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Separable homology of graphs and the separability complex

Becky Eastham

TL;DR

The paper introduces the separability complex $\mathrm{SC}(\Gamma)$ for finite regular covers of the rose $\mathbf{R}_n$ and proves a connectivity criterion: $\mathrm{SC}(\Gamma)$ is connected iff $\pi_1(\Gamma)$ is generated by $\pi_1(\Gamma) \cap \mathcal{C}^{\mathrm{sep}}$, where $\mathcal{C}^{\mathrm{sep}}$ denotes separable elements. It shows that $\mathrm{SC}(\mathbf{R}_n)$ has infinite diameter and is nonhyperbolic, leading to nonhyperbolicity of the Cayley graphs $\mathrm{Cay}(\mathbf{F}_n, \mathcal{C}^{\mathrm{sep}})$ and $\mathrm{Cay}(\mathbf{F}_n, \mathcal{C}^{\mathrm{prim}})$ and implying limitations on quasi-isometries with the free factor and free splitting complexes. The work develops a detailed combinatorial framework based on Van Kampen diagrams and Whitehead’s algorithm to analyze separability, and it introduces a homology version $\mathrm{HSC}_1(\Gamma)$ whose connectedness corresponds to the generation of $H_1(\Gamma;\mathbb{Z})$ by separable cycles; finitely many components follow from ongoing work of Boggi–Putman–Salter and pant-curve results. Overall, the paper provides new coarse-geometric invariants for free-group covers, connects separability to homology generation, and yields nonhyperbolicity results that distinguish these complexes from classical $Out(\mathbf{F}_n)$-complexes.

Abstract

We introduce the separability complex, a one-complex associated to a finite regular cover of the rose and show that it is connected if and only if the fundamental group of the associated cover is generated by its intersection with the set of elements in proper free factors of $\mathbf{F}_n$. The separability complex admits an action of $\mathrm{Out}(\mathbf{F}_n)$ by isometries if the associated cover corresponds to a characteristic subgroup of $\mathbf{F}_n$. We prove that the separability complex of the rose has infinite diameter and is nonhyperbolic, implying it is not quasi-isometric to the free splitting complex or the free factor complex. As a consequence, we obtain that the Cayley graph of $\mathbf{F}_n$ with generating set consisting of all primitive elements of $\mathbf{F}_n$ is nonhyperbolic.

Separable homology of graphs and the separability complex

TL;DR

The paper introduces the separability complex for finite regular covers of the rose and proves a connectivity criterion: is connected iff is generated by , where denotes separable elements. It shows that has infinite diameter and is nonhyperbolic, leading to nonhyperbolicity of the Cayley graphs and and implying limitations on quasi-isometries with the free factor and free splitting complexes. The work develops a detailed combinatorial framework based on Van Kampen diagrams and Whitehead’s algorithm to analyze separability, and it introduces a homology version whose connectedness corresponds to the generation of by separable cycles; finitely many components follow from ongoing work of Boggi–Putman–Salter and pant-curve results. Overall, the paper provides new coarse-geometric invariants for free-group covers, connects separability to homology generation, and yields nonhyperbolicity results that distinguish these complexes from classical -complexes.

Abstract

We introduce the separability complex, a one-complex associated to a finite regular cover of the rose and show that it is connected if and only if the fundamental group of the associated cover is generated by its intersection with the set of elements in proper free factors of . The separability complex admits an action of by isometries if the associated cover corresponds to a characteristic subgroup of . We prove that the separability complex of the rose has infinite diameter and is nonhyperbolic, implying it is not quasi-isometric to the free splitting complex or the free factor complex. As a consequence, we obtain that the Cayley graph of with generating set consisting of all primitive elements of is nonhyperbolic.
Paper Structure (14 sections, 35 theorems, 119 equations, 18 figures)

This paper contains 14 sections, 35 theorems, 119 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Graphs and Free Groups
  4. Van Kampen diagrams
  5. Whitehead's algorithm for separable words
  6. The separability complex
  7. Diameter of the separability complex
  8. Nonhyperbolicity of the separability complex
  9. Step 1: Properties of $\mathbf{r_i(n)}$
  10. Step 2: Separable length of $\mathbf{w(a, k, s, n)}$
  11. Step 3: Computing $\mathbf{d_{\mathrm{sep}}([w(a, k, s, n)], [w(b, k, s, n)])}$.
  12. Step 4: Fat triangles.
  13. Homology version of the separability complex
  14. Appendix

Key Result

Proposition 1.1

Given a finite-index normal subgroup $N$ of $\mathbf{F}_n$, let $\Gamma$ be the finite regular cover of $\mathbf{R}_n$ corresponding to $N.$ Then $\mathrm{SC}(\Gamma)$ is connected if and only if $N$ is generated by $N\cap\mathcal{C}^{\mathrm{sep}}$.

Figures (18)

  • Figure 1: A Folding
  • Figure 2: Folding to obtain a core graph for $G=\langle a, b\mid aba^{-1}, [a, b]\rangle$
  • Figure 3: Three annular diagrams with inner boundary $w_1$ and outer boundary $w_2$. Note that in the middle and right diagrams, the inner and outer boundaries overlap in certain edges and vertices.
  • Figure 4: The Whitehead graph of $ab^2c^{-1}dab^{-1}d^3(c^2a)^{-1}\in \mathbf{F}_4$.
  • Figure 5: On the left is the Whitehead graph for $w=ab^{-1}cba^{-1}\,c^{-1}a$; on the right, the Whitehead graph for $\phi(w)=ab^{-1}cbc^{-1}$.
  • ...and 13 more figures

Theorems & Definitions (103)

  • Proposition 1.1
  • Theorem 1.2
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.3
  • Proposition 1.5
  • Proposition 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 93 more