Texas A&M University, Department of
Mathematics, 216 Milner Hall, 12th of November 2003, 3:00-4:00
Groups and Dynamcs Seminar
Some solvable groups of automata
Zoran Sunik of Texas A&M University
Automaton groups appear as solutions to certain recursive equations
involving wreath product decompositions. More concretely, they are
defined and realized through actions of finite transducers on rooted
regular trees by automorphisms.
Examples of solvable groups realized by automata are the lamplighter
groups. The construction involves power series over finite rings.
Further, the Baumslag-Solitar group BS(1,n) is realized by an n-state
automaton acting on the m-ary rooted tree, for any m that is relatively
prime to n. A dualization procedure leads to a realization of BS(1,m)
as a group af an m-state automaton acting on the n-ary rooted tree.
This shows that there exists an interesting duality between the
division by n in m-adic integers and the division by m in n-adic
integers.
More generally, some ascending HNN extensions of free abelian groups
are realized by automata. The construction involves the action of an
integer matrix on a free R-module, where R is the ring of m-adic
integers.