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Abstract Let be a submonoid of a free Abelian group of finite rank. We show that if is a field of prime characteristic such that the monoid ‐algebra is , then is a finitely generated ‐algebra, or equivalently, that is a finitely generated monoid. Split‐‐regular rings are possibly non‐Noetherian or non‐‐finite rings that satisfy the defining property of strongly ‐regular rings from the theories of tight closure and ‐singularities. Our finite generation result provides evidence in favor of the conjecture that rings in function fields over have to be Noetherian. The key tool is Diophantine approximation from convex geometry.
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The automorphism group of a shift of slow growth is amenable
Suppose $$(X,\unicode[STIX]{x1D70E})$$ is a subshift, $$P_{X}(n)$$ is the word complexity function of $$X$$ , and $$\text{Aut}(X)$$ is the group of automorphisms of $$X$$ . We show that if $$P_{X}(n)=o(n^{2}/\log ^{2}n)$$ , then $$\text{Aut}(X)$$ is amenable (as a countable, discrete group). We further show that if $$P_{X}(n)=o(n^{2})$$ , then $$\text{Aut}(X)$$ can never contain a non-abelian free monoid (and, in particular, can never contain a non-abelian free subgroup). This is in contrast to recent examples, due to Salo and Schraudner, of subshifts with quadratic complexity that do contain such a monoid.
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- Award ID(s):
- 1500670
- PAR ID:
- 10317801
- Date Published:
- Journal Name:
- Ergodic Theory and Dynamical Systems
- Volume:
- 40
- Issue:
- 7
- ISSN:
- 0143-3857
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/etds.2018.126
