Geodesic
Classification of (1,2)-reflective anisotropic hyperbolic lattices of rank~4
Izvestiya. Mathematics , Tome 83 (2019) no. 1, pp. 1-19

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A hyperbolic lattice is said to be $(1{,}{\kern1pt}2)$-reflective if its automorphism group is generated by $1$- and $2$-reflections up to finite index. We prove that the fundamental polyhedron of a $\mathbb{Q}$-arithmetic cocompact reflection group in three-dimensional Lobachevsky space contains an edge with sufficiently small distance between its framing faces. Using this fact, we obtain a classification of $(1{,}{\kern1pt}2)$-reflective anisotropic hyperbolic lattices of rank $4$.
Keywords: reflective hyperbolic lattices, roots, reflection groups, fundamental polyhedra, Coxeter polyhedra. 
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     author = {N. V. Bogachev},
     title = {Classification of (1,2)-reflective anisotropic hyperbolic lattices of rank~4},
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     url = {http://geodesic.mathdoc.fr/item/IM2_2019_83_1_a0/}
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N. V. Bogachev. Classification of (1,2)-reflective anisotropic hyperbolic lattices of rank~4. Izvestiya. Mathematics , Tome 83 (2019) no. 1, pp. 1-19. http://geodesic.mathdoc.fr/item/IM2_2019_83_1_a0/