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Galois coverings of Schreier graphs of groups generated by bounded automata

Asif Shaikh, Daniele D'Angeli, Hemant Bhate, Dilip Sheth

TL;DR

The paper addresses coverings of Schreier graphs arising from groups generated by bounded automata, providing a precise criterion for when the levelwise coverings $\Gamma_{n+1}\to\Gamma_n$ are Galois. It introduces a generalized replacement product to model $\Gamma_{n+r}$ as a combinatorial product $\Gamma_n \textcircled{g} \Gamma_r$ and proves that such constructions yield unramified (and cyclic, under a root-permutation condition) coverings. The authors develop an Artin-L-functions framework for these Galois covers, giving determinant formulas and a product decomposition for Ihara zeta functions, with explicit computations in examples like Grigorchuk, Gupta–Sidki, Fabrykowski–Gupta, Basilica, and BSV. The work also discusses non-bounded automaton groups and poses an open question about the extent to which the bounded-case cyclic-covering phenomenon extends to non-bounded settings.

Abstract

We give a characterization of the covering Schreier graphs of groups generated by bounded automata to be Galois. We also investigate the zeta and L functions of Schreier graphs of few groups namely the Grigorchuk group, Gupta-Sidki p group, Gupta-Fabrykowski group and BSV torsion-free group.

Galois coverings of Schreier graphs of groups generated by bounded automata

TL;DR

The paper addresses coverings of Schreier graphs arising from groups generated by bounded automata, providing a precise criterion for when the levelwise coverings are Galois. It introduces a generalized replacement product to model as a combinatorial product and proves that such constructions yield unramified (and cyclic, under a root-permutation condition) coverings. The authors develop an Artin-L-functions framework for these Galois covers, giving determinant formulas and a product decomposition for Ihara zeta functions, with explicit computations in examples like Grigorchuk, Gupta–Sidki, Fabrykowski–Gupta, Basilica, and BSV. The work also discusses non-bounded automaton groups and poses an open question about the extent to which the bounded-case cyclic-covering phenomenon extends to non-bounded settings.

Abstract

We give a characterization of the covering Schreier graphs of groups generated by bounded automata to be Galois. We also investigate the zeta and L functions of Schreier graphs of few groups namely the Grigorchuk group, Gupta-Sidki p group, Gupta-Fabrykowski group and BSV torsion-free group.

Paper Structure

This paper contains 14 sections, 8 theorems, 39 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Galois coverings of graphs
  4. Groups generated by bounded automata
  5. Schreier graphs of group actions
  6. Inflation of graphs
  7. Examples
  8. Generalized Replacement Product
  9. Galois coverings of Schreier graphs
  10. Examples
  11. $L$ functions and zeta functions of graphs
  12. Examples
  13. Non-bounded Automaton Groups
  14. Open Question

Key Result

Theorem 3.1

Inflation process: To construct $\Gamma_{n+1}'$, take $d$ copies of $\Gamma_n'$ and label each copy by a letter in $X$. The vertex set of the $x$-th copy $\Gamma_n' \times x$, can be identified with the set $X^n \times \{x\}.$ Two vertices $(u,x)$ and $(v,y)$ are connected by an edge if and only if

Figures (12)

  • Figure 1: Binary rooted tree.
  • Figure 2: The automaton generating the Grigorchuk group.
  • Figure 3: The Basilica group $\mathcal{B}$ with post-critical set $\mathcal{P_B} = \bigl\{ p_1 = 0^{-\omega}, p_2 = (01)^{-\omega}, p_3 = (10)^{-\omega}\bigr\}$
  • Figure 4: The graphs $\Gamma_1'^{\mathcal{B}}$, $\Gamma_2'^{\mathcal{B}}$, and $\Gamma_3'^{\mathcal{B}}$ are the Tile graphs of $\mathcal{B}$ over $X$, $X^2$, and $X^3$, respectively. The corresponding edge set is $E_M^B = \biggl\{\{p_{11}, p_{30}\}, \{p_{11}, p_{21}\}\biggr\}$.
  • Figure 5: The graphs $\Gamma_1^{\mathcal{B}},\Gamma_2^{\mathcal{B}}$ and $\Gamma_3^{\mathcal{B}}$ are the Schreier graphs of the Basilica group over $X, X^2$ and $X^3$ respectively.
  • ...and 7 more figures

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • ...and 31 more