Polyhedrality of Novikov Invertibility Loci

Definitions and mathematical statement

Let $G$ be a finitely generated group, let $p\geq1$ be a matrix size, and fix $A\in M_p({\mathbb{Z}}[G])$. Here $p$ is not required to be prime. Put $V=\operatorname{Hom}(G,({\mathbb{R}},+))$. For $R={\mathbb{Z}}$ or ${\mathbb{Q}}$ and $0\ne\xi\in V$, one sign convention for the Novikov completion is \[ R[G]_\xi=\left\{\sum_{g\in G}a_g g: \text{for every }C\in{\mathbb{R}},\quad \#\{g:a_g\ne0,\ \xi(g)\leq C\}<\infty\right\}, \] with convolution multiplication. Using the opposite support convention replaces $\xi$ by $-\xi$; the report's sign convention is not given in the extraction.

Keep the coefficient choices separate by defining \[ \mathcal N_R(A)=\{0\ne\xi\in V: \exists B\in M_p(R[G]_\xi),\quad AB=BA=I_p\}. \] The question is whether these invertibility loci admit descriptions by finitely many integral linear inequalities on the character space. More explicitly, let $H=G_{\mathrm{ab}}/\text{torsion}$, a free abelian group of finite rank. An integral homogeneous linear form on $V=\operatorname{Hom}(H,{\mathbb{R}})$ is $\ell_h(\xi)=\xi(h)$ for $h\in H$. The requested inequalities would have such integral coefficients after choosing an integral basis of $H$.

The extraction does not say whether the permitted description is a single simultaneous system of inequalities, a finite union of such systems, or a finite Boolean combination, nor whether inequalities are strict or weak. These yield different polyhedrality claims. Thus the exact intended description must be recovered before promoting one of them as the conjecture. Likewise, "${\mathbb{Z}}[G]_\xi$ or ${\mathbb{Q}}[G]_\xi$" is retained as two coefficient versions, not silently replaced by equality of their loci or by their union. The sought finite description may depend on $G,A$, and the coefficient ring.

Short English statement

For a fixed group-ring matrix, is the set of characters making it invertible over a Novikov completion describable by finitely many integral inequalities? The coefficient ring and the permitted combinations of inequalities must be specified.

Sources

• Local UnsolvedMath registry, permanent ID $30000821$, source code catalogue entry, "Polyhedrality of Novikov Invertibility Loci." No individual proposer is supplied.
• Mini-Workshop: Topology of closed one-forms and Cohomology Jumping Loci (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/40, pp. 2321--2360: https://doi.org/10.4171/owr/2007/40.
• Supplied reference, "Novikov-question source": https://ems.press/content/serial-article-files/46128?nt=1.
• The literature triage checked 2026-08-21 reports only special cases such as left-orderable groups and labels the general question open. Exact hypotheses of those special cases and current status have not been independently checked.