Double-coset zeta functions for groups acting on trees

Bianca Marchionna
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Abstract

We study the double-coset zeta functions for groups acting on trees, focusing mainly on weakly locally $\infty$-transitive or (P)-closed actions. After giving a geometric characterisation of convergence for the defining series, we provide explicit determinant formulae for the relevant zeta functions in terms of local data of the action. Moreover, we prove that evaluation at $-1$ satisfies the expected identity with the Euler-Poincar\'e characteristic of the group. The behaviour at $-1$ also sheds light on a connection with the Ihara zeta function of a weighted graph introduced by A. Deitmar.
作用于树的群的双套zeta函数
我们研究了作用于树的群的双套zeta函数,主要集中于弱局部$\infty$-transitive或(P)-closed作用。在给出定义序列收敛的几何特征之后,我们根据作用的局部数据为相关zeta函数提供了明确的行列式。此外,我们还证明了在 $-1$ 处的求值满足与群的 Euler-Poincar\'e 特性的预期同一性。在 $-1$ 处的行为还揭示了与 A. Deitmar 提出的加权图的 Iharazeta 函数之间的联系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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