Growth of étale groupoids and simple algebras. (English) Zbl 1366.16016
The étale groupoids appear as a generalization of actions of discrete groups on topological spaces. The author studies growth and complexity of étale groupoids in relation to growth of their convolution algebras. As an application, the author constructs simple finitely generated algebras of arbitrary Gelfand-Kirillov dimension \(d\geq2\) and simple finitely generated algebras of quadratic growth over arbitrary fields. These algebras are constructed as matrices of countable size related to Sturmian words and Toeplitz sequences.
The author also studies groupoids associated with groups acting on a rooted tree. The respective convolution algebras are related to so called thinned algebras. In the case of a contracting self-similar group, the author proves a result of L. Bartholdi on an estimate of Gelfand-Kirillov dimension for the thinned algebras of contracting self-similar groups [Isr. J. Math. 154, 93–139 (2006; Zbl 1173.16303)].
The author also studies groupoids associated with groups acting on a rooted tree. The respective convolution algebras are related to so called thinned algebras. In the case of a contracting self-similar group, the author proves a result of L. Bartholdi on an estimate of Gelfand-Kirillov dimension for the thinned algebras of contracting self-similar groups [Isr. J. Math. 154, 93–139 (2006; Zbl 1173.16303)].
Reviewer: Victor Petrogradsky (Brasilia)
MSC:
| 16P90 | Growth rate, Gelfand-Kirillov dimension |
| 22A22 | Topological groupoids (including differentiable and Lie groupoids) |
| 20L05 | Groupoids (i.e. small categories in which all morphisms are isomorphisms) |
Keywords:
growth of algebras; Gelfand-Kirillov dimension; étale groupoids; thinned algebras; Sturmian wordsCitations:
Zbl 1173.16303References:
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