On a finite state representation of \(GL(n,\mathbb{Z})\)

Abstract

It is examined finite state automorphisms of regular rooted trees constructed in [6] to represent groups \(GL(n,\mathbb{Z})\).  The number of states of automorphisms that correspond to elementary matrices is computed. Using the representation of \(GL(2,\mathbb{Z})\) over an alphabet of size \(4\)  a finite state  representation of the free group of rank \(2\) over binary alphabet is constructed.