Arithmetic Groups

The Markoff equation $x^{2} + y^{2} + z^{2}=3xyz$, which arose in his spectacular thesis (1879), is ubiquitous in a tremendous variety of contexts.  After reviewing some of these, we will discuss joint work with Bourgain and Sarnak establishing forms of strong approximation on varieties of Markoff type, as well as ensuing implications, diophantine and dynamical.