Variations on a theme by Higman

TL;DR

This paper introduces explicit group presentations of non-amenable, SQ-universal groups with specific properties like finite classifying spaces and absence of Kazhdan and Haagerup quotients.

Contribution

It provides new elementary constructions of groups with unique properties, expanding understanding of group structures with no amenable quotients.

Findings

Groups with finite classifying spaces constructed

Examples lack Kazhdan subgroups and Haagerup quotients

Explicit presentations of such groups are provided

Abstract

We propose elementary and explicit presentations of groups that have no amenable quotients and yet are SQ-universal. Examples include groups with a finite classifying space, no Kazhdan subgroups and no Haagerup quotients.