Homological filling functions with coefficients

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 $n\ge 1$ 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.