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arXiv · 1409.5027

Self-similar groups, automatic sequences, and unitriangular representations

R. Grigorchuk·Y. Leonov·V. Nekrashevych·V. Sushchansky
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Abstract

We study natural linear representations of self-similar groups over finite fields. In particular, we show that if the group is generated by a finite automaton, then obtained matrices are automatic. This shows a new relation between two separate notions of automaticity: groups generated by automata and automatic sequences. We also show that if the group acts on the tree by $p$-adic automorphisms, then the corresponding linear representation is a representation by infinite triangular matrices. We relate this observation with the notion of height of an automorphism of a rooted tree due to L.Kaloujnine.

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R. Grigorchuk, Y. Leonov, V. Nekrashevych, V. Sushchansky. 2014-09-17. Self-similar groups, automatic sequences, and unitriangular representations. https://arxiv.org/abs/1409.5027

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