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6. BNS invariants

Let $G$ be a group and $X$ a Cayley graph of $G$ with respect to some finite generating set. Each character $\chi \colon G \to \mathbb{R}$ linearly extends to a $G$-equivariant map $\widetilde{\chi} \colon X \to \mathbb{R}.$ We define $X_{\chi}^{+}$ to be maximal subgraph of $X$ contained in $\widetilde{\chi}^{-1}([0, \infty))$.

The character $\chi \colon G \to \mathbb{R}$ is contained in the Bieri–Neumann–Strebel (BNS) invariant of $G$ if and only if $X_{\chi}^{+}$ is connected.


    1. Add remark

      Problem 6.1.

      [Sam Taylor] Let $G$ be a free-by-cyclic group and $F$ a component of the BNS invariant of $G$. When is it the case that the stretch factor functions $\lambda_{+}, \lambda_{-} \colon F \to \mathbb{R}$ agree on a convex subset of $F$?


        • Add remark

          Problem 6.2.

          Find families of examples of free-by-cyclic groups $G$ such that $b_1(G) > 2$ and the BNS invariant of $G$ has many components (up to the action of $\mathrm{Out}(G)$).


            • Add remark

              Problem 6.3.

              [Rylee Lyman] If $P$ is a polytope in $\mathbb{R}^n$ with $n \geq 2$ with a marking on its top dimensional faces, determine whether the cone over the marked faces of $P$ corresponds to a BNS invariant of a (free-by-cyclic) group.

                  Cite this as: AimPL: Rigidity properties of free-by-cyclic groups, available at http://aimpl.org/freebycyclic.