Normalizers of Congruence Groups in $SL_{2}(\mathbb{R})$ and Automorphisms of Lattices

TL;DR

This paper investigates the normalizers of certain congruence subgroups within SL(2,R) and applies these findings to analyze automorphism groups and discriminant kernels of isotropic lattices of signature (2,1).

Contribution

It provides a detailed determination of the normalizers of various congruence subgroups in SL(2,R) and links these results to automorphism groups of specific lattices.

Findings

Normalizers of several congruence subgroups in SL(2,R) are explicitly determined.

Tools developed enable evaluation of automorphism groups of isotropic lattices.

Results contribute to understanding the structure of lattices of signature (2,1).

Abstract

We determine the normalizer in $SL_{2}(\mathbb{R})$ of several families of congruence subgroups of $SL_{2}(\mathbb{Z})$. 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).