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Double-coset zeta functions for groups acting on trees

Bianca Marchionna

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-Poincaré 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.

Double-coset zeta functions for groups acting on trees

Abstract

We study the double-coset zeta functions for groups acting on trees, focusing mainly on weakly locally -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 satisfies the expected identity with the Euler-Poincaré characteristic of the group. The behaviour at also sheds light on a connection with the Ihara zeta function of a weighted graph introduced by A. Deitmar.
Paper Structure (27 sections, 42 theorems, 238 equations)

This paper contains 27 sections, 42 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Generalities
  4. Graphs
  5. Group actions on trees
  6. Group actions on trees
  7. The quotient graph and its standard edge weight
  8. (P)-closed actions on trees
  9. Local action diagrams and their associated universal groups
  10. The standard $\Delta$-tree associated to $\iota$ and $c_0$
  11. The universal group $U(\Delta,\mathbb{T})$
  12. Weakly locally $\infty$-transitive actions on trees
  13. Cosets and double-cosets for groups acting on trees
  14. Cosets and geodesics
  15. The case of weakly locally $\infty$-transitive actions on trees
  16. ...and 12 more sections

Key Result

Lemma 3.1

Let $(G,T)$ be a (P)-closed action on a tree and $(e_1,\ldots, e_n)$ be a geodesic in $T$ of length $n\geq 2$. Then, for every $k<n$, we have

Theorems & Definitions (126)

  • Remark 1.0.1
  • Example 1.1
  • Remark 2.1.1
  • Lemma 3.1
  • proof
  • Example 3.3
  • Definition 3.4
  • Remark 3.4.1
  • Remark 3.4.2
  • Remark 3.4.3
  • ...and 116 more