soficity

Soficity is a concept in mathematics, specifically in the field of group theory, that generalizes the notion of amenability. A group is said to be sofic if it can be approximated by finite groups in a certain sense. This approximation is formalized through the notion of a sequence of finite approximations, which are essentially finite groups that capture the structure of the original group.

The term "sofic" is derived from the word "sof" (short for "sofie"), which is a term used in the context of finite approximations. A group is sofic if there exists a sequence of finite groups and homomorphisms between them that approximate the group's structure in a way that allows for certain properties to be preserved.

Soficity was introduced by Gaboriau and Monod in 2006 as a way to study groups that are not necessarily amenable but still exhibit some form of regularity. Amenable groups are known for their property of having invariant means, which is a crucial tool in ergodic theory and harmonic analysis. Sofic groups, on the other hand, do not necessarily have invariant means but can be approximated by finite groups that do.

The study of sofic groups has led to the development of new techniques and tools in group theory, and it has applications in various areas of mathematics, including ergodic theory, topological dynamics, and geometric group theory. Soficity provides a bridge between the study of finite and infinite groups, allowing for the transfer of techniques and results between these two realms.