A survey of configurations

TL;DR

This paper surveys the concept of configurations in groups, exploring their role in understanding properties like amenability and paradoxical decompositions, and providing an exposition of related group theoretical topics.

Contribution

It offers a comprehensive overview of configurations and their applications in analyzing group properties, connecting various concepts in group theory.

Findings

Configurations characterize group behaviors in partitions.

Configurations relate to amenability and paradoxical decompositions.

The paper clarifies the role of configurations in group theory.

Abstract

To any finite ordered subset and any finite partition of a group a set of tuples of positive integers, named as configurations, is associated that describes the group's behavior. The present paper provides an exposition of this notion and related topics, including amenability, paradoxical decomposition and the group theoretical properties.