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09 February 2025
 
  » arxiv » 0708.4234

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Algebraic G-functions associated to matrices over a group-ring
Jean Bellissard ; Stavros Garoufalidi ;
Date 3 Sep 2007
AbstractGiven 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. 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 holonomic $G$-function. We ask whether the latter holds for general hyperbolic groups.
Source arXiv, 0708.4234
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