Random walks on automaton groups

Automaton groups comprise the algebraic theory of fractals. The Sierpinski gasket, Julia sets, and the Hanoi towers game are all described by polynomial automaton groups. Well-known examples include the Grigorchuk group and the lamplighter group.
I will describe how the asymptotic speed of random walks on automaton groups can be understood, and how this can be used to resolve an open problem from the Kourovka notebook.