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

TL;DR

This paper explores finite state automorphisms representing groups like $GL(n,Z)$, computes the number of states for elementary matrices, and constructs a finite state model of a free group of rank 2.

Contribution

It introduces a finite state automorphism framework for $GL(n,Z)$ and constructs a finite state representation of a free group of rank 2.

Findings

Number of states for elementary matrix automorphisms computed

Finite state representation of free group of rank 2 constructed

Representation of $GL(2,Z)$ over a 4-letter alphabet demonstrated

Abstract

It is examined finite state automorphisms of regular rooted trees constructed 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.