  Given a square matrix with elements in the group-ring of a group, one can
consider the sequence formed by the trace (in the sense of the group-ring) of
its powers. We prove that the corresponding generating series is an algebraic
$G$-function (in the sense of Siegel) when the group is free of finite rank.
Consequently, it follows that the norm of such elements is an exactly
computable algebraic number, and their Green function is algebraic. Our proof
uses the notion of rational and algebraic power series in non-commuting
variables and is an easy application of a theorem of Haiman. Haiman's theorem
uses results of linguistics regarding regular and context-free language. On the
other hand, when the group is free abelian of finite rank, then the
corresponding generating series is a $G$-function. We ask whether the latter
holds for general hyperbolic groups. This version has an expanded introduction
following suggestions from Lehner, Voiculescu and others.

Combinatorics;Algebraic Geometry