Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1607.07650

The concept of self-similar automata over a changing alphabet and lamplighter groups generated by such automata

Adam Woryna
View original source ↗Read source document ↗Export metadata ↓

Abstract

Generalizing the idea of self-similar groups defined by Mealy automata, we itroduce the notion of a self-similar automaton and a self-similar group over a changing alphabet. We show that every finitely generated residually-finite group is self-similar over an arbitrary unbounded changing alphabet. We construct some naturally defined self-similar automaton representations over an unbounded changing alphabet for any lamplighter group $K\wr \mathbb{Z}$ with an arbitrary finitely generated (finite or infinite) abelian group $K$.

Explore related subjects

math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Adam Woryna. 2016-07-26. The concept of self-similar automata over a changing alphabet and lamplighter groups generated by such automata. https://doi.org/10.1016/j.tcs.2013.02.027

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Word Measures on Wreath Products II

Every word $w$ in $F_r$, the free group of rank $r$, induces a probability measure (the $w$-measure) on every finite group $G$, by substitution of random $G$-elements in the letters. This measure is determined by its Fourier coefficients: the $w$-expectations $E_w[χ]$ of the irreducible characters of $G$. For every finite group $G$, every stable character $χ$ of $G\wr S_n$ (trace of a finitely generated $FI_G$-module), and every word $w\in F_r$, we approximate $E_w[χ]$ up to an error term of $O(n^{-π(w)})$, where $π(w)$ is the primitivity rank of $w$. This generalizes previous works by Puder, Hanany, Magee and the author. As an application we show that random Schreier graphs of representation-stable actions of $G\wr S_n$ are close-to-optimal expanders. The paper reveals a surprising relation between stable representation theory of wreath products and not-necessarily connected Stallings core graphs.

math.GR↗
arXiv2026

Robust quasi-isometric embeddings inapproximable by Anosov representations

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. For all but finitely many $m\in \mathbb{N}$, we exhibit the first examples of non-locally rigid, Zariski dense, robust quasi-isometric embeddings of hyperbolic groups in $\mathsf{SL}_m(\mathbb{K})$ which are not limits of Anosov representations. As a consequence, we show that higher rank analogues of Sullivan's structural stabilty theorem and of the density theorem for Kleinian groups fail for Anosov representations in $\mathsf{SL}_m(\mathbb{C}), m\geq 30$.

math.GR↗
arXiv2026

$\mathcal{C}$-Hereditarily conjugacy separable groups and wreath products

We provide a necessary and sufficient condition for the restricted wreath product $A\wr B$ to be $\mathcal{C}$-hereditarily conjugacy separable where $\mathcal{C}$ is an extension-closed pseudovariety of finite groups. Moreover, we prove that the Grigorchuk group is 2-hereditarily conjugacy separable. As an application, we demonstrate that the lamplighter groups and $\mathbb{Z} \wr \mathbb{Z}$ are hereditarily conjugacy separable (but not $p$-conjugacy separable for any prime $p$). This provides infinitely many new examples of solvable, non-polycyclic hereditarily conjugacy separable groups. Furthermore, we study wreath products of cyclic subgroup separable groups and the derived length of iterated wreath products of solvable groups with an abelian base group and, as an application, we give an explicit construction of non-polycyclic hereditarily conjugacy separable groups of arbitrary derived length as iterated wreath products of abelian groups.

math.GR↗