Projective resolutions for modules over infinite groups

TL;DR

This paper introduces a new notion of module complexity for infinite groups, generalizes a classical theorem, and applies it to construct projective resolutions for groups like SL(n,Z).

Contribution

It extends the concept of complexity from finite to infinite groups and generalizes the Alperin-Evens Theorem, enabling new constructions of projective resolutions.

Findings

Complexity can be controlled via finite index subgroups.

Generalization of Alperin-Evens Theorem to infinite groups.

Construction of projective resolutions for SL(n,Z).

Abstract

We define a notion of complexity for modules over infinite groups. We show that if $M$ is a module over the group ring $kG$, and $M$ has complexity $\leq f$ (where $f$ is some complexity function) over some set of finite index subgroups of $G$, then $M$ has complexity $\leq f$ over $G$ (up to a direct summand). This generalizes the Alperin-Evens Theorem, which states that if the group $G$ is finite then the complexity of $M$ over $G$ is the maximal complexity of $M$ over an elementary abelian subgroup of $G$. We also show how we can use this generalization in order to construct projective resolutions for the integral special linear groups, $SL(n,\Z)$, where $n\geq 2$.