Infinite characters on $G{L}_n(\mathbf{Q})$, on $S{L}_n(\mathbf{Z}),$ and on groups acting on trees

Abstract

Answering a question of J. Rosenberg, we construct the first examples of infinite characters on $G{L}_n(\mathbf{K})$ for a global field $\mathbf{K}$ and $n\ge 2.$ The case $n=2$ is deduced from the following more general result. Let $G$ a non amenable countable subgroup acting on locally finite tree $X$. Assume either that the stabilizer in $G$ of every vertex of $X$ is finite or that the closure of the image of $G$ in $\mathrm{Aut}(X)$ is not amenable. We show that $G$ has uncountably many infinite dimensional irreducible unitary representations $(\pi ,\mathcal{H})$ of $G$ which are traceable, that is, such that the $C^{\ast }$-subalgebra of $\mathcal{B}(\mathcal{H})$ generated by $\pi (G)$ contains the algebra of the compact operators on $\mathcal{H}.$ In the case $n\ge 3,$ we prove the existence of infinitely many characters for $G=S{L}_n(R)$, where $n\ge 3$ and $R$ is an integral domain such that $G$ is not amenable. In particular, the group $S{L}_n(\mathbf{Z})$ has infinitely many such characters for $n\ge 2.$

20 pages