Tilings and Submonoids of Metabelian Groups

TL;DR

This paper demonstrates the undecidability of membership problems in certain algebraic structures like free metabelian groups and lamplighter groups, using tiling encodings of Turing machines.

Contribution

It establishes new undecidability results for submonoid and subsemimodule membership in specific groups and modules, employing tiling-based encodings.

Findings

Membership in finitely generated submonoids is undecidable for free metabelian groups of rank 2.

Subsemimodule membership is undecidable for finite rank free modules over Z×Z.

Rational subset membership is undecidable for two-dimensional lamplighter groups.

Abstract

In this paper we show that membership in finitely generated submonoids is undecidable for the free metabelian group of rank 2 and for the wreath product $\mathbb Z\wr (\mathbb Z\times \mathbb Z)$. We also show that subsemimodule membership is undecidable for finite rank free $(\mathbb Z\times \mathbb Z)$-modules. The proof involves an encoding of Turing machines via tilings. We also show that rational subset membership is undecidable for two-dimensional lamplighter groups.