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Article overview
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Algebraic G-functions associated to matrices over a group-ring | Jean Bellissard
; Stavros Garoufalidi
; | Date: |
3 Sep 2007 | Abstract: | 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.
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 | Services: | Forum | Review | PDF | Favorites |
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