The Twisted Twin of the Grigorchuk Group

TL;DR

This paper investigates a twisted variant of Grigorchuk's first group, revealing its algebraic properties such as finite endomorphic presentation, infinite-rank multiplier, and lack of the congruence property.

Contribution

It introduces a new twisted group related to Grigorchuk's group and analyzes its algebraic structure and properties.

Findings

Admits a finite endomorphic presentation

Has infinite-rank multiplier

Lacks the congruence property

Abstract

We study a twisted version of Grigorchuk's first group, and stress its similarities and differences to its model. In particular, we show that it admits a finite endomorphic presentation, has infinite-rank multiplier, and does not have the congruence property.