Rigidity properties of free-by-cyclic groups

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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.

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.