2(E))}$. A cell $\sigma\in \Sigma_i^{\hat{Y}}$ is called 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 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 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

      Problem 4.1.

      [Igor Mineyev] 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 $\mathbb{C}$.

          Cite this as: AimPL: $L^2$ invariants for finitely generated groups, available at http://aimpl.org/l2invariantsgroups.