p-quotients of the G.Higman group

TL;DR

This paper explores the structure of p-quotients of the Graham Higman group, focusing on cases where p divides (k-1), providing insights into their algebraic properties within measured group theory.

Contribution

It characterizes p-quotients of the Higman group H(k) under specific divisibility conditions, expanding understanding of their algebraic structure.

Findings

p-quotients exist only when p divides (k-1)

Provides explicit descriptions of these quotients

Enhances understanding of Higman group's algebraic properties

Abstract

These notes are based on the mini-course "On the Graham Higman group", given at the Erwin Schr\"odinger Institute in Vienna, January 20, 22, 27 and 29, 2016, as a part of the Measured Group Theory program. The main purpose is to describe p-quotients of the Higman group $H(k)$ for $p|(k-1)$. (One may check that the condition $p|(k-1)$ is necessary for the existence of such quotients.)