Monadic second-order logic and the domino problem on self-similar graphs

TL;DR

This paper investigates the decidability of monadic second-order logic and the domino problem on Schreier graphs of self-similar groups, showing decidability for bounded groups and undecidability in certain cases.

Contribution

It establishes decidability results for bounded self-similar groups and proves undecidability for some groups, answering open questions in the field.

Findings

Decidability of monadic second-order logic for bounded self-similar groups.

Undecidability of the domino problem for certain self-similar groups.

Examples include Sierpiński gasket graphs and the Basilica group.

Abstract

We consider the domino problem on Schreier graphs of self-similar groups, and more generally their monadic second-order logic. On the one hand, we prove that if the group is bounded then the graph's monadic second-order logic is decidable. This covers, for example, the Sierpi\'nski gasket graphs and the Schreier graphs of the Basilica group. On the other hand, we already prove undecidability of the domino problem for a class of self-similar groups, answering a question by Barbieri and Sablik, and some examples including one of linear growth.