Metric ultraproducts of groups -- simplicity, perfectness and torsion

TL;DR

This paper characterizes when metric ultraproducts of groups are simple and introduces new examples of simple groups, exploring properties like perfectness and torsion using ultraproduct constructions.

Contribution

It provides a characterization of simplicity in metric ultraproducts and presents new examples of simple groups derived from various classes of groups.

Findings

Characterization of simplicity in metric ultraproducts

New examples of simple groups from symmetric and linear groups

Analysis of properties like perfectness and torsion in ultraproducts

Abstract

We characterise the simplicity of metric ultraproducts of a family of metric groups. We also present several new examples of simple groups, such as metric ultraproducts of finite and infinite symmetric groups, linear groups, and interval exchange transformation groups. Using similar methods, we also examine concepts such as genericity, perfectness, and torsion.