turkmath.org

Automorphism group of a regular k-valent tree $T(k) (k > 2)$ is a locally compact group acting by homeomorphisms on its boundary, a compact totally disconnected Hausdorff space homeomorphic to the Cantor set. Given a ribbon structure of our tree (equivalently a planar embedding), a canonically defined quotient space of this boundary is homeomorphic to the circle, which we call the boundary circle. Most of the automorphisms of $T(k)$ does not respect the ribbon structure and does not pass to an action on the boundary circle. On the other hand, there are certain "automorphims at infinity" of $T(k)$ which respect the ribbon structure. Hence this action passes to the boundary circle where the action is by homeomorphisms. I will concentrate on the case $k = 3$ (which corresponds to the modular group $PSL2(Z)$ in some precise sense) and try to explain why these ideas are of importance in several fields of mathematics. In this case the homeomorphism which identifies the boundary circle with the standard circle is the continued fraction map and the group in question is Thompson's group.