Snakes can be fooled into thinking they live in a tree

TL;DR

This paper constructs a specific finitely generated group with a decidable snake tiling problem that challenges existing conjectures about the relationship between group properties and tiling problem decidability.

Contribution

It provides a counterexample to the conjecture that only virtually free groups have decidable domino problems, or shows a discrepancy between domino and snake problem complexities.

Findings

A finitely generated group with decidable snake tiling problem but not virtually free.

Counterexample to the conjecture linking group structure to domino problem decidability.

Evidence that snake and domino problems can differ in difficulty for the same group.

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).