Stability results for Houghton groups

TL;DR

This paper establishes homological stability for twisted Houghton groups and their multidimensional analogues, enabling a detailed description of their homology as finitely generated $FI$-modules over noetherian coefficients.

Contribution

It introduces homological stability results for twisted Houghton groups and their multidimensional versions, providing a new understanding of their homology structure.

Findings

Homological stability for twisted Houghton groups

Description of homology as finitely generated $FI$-modules

Homology over noetherian coefficients is finitely generated

Abstract

We prove homological stability for a twisted version of the Houghton groups and their multidimensional analogues. Based on this, we can describe the homology of the Houghton groups and that of their multidimensional analogues over constant noetherian coefficients as an essentially finitely generated $FI$-module.