Analytic properties of bivariate representation and conjugacy class zeta functions of finitely generated nilpotent groups

Araujo, Paula Macedo Lins de

doi:10.48550/arxiv.1807.05577

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“…Class counting zeta functions were introduced by du Sautoy [23] for p-adic linear groups. They were further studied by Berman et al [8] for Chevalley groups and by the first author [57,60] and Lins [46][47][48] for unipotent groups; other names for these functions in the literature are "conjugacy class zeta functions" and "class number zeta functions". The use of zeta functions as a tool in group theory was pioneered by Grunewald et al [33].…”

Section: Zeta Functionsmentioning

confidence: 99%

Groups, graphs, and hypergraphs: average sizes of kernels of generic matrices with support constraints

Rossmann,

Voll

2019

Preprint

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We develop a theory of average sizes of kernels of generic matrices with support constraints defined in terms of graphs and hypergraphs. We apply this theory to study unipotent groups associated with graphs. In particular, we establish strong uniformity results pertaining to zeta functions enumerating conjugacy classes of these groups. We deduce that the numbers of conjugacy classes of F q -points of the groups under consideration depend polynomially on q. Our approach combines group theory, graph theory, toric geometry, and p-adic integration.Our uniformity results are in line with a conjecture of Higman on the numbers of conjugacy classes of unitriangular matrix groups. Our findings are, however, in stark contrast to related results by Belkale and Brosnan on the numbers of generic symmetric matrices of given rank associated with graphs.

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