Geodesic
The smallest field of definition of a~subgroup of the~group $\mathrm{PSL}_2$
Sbornik. Mathematics, Tome 80 (1995) no. 1, pp. 179-190

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As previously proved by the author, for each semisimple algebraic group of adjoint type that is dense in the Zariski topology there exists a smallest field of definition which is an invariant of the class of commensurable subgroups. In the present paper an algorithm is given for finding the smallest field of definition of a dense finitely generated subgroup of the group $\mathrm{PSL}_2(\mathbb{C})$. A criterion for arithmeticity of a lattice in $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{PSL}_2(\mathbb{C})$ in terms of this field is presented. 
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     title = {The smallest field of definition of a~subgroup of the~group $\mathrm{PSL}_2$},
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È. B. Vinberg. The smallest field of definition of a~subgroup of the~group $\mathrm{PSL}_2$. Sbornik. Mathematics, Tome 80 (1995) no. 1, pp. 179-190. http://geodesic.mathdoc.fr/item/SM_1995_80_1_a8/