Combinatorial isoperimetric inequality for the free factor complex

TL;DR

The paper demonstrates that the free factor complex of free groups of rank ≥ 4 lacks a combinatorial isoperimetric inequality by constructing loops with fillings growing linearly, using a coarsely Lipschitz function.

Contribution

It establishes the non-existence of a combinatorial isoperimetric inequality for the free factor complex of certain free groups, introducing a new construction involving Lipschitz functions.

Findings

Loops in the free factor complex require linearly growing fillings.

The free factor complex does not satisfy a combinatorial isoperimetric inequality.

A coarsely Lipschitz function from the upward link to integers is constructed.

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.