Subtle point in definition of BNS invariant

Question

Let $G$ be a finitely generated group. Let $S(G)$ be the quotient of $\text{Hom}(G,\mathbb{R}) \setminus \{0\}$ by the equivalence relation that identifies two homomorphisms if they differ by scaling by a positive constant. For a nonzero $\phi \in \text{Hom}(G,\mathbb{R})$, write $[\phi] \in S(G)$ for the associated equivalence class.

The Bieri-Neumann-Strebel (BNS) invariant of $G$ is a certain subset of $S(G)$. The definition in the original paper is hard to parse, but in several places I have seen the following definition of it:

Fix a generating set $S$ of $G$. Let $\text{Cay}(G,S)$ be the Cayley graph of $G$ with respect to $S$. Consider a nonzero $\phi \in \text{Hom}(G,\mathbb{R})$. Define $X_{\phi} = \{\text{$g \in G$ $|$ $\phi(g) \geq 0$}\}$. Then $[\phi] \in S(G)$ is in the BNS invariant if and only if the full subgraph of $\text{Cay}(G,S)$ spanned by $X_{\phi}$ is connected.

I have seen it asserted without proof or reference that the above property does not depend on the choice of generating set $S$.

Question 1: Can someone give me either a proof or reference for this?

Question 2: Can someone explain how to relate this to the definition in the original paper defining the BNS invariant? A reference here would also be fine.

Answer 1

Let $S$ and $T$ be generating sets for $G$, and suppose that $X_\phi$ is $T$-connected (i.e., spans a connected subgraph in the Cayley graph of $G$ with respect to $T$.) Let $[X_\phi]_S$ be the subgraph of $\text{Cay}(G,S)$ spanned by $X_\phi$. We must show that $[X_\phi]_S$ is connected.

Claim 1: For any $n\in\mathbb{Z}$, the set $\{g\in G:\phi(g)\geq n\}$ is $T$-connected.

Proof: This follows, by translation, from the fact that $X_\phi$ is $T$-connected.

Claim 2: There exists some $n$ such that for any $g\in G$ and $t\in T$, there is a path in $\text{Cay}(G,S)$ from $g$ to $gt$ such that if $v$ is a vertex of this path, then $\phi(v)>\phi(g)-n$ and $\phi(v)>\phi(gt)-n$.

Proof: For each $t\in T$, choose some word $w_t$ in the alphabet $S$ representing $t$. Choose $n$ so that it is larger than $|\phi(w)|$ whenever $w$ is a prefix or suffix of any $w_t$. The claim follows by connecting $g$ and $gt$ via the path given by $g \cdot w_t$.

If $n$ is as in Claim 2, it follows that any two points of $\{g\in G:\phi(g)\geq n\}$ may be connected by a path in $[X_\phi]_S$.

Claim 3: there is some $s \in S \cup S^{-1}$ such that $\phi(s) > 0$.

Proof: this follows from the fact that $\phi$ is not identically $0$, by definition.

By Claim 3 we see that, for any $g \in X_\phi$, there is a path $g,gs,gs^2,\ldots$ in $[X_\phi]_S$ from $g$ to $\{g\in G:\phi(g)\geq n\}$. Since any two vertices of $\{g\in G:\phi(g)\geq n\}$ may be connected by a path in $[X_\phi]_S$, it follows that $[X_\phi]_S$ is connected.

Answer 2

This is a reference of question 2.

Various definitions (including the original one and the one you have given) of BNS invariant are discussed in Chapter C of this lecture notes by Ralph Strebel who call the BNS invariant as "Sigma invariants".