The bizarre world of Thompson's groups

Abstract

In the early 1960's Richard J. Thompson discovered a fascinating family of infinite groups in connection with his work in logic. These groups have reappeared in a wide variety of settings, including homotopy theory, measure theory of discrete groups, non-associative algebras, dynamical systems and geometric group theory. Thompson's group \(F\) is the simplest known example of a variety of unusual group-theoretic phenomena and has been the subject of a great deal of study. I will describe these groups from several different perspectives and discuss some of their remarkable properties, particularly some unusual aspects of the geometry of their standard Cayley graphs and the first known non-trivial self quasi-isometries which arise as commensurations of \(F\).