2007/08/30 by Jean Bellissard, Bellissard, Jean, Stavros Garoufalidis +1
Computer Science · Mathematics · #57M25 #57N10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.0708.4234
openalex publication_date 2007/08/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.