Infinite characters on $GL_n(\mathbf{Q})$, on $SL_n(\mathbf{Z}),$ and on groups acting on trees

Answering a question of J. Rosenberg, we construct the first examples of infinite characters on $GL_n(\mathbf{K})$ for a global field $\mathbf{K}$ and $n\geq 2.$ 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 ${\rm Aut}(X)$ 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 $C^*$-subalgebra of $\mathcal{B}(\mathcal{H})$ generated by $\pi(G)$ contains the algebra of the compact operators on $\mathcal{H}.$ In the case $n\geq 3,$ we prove the existence of infinitely many characters for $G=SL_n(R)$, where $n\geq 3$ and $R$ is an integral domain such that $G$ is not amenable. In particular, the group $SL_n(\mathbf{Z})$ has infinitely many such characters for $n\geq 2.$