Abstract

Arithmetic subgroups whose representations all map into GL_n(Z)

Dave Witte Morris
University of Lethbridge

Suppose $\Gamma$ is an arithmetic subgroup of a semisimple Lie group $G$. For any finite-dimensional representation $\rho: G \to GL_n(\mathbb{R})$, a classical paper of J. Tits determines whether $\rho(\Gamma)$ is conjugate to a subgroup of $GL_n(\mathbb{Z})$. Combining this with a well-known surjectivity result in Galois cohomology provides a short proof of the known fact that every $G$ has an arithmetic subgroup $\Gamma$, such that the containment is true for \emph{every} representation $\rho$. We will not assume the audience is acquainted with Galois cohomology or the theorem of Tits.