We examine the geometry of the Cayley graph of the lamplighter group with respect to the generating set rising from its interpretation as an automata group due to Grigorchuk and Zuk. We find some metric behavior with respect to this generating set analogous to the metric behavior in the standard wreath product generating set. The similar metric behavior includes expressions for geodesic paths and families of `dead-end' elements, which are endpoints of terminating geodesic rays. We also exhibit some different metric behavior between these two generating sets related to the existence of `seesaw' elements.
We consider the two generalizations of lamplighter groups: automata groups generated by Cayley machine and cross-wired lamplighter groups. For a finite step two nilpotent group with central squares, we study its associated Cayley machine and give a presentation of the corresponding automata group. We show the automata group is a cross-wired lamplighter group and does not embed in the wreath product of a finite group with a torsion free group. For a subfamily of such finite step two nilpotent groups, we prove that their associated automata groups are linear.
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 $K$.
We realize lamplighter groups $A\wr \mathbb Z$, with $A$ a finite abelian group, as automaton groups via affine transformations of power series rings with coefficients in a finite commutative ring. Our methods can realize $A\wr \mathbb Z$ as a bireversible automaton group if and only if the $2$-Sylow subgroup of $A$ has no multiplicity one summands in its expression as a direct sum of cyclic groups of order a power of $2$.
We show presentations of automata groups generated by Cayley machines of finite groups of nilpotency class two and these automata groups are all cross-wired lamplighters.
We calculate the spectra and spectral measures associated to random walks on restricted wreath products of finite groups with the infinite cyclic group, by calculating the Kesten-von Neumann-Serre spectral measures for the random walks on Schreier graphs of certain groups generated by automata. This generalises the work of Grigorchuk and Zuk on the lamplighter group. In the process we characterise when the usual spectral measure for a group generated by automata coincides with the Kesten-von Neumann-Serre spectral measure.
Given a finite group $H$ and a free group $F_n$, we prove that the wreath product $H\wr F_n$ admits a metrically proper, isometric action on a Hilbert space.
We observe that embeddings into random metrics can be fruitfully used to study the $L_1$-embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new upper bound estimates on the $L_1$-distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on a $n$-point metric space embeds bi-Lipschitzly into $L_1$ with distortion $O(\log n)$. In particular, for every finite group $G$ the lamplighter group $H = \m…
We give the automata which describe time evolution rules of the box-ball system (BBS) with a carrier. It can be shown by use of tropical geometry, such systems are ultradiscrete analogues of KdV equation. We discuss their relation with the lamplighter group generated by an automaton. We present spectral analysis of the stochastic matrices induced by these automata, and verify their spectral coincidence.
It is shown that certain ascending HNN extensions of free abelian groups of finite rank, as well as various lamplighter groups, can be realized as automaton groups, i.e., can be given a self-similar structure. This includes the solvable Baumslag-Solitar groups BS(1,m), for m not +/-1. In addition, it is shown that, for any relatively prime integers m,n>=2, the pair of Baumslag-Solitar groups BS(1,m) and BS(1,n) can be realized by a pair of dual automata. The examples are then used to illustrate more general connections between Schreier graphs, composition of automata and dual automata.
We fully describe the horofunction boundary $\partial_h L_2$ with the word metric associated with the generating set $\{t,at\}$ (i.e the metric arising in the Diestel-Leader graph $\text{DL}(2,2)$). The visual boundary $\partial_\infty L_2$ with this metric is a subset of $\partial_h L_2$. Although $\partial_\infty L_2$ does not embed continuously in $\partial_h L_2$, it naturally splits into two subspaces, each of which is a punctured Cantor set and does embed continuously. The height function on $\text{DL}(2,2)$ provides a natural stratification of $\partial_h L_2$, in which countably-many n…
We introduce the discrete affine group of a regular tree as a finitely generated subgroup of the affine group. We describe the Poisson boundary of random walks on it as a space of configurations. We compute isoperimetric profile and Hilbert compression exponent of the group. We also discuss metric relationship with some lamplighter groups and lamplighter graphs.
In this paper we investigate metric properties of the groups $Γ_d(q)$ whose Cayley graphs are the Diestel-Leader graphs $DL_d(q)$ with respect to a given generating set $S_{d,q}$. These groups provide a geometric generalization of the family of lamplighter groups, whose Cayley graphs with respect to a certain generating set are the Diestel-Leader graphs $DL_2(q)$. Bartholdi, Neuhauser and Woess in \cite{BNW} show that for $d \geq 3$, $Γ_d(q)$ is of type $F_{d-1}$ but not $F_d$. We show below that these groups have dead end elements of arbitrary depth with respect to the generating set $S_{d,q}…
We study an abstract group of reversible Turing machines. In our model, each machine is interpreted as a homeomorphism over a space which represents a tape filled with symbols and a head carrying a state. These homeomorphisms can only modify the tape at a bounded distance around the head, change the state and move the head in a bounded way. We study three natural subgroups arising in this model: the group of finite-state automata, which generalizes the topological full groups studied in topological dynamics and the theory of orbit-equivalence; the group of oblivious Turing machines whose movem…
Given a metric space $(X,d)$, the wobbling group of $X$ is the group of bijections $g:X\rightarrow X$ satisfying $\sup\limits_{x\in X} d(g(x),x)<\infty$. We study algebraic and analytic properties of $W(X)$ in relation with the metric space structure of $X$, such as amenability of the action of the lamplighter group $ \bigoplus_{X} \mathbf Z/2\mathbf Z \rtimes W(X)$ on $\bigoplus_{X} \mathbf Z/2\mathbf Z$ and property (T).
In this paper, we introduce a geometric statistic called the "sprawl" of a group with respect to a generating set, based on the average distance in the word metric between pairs of words of equal length. The sprawl quantifies a certain obstruction to hyperbolicity. Group presentations with maximum sprawl (i.e., without this obstruction) are called statistically hyperbolic. We first relate sprawl to curvature and show that nonelementary hyperbolic groups are statistically hyperbolic, then give some results for products, for Diestel-Leader graphs and lamplighter groups. In free abelian groups, t…
