Geodesic
Non-Free Groups Generated by Two 2 X 2 Matrices
Canadian journal of mathematics, Tome 27 (1975) no. 2, pp. 237-245

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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. 
Brenner, J. L.; Macleod, R. A.; Olesky, D. D. Non-Free Groups Generated by Two 2 X 2 Matrices. Canadian journal of mathematics, Tome 27 (1975) no. 2, pp. 237-245. doi: 10.4153/CJM-1975-029-5
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[1] 1. Brenner, J. L., Quelques groupes libres de matrices, C. R. Acad. Sci. Paris Sér. A-B. 2-1 (1955), 1689–1691. Google Scholar

[2] 2. Chang, B., Jennings, S. A., and Ree, R., On certain matrices which generate free groups, Can. J. Math. 10 (1958), 279–284. Google Scholar

[3] 3. Fuchs-Rabinowitsch, D. J., On a certain representation of a free group, Leningrad State University Annals (Uchenye Zapiski), Math. Ser. 10 (1940), 154–157. Google Scholar

[4] 4. Lyndon, R. C. and Ullman, J. L., Groups generated by two parabolic linear fractional transformations, Can. J. Math. 21 (1969), 1388–1403. Google Scholar

[5] 5. Ree, R., On certain pairs of matrices which do not generate a free group, Can. Math. Bull. 4 (1961), 49–52. Google Scholar

[6] 6. Sanov, I. N., A property of a representation of a free group, Dokl. Akad. Nauk SSSR57 (1947), 657–659. Google Scholar

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