Is Unreasonable Slightness a General Phenomenon?

Abstract

We consider the problem of determining when certain types of arithmetic groups like SLp2, Oq are generated by their elementary matrices. We give a simple, geometric criterion with a similarly simple proof based on finding fundamental domains for Kleinian groups. This both gives a new geometric proof for older results of Cohn, Nica, and others, and allows us to extend some of those old ideas in new directions.