Normalizers of Congruence Groups in $S{L}_2(\mathbb{R})$ and Automorphisms of Lattices

We determine the normalizer in $S{L}_2(\mathbb{R})$ of several families of congruence subgroups of $S{L}_2(\mathbb{Z})$. In addition, we show how these tools can be used to evaluate the groups of automorphisms and the discriminant kernels of a large family of isotropic lattices of signature (2,1).