Detecting products of elementary matrices in 𝐺𝐿₂(𝑍[√𝑑])

Tuler, Randy

doi:10.1090/S0002-9939-1983-0706508-3

Detecting products of elementary matrices in $\textrm {GL}_{2}(\textbf {Z}[\sqrt {d}])$
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Proc. Amer. Math. Soc. 89 (1983), 45-48 Request permission

Abstract:

An elementary $n \times n$ matrix over a ring $R$ has 1 in each diagonal position and at most one additional nonzero element. Let $R = {\mathbf {Z}}[\sqrt d ]$ where $d$ is an integer less than $- 4$. We give an algorithm for determining whether or not a $2 \times 2$ invertible matrix over $R$ is generated by elementary matrices. This is connected with the theory of integral binary quadratic forms.

References

P. M. Cohn, On the structure of the $\textrm {GL}_{2}$ of a ring, Inst. Hautes Études Sci. Publ. Math. 30 (1966), 5–53. MR 207856, DOI 10.1007/BF02684355
Burton W. Jones, The Arithmetic Theory of Quadratic Forms, Carcus Monograph Series, no. 10, Mathematical Association of America, Buffalo, N.Y., 1950. MR 0037321, DOI 10.5948/UPO9781614440109
Robert Riley, Applications of a computer implementation of Poincaré’s theorem on fundamental polyhedra, Math. Comp. 40 (1983), no. 162, 607–632. MR 689477, DOI 10.1090/S0025-5718-1983-0689477-2
Richard G. Swan, Generators and relations for certain special linear groups, Advances in Math. 6 (1971), 1–77 (1971). MR 284516, DOI 10.1016/0001-8708(71)90027-2
Randy Tuler, Subgroups of $\textrm {SL}_{2}\textbf {Z}$ generated by elementary matrices, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), no. 1-2, 43–47. MR 611299, DOI 10.1017/S0308210500017273