Homological filling functions with coefficients

[摘要] 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 RRR. Our main theorem is that the coefficients make a difference. That is, for every n≥1n \geq 1n≥1 and every pair of coefficient groups A,B∈{Z,Q}∪{Z/pZ ⁣:p prime}A, B \in \{\mathbb{Z},\mathbb{Q}\} \cup \{\mathbb{Z}/p\mathbb{Z}\colon p\text{ prime}\}A,B∈{Z,Q}∪{Z/pZ:p prime}, there is a group whose filling functions for nnn-cycles with coefficients in AAA and BBB have different asymptotic behavior.

[学科分类] 神经科学

[关键词] Homological filling functions;isoperimetric functions;Dehn functions;discrete Morse theory