Tilings and Submonoids of Metabelian Groups

Authors:
Markus Lohrey;Benjamin Steinberg
Affiliations:
Universität Leipzig, Institut für Informatik, Leipzig, Germany;Carleton University, School of Mathematics and Statistics, Ottawa, ON, Canada
Venue:
Theory of Computing Systems
Year:
2011

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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 ℤ≀(ℤ×ℤ). We also show that subsemimodule membership is undecidable for finite rank free (ℤ×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.