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Snakes can be fooled into thinking they live in a tree

Laurent Bartholdi, Ville Salo

Abstract

We construct a finitely generated group which is not virtually free, yet has decidable snake tiling problem. This shows that either a long-standing conjecture by Ballier and Stein (the characterization of groups with decidable domino problem as those virtually free ones) is false, or a question by Aubrun and Bitar has a positive answer (there exists a group for which the domino and snake problems are of different difficulty).

Snakes can be fooled into thinking they live in a tree

Abstract

We construct a finitely generated group which is not virtually free, yet has decidable snake tiling problem. This shows that either a long-standing conjecture by Ballier and Stein (the characterization of groups with decidable domino problem as those virtually free ones) is false, or a question by Aubrun and Bitar has a positive answer (there exists a group for which the domino and snake problems are of different difficulty).
Paper Structure (7 sections, 14 theorems, 5 equations)

This paper contains 7 sections, 14 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Snake problems
  3. Amalgamated free products and HNN extensions
  4. A tower of HNN extensions
  5. Marked groups
  6. Tree automata
  7. Fooling snakes

Key Result

Theorem 1.3

There exists a finitely generated group which is not virtually free, but which has decidable snake tiling problem for any generating set.

Theorems & Definitions (36)

  • Conjecture 1.1: Ballier & Stein BS18
  • Theorem 1.3
  • definition 1: Tileset
  • definition 2: Domino problem
  • definition 3: Snake problem
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • ...and 26 more