BNS Invariants and First Resonance Varieties

Definitions and mathematical statement

Let $G$ be finitely generated, and identify $H^1(G;{\mathbb{R}})$ with $\operatorname{Hom}(G,{\mathbb{R}})$. Its character sphere is $S(G)=(H^1(G;{\mathbb{R}})\setminus\{0\})/{\mathbb{R}}_{>0}$. For a nonzero character $\chi$, the Bieri--Neumann--Strebel invariant contains $[\chi]$ when the full subgraph on vertices $\{g\in G:\chi(g)\geq0\}$ of a Cayley graph for a finite generating set is connected. To compare with a subset of cohomology, use its conical lift \[\widetilde\Sigma^1(G)=\{\chi\ne0:[\chi]\in\Sigma^1(G)\}\subseteq H^1(G;{\mathbb{R}}).\] Define the real first resonance variety by \[\mathcal R_1^1(G;{\mathbb{R}})=\{a\in H^1(G;{\mathbb{R}}):\dim_{\mathbb{R}} H^1(H^*(G;{\mathbb{R}}),a\smile-)\geq1\}.\] The differential is multiplication by $a$ in the cohomology ring; its square is zero over ${\mathbb{R}}$. Equivalently, at degree one this cohomology is the kernel of $a\smile-:H^1\to H^2$ modulo the image of $a\smile-:H^0\to H^1$. If the source uses complex resonance, its real locus must be used for the displayed comparison. The source asks for the relationship between these invariants and, specifically, for conditions under which \[\widetilde\Sigma^1(G)\cup\left(-\widetilde\Sigma^1(G)\right)=H^1(G;{\mathbb{R}})\setminus\mathcal R_1^1(G;{\mathbb{R}}).\] This retains the source's union, not an intersection. The conical lift resolves its suppressed sphere-versus-vector-space convention; equivalently one can compare nonzero rays after projectivizing both sides. The zero class needs a convention when $H^1(G;{\mathbb{R}})=0$: with the displayed resonance definition the resonance set is empty, so the right side contains zero while the conical left side is empty. The literal equality and its nonzero-ray version then differ. No unprovided formality, finite-presentation, or finiteness-of-classifying-space assumption is imposed. The supplied 2026-08-21 triage labels the question open and reports Papadima--Suciu bounds involving BNSR invariants and homology jumping loci, with resonance descriptions under additional hypotheses, without a general characterization for all finitely generated groups.

Short English statement

Determine when the union of the BNS character cone and its negative is exactly the complement of first resonance. The sphere/cone convention, real resonance locus, and treatment of the zero character must be made explicit.

Sources

• Local UnsolvedMath registry, permanent ID 30000825, source problem catalogue entry, BNS Invariants and First Resonance Varieties, proposed year 2007; no individual proposer supplied.
• Supplied citation: 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 literature label: original Oberwolfach problem.
• Supplied literature label: Papadima--Suciu, https://arxiv.org/abs/0812.2660.