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Simplicity of Nekrashevych algebras of contracting self-similar groups
14.1.2021, 16:15 - 17:15
Speaker:Dr. Nóra Szakács, University of York
Location:Mathematisches Institut, Bunsenstr 3-5Online über Zoom:
Organizer:Mathematisches Institut
A self-similar group is a group G acting on the infinite |X|-regular rooted tree by automorphisms in such a way that the self-similarity of the tree is reflected in the group. The most common examples are generated by the states of a finite automaton. Many famous groups like Grigorchuk's 2-group of intermediate growth are of this form. Nekrashevych associated C*-algebras and algebras with coefficients in a field to self-similar groups. In the case G is trivial, the algebra is the classical Leavitt algebra. Nekrashevych showed that the algebra associated to the Grigorchuk group is not simple in characteristic 2, but Clark, Exel, Pardo, Sims and Starling showed its Nekrashevych algebra is simple over all other fields. Nekrashevych then showed that the algebra associated to the Grigorchuk-Erschler group is not simple over any field (the first such example). The Grigorchuk and Grigorchuk-Erschler groups are contracting self-similar groups. This important class of self-similar groups includes Gupta-Sidki p-groups and many iterated monodromy groups like the Basilica group. Nekrashevych proved algebras associated to contracting groups are finitely presented.
In this talk we discuss the simplicity of Nekrashevych algebras of contracting groups. In particular, we give an algorithm which, given an automaton generating the group, outputs the characteristics over which the algebra is non-simple. We apply our results to several families of contracting groups like Sunic's generalizations of Grigorchuk's group associated to polynomials over finite fields.
This work is joint with Benjamin Steinberg (City College of New York).
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