Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Lamplighter groups have infinite weak cop number

Anders Cornect, Eduardo Martínez-Pedroza

TL;DR

The paper addresses whether there exist locally finite vertex-transitive graphs with finite, nontrivial weak cop number and answers affirmatively for a broad class of wreath-product Cayley graphs by proving $\mathop{\mathrm{wCop}}(G \wr H)=\infty$ whenever $G$ is nontrivial and $H$ is infinite. A key contribution is the Lamplighter game, a new pursuit-evasion framework that models lamp-state configurations on a streetmap and yields a direct link to the weak cops and robbers game via a graph isomorphism to the wreath product $\Omega \wr \Lambda$. The authors show that for infinite-diameter $\Lambda$ and nontrivial $\Omega$, the lamplighter can defeat any finite number of copiers, implying $\mathop{\mathrm{wCop}}^{*}(M)=\infty$ and, consequently, $\mathop{\mathrm{wCop}}(G \wr H)=\infty$ for the corresponding Cayley graphs; they also deduce that Thompson's group $F$ has infinite weak cop number by exhibiting a retract onto $\mathbb{Z}^2$. These results illuminate the landscape of weak cop numbers on graphs with exotic geometries and strengthen the connection between pursuit-evasion games and geometric group theory, particularly regarding quasi-isometric invariants and wreath-product structures.

Abstract

The weak-cop number of a graph, a variation of the cop number, is an invariant suitable for infinite graphs and is a quasi-isometric invariant. While for any $m\in\mathbb{Z}_+\cup\{\infty\}$ there exist locally finite infinite graphs with weak-cop number $m$, it is an open question whether there exists locally finite vertex transitive graphs whose weak-cop number is different than $1$ and $\infty$. We test this question on Cayley graphs of wreath products, these are objects known for their exotic geometries. We prove that Cayley graphs of wreath products of nontrivial groups by infinite groups have infinite weak-cop number. The result is proved by defining a new pursuit and evasion game and proving the existence of strategies for the evader. We also include a short argument that Cayley graphs of Thompson's group $F$ have infinite weak cop number.

The Lamplighter groups have infinite weak cop number

TL;DR

The paper addresses whether there exist locally finite vertex-transitive graphs with finite, nontrivial weak cop number and answers affirmatively for a broad class of wreath-product Cayley graphs by proving whenever is nontrivial and is infinite. A key contribution is the Lamplighter game, a new pursuit-evasion framework that models lamp-state configurations on a streetmap and yields a direct link to the weak cops and robbers game via a graph isomorphism to the wreath product . The authors show that for infinite-diameter and nontrivial , the lamplighter can defeat any finite number of copiers, implying and, consequently, for the corresponding Cayley graphs; they also deduce that Thompson's group has infinite weak cop number by exhibiting a retract onto . These results illuminate the landscape of weak cop numbers on graphs with exotic geometries and strengthen the connection between pursuit-evasion games and geometric group theory, particularly regarding quasi-isometric invariants and wreath-product structures.

Abstract

The weak-cop number of a graph, a variation of the cop number, is an invariant suitable for infinite graphs and is a quasi-isometric invariant. While for any there exist locally finite infinite graphs with weak-cop number , it is an open question whether there exists locally finite vertex transitive graphs whose weak-cop number is different than and . We test this question on Cayley graphs of wreath products, these are objects known for their exotic geometries. We prove that Cayley graphs of wreath products of nontrivial groups by infinite groups have infinite weak-cop number. The result is proved by defining a new pursuit and evasion game and proving the existence of strategies for the evader. We also include a short argument that Cayley graphs of Thompson's group have infinite weak cop number.
Paper Structure (16 sections, 11 theorems, 9 equations, 4 figures)

This paper contains 16 sections, 11 theorems, 9 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph theory language
  4. Definition of the weak cop number
  5. The weak cop number of wreath products
  6. The Lamplighter game
  7. Initial setup of the Lamplighter game.
  8. Turns and player movement
  9. Lamplighter move restriction
  10. Win conditions
  11. The copier-win number $\mathop{\mathrm{wCop}}\nolimits^*(M)$
  12. General winning strategy for the Lamplighter
  13. The wreath product of graphs
  14. Relation Between Lamplighter and Weak cops and Robbers
  15. Weak cop number of wreath-products of groups
  16. ...and 1 more sections

Key Result

Theorem 1.5

The restricted wreath product $G \wr H$ of finitely generated groups has infinite weak cop number if $G$ is non-trivial and $H$ is infinite.

Figures (4)

  • Figure 1: An illustration of an $M$-board over the streetmap $M=(C_5,a, P_\infty)$.
  • Figure 2: An example of a turn in Lamplighter. In this case there is a move of type 1, follow by a move of type 2, and then a move of type 1 again.
  • Figure 3: An example of a game state in Lamplighter. If in this game $\rho \ge 3$, then the copiers have won.
  • Figure 4: Two examples of the wreath products of small graphs.

Theorems & Definitions (26)

  • Definition 1.2: Weak cop number of a group
  • Theorem 1.5
  • Theorem 1.7
  • Definition 3.1: Streetmap
  • Definition 3.2: Board
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Theorem 3.6
  • proof
  • ...and 16 more