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Injective linear cellular automata and sofic groups

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Abstract

Let V be a finite-dimensional vector space over a field \(\mathbb{K}\) and let G be a sofic group. We show that every injective linear cellular automaton τ: V G → V G is surjective. As an application, we obtain a new proof of the stable finiteness of group rings of sofic groups, a result previously established by G. Elek and A. Szabó using different methods.

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Authors and Affiliations

  1. Dipartimento di Ingegneria, Università del Sannio, C.so Garibaldi 107, 82100, Benevento, Italy

    Tullio Ceccherini-Silberstein

  2. Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, 7 rue René Descartes, 67084, Strasbourg Cedex, France

    Michel Coornaert

Authors
  1. Tullio Ceccherini-Silberstein
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  2. Michel Coornaert
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Correspondence to Tullio Ceccherini-Silberstein.

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Ceccherini-Silberstein, T., Coornaert, M. Injective linear cellular automata and sofic groups. Isr. J. Math. 161, 1–15 (2007). https://doi.org/10.1007/s11856-007-0069-8

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  • DOI: https://doi.org/10.1007/s11856-007-0069-8

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