[Paper Review] The $\Sigma$-invariants of Thompson's group $F$ via Morse theory

Stefan Witzel, Matthew C. B. Zaremsky|arXiv (Cornell University)|Jan 27, 2015

Geometric and Algebraic Topology8 references3 citations

TL;DR

This paper re-computes the BNSR-invariants $Σ^m(F)$ of Thompson's group $F$ using geometric Morse theory on the Stein–Farley $Σ(0)$ cube complex $X$, providing a self-contained, topological proof that avoids algebraic techniques. The key result confirms that $[χ] \in \Sigma^1(F)$ unless $\chi$ is a positive multiple of $\chi_0$ or $\chi_1$, and $[\chi] \in \Sigma^2(F) = \Sigma^\infty(F)$ unless $\chi$ lies in the non-negative quadrant of $\operatorname{Hom}(F,\mathbb{R}) \cong \mathbb{R}^2$. The approach relies on analyzing ascending links and superlevel sets of height functions induced by characters.

ABSTRACT

Bieri, Geoghegan and Kochloukova computed the BNSR-invariants $\\Sigma^m(F)$ of Thompson's group $F$ for all $m$. We recompute these using entirely geometric techniques, making use of the Stein--Farley CAT(0) cube complex on which $F$ acts.

Motivation & Objective

• To provide a geometric, self-contained proof of the BNSR-invariants $\Sigma^m(F)$ for Thompson's group $F$, independent of algebraic methods used in prior work.
• To apply Bestvina–Brady Morse theory to the Stein–Farley $\operatorname{CAT}(0)$ cube complex $X$ on which $F$ acts freely by isometries.
• To characterize $\Sigma^m(F)$ by analyzing the homotopy types of ascending links and superlevel sets of character height functions on $X$.
• To demonstrate that the geometric approach is viable for computing invariants of other groups with similar geometric structures.

Proposed method

• Use the Stein–Farley $\operatorname{CAT}(0)$ cube complex $X$ as the geometric model for $F$'s action, leveraging its proper, cocompact, and isometric properties.
• Define character height functions $h_\chi: X \to \mathbb{R}$ such that $h_\chi(gx) = \chi(g) + h_\chi(x)$ for all $g \in F$, $x \in X$, using the abelianization $\operatorname{Hom}(F,\mathbb{R}) \cong \mathbb{R}^2$.
• Apply a version of Bestvina–Brady Morse theory to the $F$-action on $X$, focusing on the homotopy type of ascending links at vertices and superlevel sets $X^{t \leq \chi}$.
• Model vertex links combinatorially using the structure of reduced tree-pair representatives $(T/E)$ of group elements in $F$.
• Use the Morse Lemma and connectivity criteria to analyze the essential connectivity of filtrations $X^{t \leq \chi}$, particularly for $t$ near zero.
• Construct explicit cycles in the nerve of a cover of $X^{0 \leq \chi}$ by subcomplexes $Y_{L=i}^m$ and $Y_{R=i}^m$ to show non-trivial fundamental group when $a > 0$, $b > 0$.

Experimental results

Research questions

• RQ1 Can the BNSR-invariants $\Sigma^m(F)$ be re-derived using only geometric and topological tools, without relying on algebraic properties like HNN-structure or absence of free subgroups?
• RQ2 What is the homotopy type of the ascending links and superlevel sets of height functions on the Stein–Farley $\operatorname{CAT}(0)$ cube complex $X$?
• RQ3 How does the geometry of the $F$-action on $X$ reflect the finiteness properties of $F$'s normal subgroups via the BNSR-invariants?
• RQ4 Can the nerve of a cover of $X^{0 \leq \chi}$ by $Y_{L=i}^m$ and $Y_{R=i}^m$ components detect non-trivial fundamental group in the superlevel set?
• RQ5 What conditions on the character $\chi = a\chi_0 + b\chi_1$ determine whether $[\chi] \in \Sigma^1(F)$ or $\Sigma^2(F)$, and how is this reflected in the geometry of $X$?

Key findings

• The invariant $\Sigma^1(F)$ consists of all $[\chi] \in S^1$ except those in the positive directions of $\chi_0$ and $\chi_1$, i.e., $[\chi] \notin \Sigma^1(F)$ iff $\chi$ is a positive multiple of $\chi_0$ or $\chi_1$.
• The invariant $\Sigma^2(F) = \Sigma^\infty(F)$ consists of all $[\chi] \in S^1$ such that $\chi = a\chi_0 + b\chi_1$ with $a \geq 0$, $b \geq 0$, and $[\chi] \notin \Sigma^2(F)$ iff $a > 0$, $b > 0$.
• For $a > 0$, $b > 0$, the superlevel set $X^{0 \leq \chi}$ is not simply connected, as shown by constructing a non-trivial cycle in the nerve of the cover by $Y_{L=i}^m$ and $Y_{R=i}^m$ components.
• The filtration $(X^{t \leq \chi})_{t \in \mathbb{R}}$ is essentially connected for all $\chi$, but not essentially simply connected when $a > 0$, $b > 0$, confirming $[\chi] \notin \Sigma^2(F)$ in this case.
• The ascending links of vertices in $X$ are shown to be contractible or homotopy-equivalent to a point when $\chi$ is not aligned with $\chi_0$ or $\chi_1$, supporting the connectivity of the level sets.
• The proof establishes that $\Sigma^1(F) \setminus \Sigma^2(F)$ consists of characters $\chi = a\chi_0 + b\chi_1$ with $a > 0$, $b > 0$, confirming the geometric origin of the invariant boundary.