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Non-Free Groups Generated by Two 2 X 2 Matrices

Published online by Cambridge University Press:  20 November 2018

J. L. Brenner ,
R. A. Macleod  and
D. D. Olesky
J. L. Brenner
Affiliation:
10 Phillips Road, Palo Alto, California;
R. A. Macleod
Affiliation:
10 Phillips Road, Palo Alto, California;
D. D. Olesky
Affiliation:
University of Victoria, Victoria, British Columbia
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Let m be any real or complex number, and let Gm be the group generated by the 2 X 2 matrices A = (1, m\ 0, 1) and B = (1, 0; m, 1), where we use the notation (C11, C12; c21, C22) to denote (by rows) the elements of a 2 X 2 matrix C. Thus, Gm is the set of all finite products (or words) of the form

… Ah(3)Bh(2)Ah(1)

where the h(i) are nonzero integers with h﹛\) possibly zero. If a non-trivial word of this form equals the identity / = (1, 0; 0, 1), then Gm is non-free; otherwise, Gm is free.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 2 , 01 April 1975 , pp. 237 - 245
Copyright
Copyright © Canadian Mathematical Society 1975

References

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