What is a superrigid subgroup?

TL;DR

This expository paper explores the concept of superrigid subgroups, illustrating how certain discrete subgroups allow homomorphisms to extend to larger groups, with connections to classical geometry and linkages.

Contribution

It provides an overview of superrigid subgroups and their extension properties, highlighting classical geometric connections and examples.

Findings

Homomorphisms from discrete subgroups can often extend to larger groups.

Examples include Z^k to R^d homomorphisms and their extensions.

Connections to classical topics like linkages in geometry.

Abstract

This is an expository paper. It is well known that a linear transformation can be defined to have any desired action on a basis. From this fact, one can show that every group homomorphism from Z^k to R^d extends to a homomorphism from R^k to R^d, and we will see other examples of discrete subgroups H of connected groups G, such that the homomorphisms defined on $H$ can ("almost") be extended to homomorphisms defined on all of G. This is related to a very classical topic in geometry, the study of linkages.