Pólya–Carlson dichotomy for coincidence Reidemeister zeta functions via profinite completions
Copyright policy )Pólya–Carlson dichotomy for coincidence Reidemeister zeta functions via profinite completions
We consider coincidence Reidemeister zeta functions for tame endomorphism pairs of nilpotent groups of finite rank, shedding new light on the subject by means of profinite completion techniques. In particular, we provide a closed formula for coincidence Reidemeister numbers for iterations of endomorphism pairs of torsion-free nilpotent groups of finite rank, based on a weak commutativity condition, which derives from simultaneous triangularisability on abelian sections. Furthermore, we present results in support of a P��lya-Carlson dichotomy between rationality and a natural boundary for the analytic behaviour of the zeta functions in question.
13 pages, small improvements of the exposition
- Heinrich Heine University Düsseldorf Germany
- University of Szczecin Poland
Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc., Word problems, other decision problems, connections with logic and automata (group-theoretic aspects), Group Theory (math.GR), nilpotent group of finite rank, twisted conjugacy classes, Finite nilpotent groups, \(p\)-groups, abelian group of finite rank, Nilpotent groups, Reidemeister zeta function, FOS: Mathematics, Limits, profinite groups, Mathematics - Group Theory, Residual properties and generalizations; residually finite groups, 37C25 (Primary) 37C30, 20F18, 20F69, 20K15, 20K30 (Secondary)
Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc., Word problems, other decision problems, connections with logic and automata (group-theoretic aspects), Group Theory (math.GR), nilpotent group of finite rank, twisted conjugacy classes, Finite nilpotent groups, \(p\)-groups, abelian group of finite rank, Nilpotent groups, Reidemeister zeta function, FOS: Mathematics, Limits, profinite groups, Mathematics - Group Theory, Residual properties and generalizations; residually finite groups, 37C25 (Primary) 37C30, 20F18, 20F69, 20K15, 20K30 (Secondary)
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