Vol. 77, No. 1, 1978

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Free semigroups of 2 × 2 matrices

J. L. Brenner and Allen Kenneth Charnow

Vol. 77 (1978), No. 1, 57–69
Abstract

Let A = [1,m;0,1], B = [1,0;m,1]. The semigroup Sm = sgpA,B(including identity) generated by A, B is nonfree if two formally different words (with positive exponents) are equal; free otherwise. Theorem. Sm is free if π∕4 arg m π∕4, |m|1.

Thus Sm can be free when Gm = gpA,Bis nonfree.

Theorem. Values of m for which Sm is nonfree are dense on the line segment joining 2i to 2i; there are nonfree values of m arbitrarily close to m = 1.

Mathematical Subject Classification 2000
Primary: 20M05
Milestones
Received: 15 August 1977
Revised: 12 December 1977
Published: 1 July 1978
Authors
J. L. Brenner
Allen Kenneth Charnow