Homology of Gaussian groups

TL;DR

This paper introduces combinatorial methods to explicitly compute the homology of Gaussian groups, including Artin groups of finite Coxeter type, by constructing free resolutions of the integers as modules over the group.

Contribution

It provides new combinatorial techniques for constructing free resolutions of Z over ZG for locally Gaussian monoids, enabling homology computations for these groups.

Findings

Explicit free resolutions constructed for Gaussian groups

New methods applied to Artin groups of finite Coxeter type

Facilitates homology calculations for these classes of groups

Abstract

We describe new combinatorial methods for constructing an explicit free resolution of Z by ZG-modules when G is a group of fractions of a monoid where enough least common multiples exist (``locally Gaussian monoid''), and, therefore, for computing the homology of G. Our constructions apply in particular to all Artin groups of finite Coxeter type, so, as a corollary, they give new ways of computing the homology of these groups.