Lie Groups, Representations and Discrete Mathematics

Let $G$ be a simple $G(\mathbf Q)$-group of $G(\mathbf Q)$-rank at least 2. In 1987 T. N. Venkataramana showed that if $\Gamma \subset G(\mathbf Z)$ is an infinite subgroup whose commensurator is a subgroup of finite index in $G(\mathbf Z)$, then $\Gamma$ itself has finite index. In alternative language, if $\Gamma$ is almost normal in a finite index subgroup, then $\Gamma$ has finite index. The argument relies on techniques from the theory of arithmetic groups.

A new proof of Venkataramana’s result will be presented that uses techniques from the structure theory of totally disconnected groups to reduce the problem to theorems about normal and congruence subgroups of $G(\mathbf Q)$. This approach in fact yields a stronger statement about any group (not just subgroups) commensurated by $G(\mathbf Z)$ and deals with small-rank groups not covered by Venkataramana.