Self-similar groups of type FP_{n}

TL;DR

This paper introduces new classes of self-similar groups, including S-arithmetic, affine, and metabelian groups, many of which are finitely presented and of type FP_{n}, expanding the understanding of their algebraic properties.

Contribution

It constructs novel self-similar groups such as S-arithmetic, affine, and metabelian groups, and analyzes their finiteness properties, especially their type FP_{n}.

Findings

Most soluble groups are finitely presented.

Many groups are of type FP_{n} for suitable n.

New classes of self-similar groups are introduced.

Abstract

We construct new classes of self-similar groups : S-aritmetic groups, affine groups and metabelian groups. Most of the soluble ones are finitely presented and of type FP_{n} for appropriate n.