FPn properties of generalized Houghton groups

TL;DR

This thesis introduces a 2D generalization of Houghton's groups, analyzing their algebraic properties and showing they are of type FP(n-1) with specific subgroup structures.

Contribution

It extends Houghton's groups to two dimensions and establishes their finiteness properties, including the existence of a normal subgroup with a particular quotient structure.

Findings

G(n) is of type FP(n-1)

G(n) contains a normal subgroup N with G(n)/N = Z^(n-1)

G(n) is not of type FPn

Abstract

This Thesis presents a 2-dimensional generalization of Houghtons' groups H_n. H_n is defined to be the group of all permutations p of a disjoint union of copies of the natural numbers N, with the property that each copy of N contains a cofinite subset on which p restricts to a translation. Our group G(n) is defined to be the group of all permutations p of a disjoint union of quadrants (i.e., copies of NxN) with the property that each quadrant contains a subquadrant on which p is a translation, while p restricted to the remaining set is piecewise isometric on a cofinite disjoint union of rays (each isometric to N). Based on K.S. Brown's treatment of the Houghton group case it is shown that G(n) is of type FP(n-1); in fact, that G(n) contains a normal subgroup N with G(n)/N = Z^(n-1) which is of type FP(n-1) and not FPn.