Eventually periodic resolutions with applications to integral group rings

TL;DR

This paper introduces a systematic method for constructing eventually periodic projective resolutions over certain quotient rings, with applications to integral group rings of groups with finite virtual cohomological dimension, enabling explicit calculations.

Contribution

It develops a general construction combining Shamash's method and iterated mapping cones to produce periodic resolutions, applied to integral group rings of specific groups.

Findings

Explicit resolutions for hyperbolic triangle groups

Resolutions for mapping class groups of punctured planes

Method's computability demonstrated through examples

Abstract

We present a general construction of eventually periodic projective resolutions for modules over quotients of rings of finite left global dimension by a regular central element. Our approach utilizes a construction of Shamash, combined with the iterated mapping cone technique, to systematically 'purge' homology from a complex. The construction is applied specifically to the integral group rings of groups with finite virtual cohomological dimension. We demonstrate the computability of our method through explicit calculations for several families of groups including hyperbolic triangle groups and mapping class groups of the punctured plane.