Geodesic
Private life of $\mathrm{GL}(5,\mathbb Z)$
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 10, Tome 305 (2003), pp. 153-162

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The paper investigates $(2,3)$-generation of $\mathrm{GL}(5,\mathbb Z)$. We reduce the problem to five specific pairs of matrices. 
@article{ZNSL_2003_305_a9,
     author = {A. Yu. Luzgarev and I. M. Pevzner},
     title = {Private life of $\mathrm{GL}(5,\mathbb Z)$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {153--162},
     publisher = {mathdoc},
     volume = {305},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_305_a9/}
}
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A. Yu. Luzgarev; I. M. Pevzner. Private life of $\mathrm{GL}(5,\mathbb Z)$. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 10, Tome 305 (2003), pp. 153-162. http://geodesic.mathdoc.fr/item/ZNSL_2003_305_a9/