Geometry and Topology

Algebraic $G$-functions associated to matrices over a group-ring

S. Garoufalidis, J. Bellissard

School of Mathematics of Georgia Institute of Technology, Atlanta, USA

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. 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.

Keywords: rational function, algebraic function, holonomic function, $G$-function, generating series, non-commuting variables, moment, hamiltonian, resolvant, regular language, context-free language, Hadamard product, group-ring, free probability, Schur complement method, free group, von Neumann algebra, polynomial Hamiltonian, spectral theory, norm.

UDC: 512.7
BBK: B144.5

Language: English