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.
Categories: math.GR math.GT