Free semigroups of 2 × 2 matrices

J. L. Brenner and Allen Kenneth Charnow

Vol. 77 (1978), No. 1, 57–69

DOI: 10.2140/pjm.1978.77.57

Abstract

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

Thus S_m can be free when G_m = gp⟨A,B⟩ is nonfree.

Theorem. Values of m for which S_m 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