11,122 research outputs found
Enumerating finite class-2-nilpotent groups on 2 generators
No full textWe compute the numbers g(n,2,2) of nilpotent groups of order n, of class atmost 2 generated by at most 2 generators, by giving an explicit formula for theDirichlet generating function \sum_{n=1}^\infty g(n,2,2)n^{-s}
Functional equations for zeta functions of groups and rings
No full textWe introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or T -)groups, and the normal zeta functions of T -groups of class 2. In particular we solve the two problems posed in [9, Section 5]. We deduce our theorems from a ‘blueprint result’ on certain p-adic integrals which generalises work of Denef and others on Igusa’s local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to ‘linearise’ the problems of counting subgroups and representations in T -groups, respectively
Representation zeta functions of compact p-adic analytic groups and arithmetic groups
No full textWe introduce new methods from p-adic integration into the study of representation zeta functions associated to compact p-adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of generic members of families of p-adic analytic pro-p groups obtained from a global, `perfect' Lie lattice satisfy functional equations. In the case of `semisimple' compact p-adic analytic groups, we exhibit a link between the relevant p-adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple Lie algebra, defined by centraliser dimension.Based on this algebro-geometric description, we compute explicit formulae for the representation zeta functions of principal congruence subgroups of the groups SL_3(O), where O is a compact discrete valuation ring of characteristic 0, and of the corresponding unitary groups. These formulae, combined with approximative Clifford theory, allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type A_2. Assuming a conjecture of Serre on the Congruence Subgroup Problem, we thereby prove a conjecture of Larsen and Lubotzky on lattices in higher-rank semisimple groups for algebraic groups of type A_2 defined over number fields
Counting subgroups in a family of nilpotent semi-direct products
No full textIn this paper we compute the subgroup zeta functions of nilpotent groups of the form , all other [,] trivial [right angle bracket] and deduce local functional equations
Normal subgroup growth in free class-2-nilpotent groups
No full textLet F2,d denote the free class-2-nilpotent group on d generators. We compute the normal zeta functions zeta^\triangleleftF2,d(s), prove that they satisfy local functional equations and determine their abscissae of convergence and pole orders
Zeta functions of groups - singular Pfaffians
No full textThe local normal zeta functions of a finitely generated, torsion-free nilpotent group G of class 2 depend on the geometry of the Pfaffian hypersurface associated to the bilinear form induced by taking commutators in G.The smallest examples of zeta functions which are not finitely uniform arise from groups whose associated Pfaffian hypersurfaces are plane curves. In this paper we study groups whose Pfaffians define singular curves, illustrating that the local normal zeta functions may indeed invoke all the degeneracy loci of the Pfaffian
Zeta functions of groups and enumeration in Bruhat-Tits buildings
No full textWe introduce a new method to calculate local normal zeta functions of finitely generated, torsion-free nilpotent groups. It is based on an enumeration of vertices in the Bruhat-Tits building for Sln(Qp). It enables us to give explicit computations for groups of class 2 with small centres and to derive local functional equations. Examples include formulae for non-uniform normal zeta functions
Zeta functions of three-dimensional p-adic Lie algebras
No full textWe give an explicit formula for the subalgebra zeta function of a general 3-dimensional Lie algebra over the p-adic integers. To this end, we associate to such a Lie algebra a ternary quadratic form over the p-adic integers. The formula for the zeta function is given in terms of Igusa’s local zeta function associated to this form
A qualitative and quantitative post-mortem analysis: Studying free-radical initiation processes via soft ionization mass spectrometry
No full textThe current article contains a review of the electrospray ionization-mass spectrometry characterization of polymers prepared via thermal- and photoinitiation processes. The used analysis method permits direct access to detailed endgroup information. For a qualitative and quantitative endgroup analysis, sophisticated methods have been developed which provide a detailed image of the incorporation propensity of thermally as well as photolytically generated radicals at the polymer chain termini. Such a post-mortem analysis of polymeric materials specifically allows for the quantification of the ability of radical fragments to initiate polymerization processes. Herein, the most recent progress in the field of mass spectrometric radical reactivity mapping is outlined and open questions as well as future directions are discussed. © 2012 Wiley Periodicals, Inc
Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B
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