Friends and relatives of BS(1,2)

Algorithms, constructions and examples are of central interest in combinatorial and geometric group theory. Teaching experience and, more recently, preparing a historical essay have led me to think the familiar group BS(1, 2) is an example of fundamental importance. The purpose of this note is to make a case for this point of view.We recall several interesting constructions and important examples of groups related to BS(1, 2), and indicate why certain of these groups played a key role in showing the word problem for nitely presented groups is unsolvable.