Homological Filling Functions with Coefficients

TL;DR

This paper introduces homological filling functions with coefficients in groups, demonstrating that the choice of coefficients significantly affects the asymptotic behavior of filling functions in groups.

Contribution

It establishes that different coefficient groups can lead to different asymptotic filling functions, highlighting the importance of coefficients in homological filling problems.

Findings

Filling functions depend on the choice of coefficient groups.

For each dimension, there exist groups with different filling behaviors for different coefficients.

Coefficients such as al, al/Q, and al/pal yield distinct asymptotic properties.

Abstract

How hard is it to fill a loop in a Cayley graph with an unoriented surface? Following a comment of Gromov in "Asymptotic invariants of infinite groups", we define homological filling functions of groups with coefficients in a group $R$. Our main theorem is that the coefficients make a difference. That is, for every $n \geq 1$ and every pair of coefficient groups $A, B \in \{\mathbb{Z},\mathbb{Q}\} \cup \{\mathbb{Z}/p\mathbb{Z} : p\text{ prime}\}$, there is a group whose filling functions for $n$-cycles with coefficients in $A$ and $B$ have different asymptotic behavior.