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Combinatorial isoperimetric inequality for the free factor complex

Radhika Gupta

TL;DR

The paper proves that the free factor complex $\mathcal{F}$ of a free group with rank $n\ge 4$ does not admit a linear combinatorial isoperimetric inequality. By constructing a family of length-4 loops $c_N$ and a coarsely Lipschitz map from the upward link of a free factor to $\mathbb{Z}$, it is shown that any disc filling $c_N$ must contain at least $(N-2C)/3$ triangles with $C=2n-4$, giving a linear lower bound that grows with $N$. The argument hinges on a careful analysis of folding paths, an optimal morphism guiding a coarsely Lipschitz twist-counting function $\Psi_{w,b,B}$, and a reduction to a sufficiently connected upward link when the corank is at least 3. The rank-3 case is handled by passing to a quasi-isometric modification $\mathcal{F}_3'$ that triangulates certain loops, preserving the lack of a linear combinatorial isoperimetric bound. Together, these results align the free factor complex with other $Out(\mathbb{F})$-complexes that fail to satisfy combinatorial isoperimetric inequalities, while leaving open whether a cocompact, curve-like complex satisfying linear bounds exists for $Out(\mathbb{F})$.

Abstract

We show that the free factor complex of the free group of rank greater than or equal to 4 does not satisfy a combinatorial isoperimetric inequality: that is, for every natural number N, there is a loop c_N of length 4 in the free factor complex such that the number of 2-simplices required to fill c_N grows at least as a linear function of N. To prove the result, we construct a coarsely Lipschitz function from the `upward link' of a free factor to the set of integers.

Combinatorial isoperimetric inequality for the free factor complex

TL;DR

The paper proves that the free factor complex of a free group with rank does not admit a linear combinatorial isoperimetric inequality. By constructing a family of length-4 loops and a coarsely Lipschitz map from the upward link of a free factor to , it is shown that any disc filling must contain at least triangles with , giving a linear lower bound that grows with . The argument hinges on a careful analysis of folding paths, an optimal morphism guiding a coarsely Lipschitz twist-counting function , and a reduction to a sufficiently connected upward link when the corank is at least 3. The rank-3 case is handled by passing to a quasi-isometric modification that triangulates certain loops, preserving the lack of a linear combinatorial isoperimetric bound. Together, these results align the free factor complex with other -complexes that fail to satisfy combinatorial isoperimetric inequalities, while leaving open whether a cocompact, curve-like complex satisfying linear bounds exists for .

Abstract

We show that the free factor complex of the free group of rank greater than or equal to 4 does not satisfy a combinatorial isoperimetric inequality: that is, for every natural number N, there is a loop c_N of length 4 in the free factor complex such that the number of 2-simplices required to fill c_N grows at least as a linear function of N. To prove the result, we construct a coarsely Lipschitz function from the `upward link' of a free factor to the set of integers.
Paper Structure (9 sections, 7 theorems, 3 equations, 5 figures)

This paper contains 9 sections, 7 theorems, 3 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Combinatorial isoperimetric inequalities
  4. Outer space
  5. Train track structure and morphism
  6. Folding
  7. Coarsely Lipschitz map to Z
  8. Proof of main theorem
  9. The case rank 3

Key Result

Theorem 1

Let $\mathcal{F}$ be the free factor complex of free group of rank $n \geq 4$. There exists a constant $C = 2n-4$ and a family of loops $c_N$, for $N \in \mathbb{N}$, of combinatorial length 4 in $\mathcal{F}^{(1)}$ such that the following holds: whenever $P$ is a triangulation of a disc and $f\colo

Figures (5)

  • Figure 1: Combinatorial vs coarse isoperimetric inequality.
  • Figure 2: The loop $c_N$.
  • Figure 3: Graph $G$ and some subgraphs as in the last case of proof of \ref{['connected']}.
  • Figure 4: In this figure $\Phi_{w,b,B}(T) = 6$.
  • Figure 5: Adding edges and 2-simplices to $\mathcal{F}_3$.

Theorems & Definitions (15)

  • Theorem 1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Example 3.4
  • Example 3.5
  • Proposition 3.6
  • ...and 5 more