ABSTRACT
This chapter is concerned with a development which began as a problem in number theory even before Markoff, but which has progressed recently to new results with implications ranging over many fields, epitomized by the use of geodesics to parallel purely arithmetic results. In the interests of clarification, it is noted that the classic work was performed by A. A. Markoff, the Russian number theorist, who is perhaps better known for his later statistical work in random variables. While there is little chance to confuse him with his younger brother, W. A. Markoff, a number theorist, there is a more understandable confusion with his son, A. A. Markov the logician and algebraist. Fricke discovered modular functions and certain Fuchsian groups whose behavior bore a curious resemblance to the identities for Markoff forms. This was first noticed in 1955 by the speaker and led to connections with such topics as integral matrix identities; matrix inequalities; complex multiplication and geodesics on a torus.
