Houghton-like groups from "shift-similar" groups

TL;DR

This paper introduces shift-similar groups and Houghton-like groups, establishing their properties and showing that any finitely generated group can embed into a finitely generated shift-similar group, highlighting their richness.

Contribution

It defines shift-similar groups and Houghton-like groups, and proves that every finitely generated group embeds into a finitely generated shift-similar group, unlike self-similar groups.

Findings

Every finitely generated group embeds into a finitely generated shift-similar group.

Uncountably many isomorphism classes of finitely generated shift-similar groups exist.

Results on finite generation and amenability of these groups.

Abstract

We introduce and study \emph{shift-similar} groups $G\le\textrm{Sym}(\mathbb{N})$, which play an analogous role in the world of Houghton groups that self-similar groups play in the world of Thompson groups. We also introduce Houghton-like groups $H_n(G)$ arising from shift-similar groups $G$, which are an analog of R\"over-Nekrashevych groups from the world of Thompson groups. We prove a variety of results about shift-similar groups and these Houghton-like groups, including results about finite generation and amenability. One prominent result is that every finitely generated group embeds as a subgroup of a finitely generated shift-similar group, in contrast to self-similar groups, where this is not the case. This establishes in particular that there exist uncountably many isomorphism classes of finitely generated shift-similar groups, again in contrast to the self-similar situation.