Lamplighter Group Research Articles

L^2-Betti numbers arising from the lamplighter group

We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute ell ^2-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational ell ^2-Betti numbers arising from the lamplighter group algebra {mathbb Q}[{mathbb Z}_2 wr {mathbb Z}]. This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational ell ^2-Betti numbers from the algebras {mathbb Q}[{mathbb Z}_n wr {mathbb Z}], where n ge 2 is a natural number. We also apply the techniques developed to the generalized odometer algebra {mathcal {O}}({overline{n}}), where {overline{n}} is a supernatural number. We compute its *-regular closure, and this allows us to fully characterize the set of {mathcal {O}}({overline{n}})-Betti numbers.

Automata groups generated by Cayley machines of groups of nilpotency class two

We build presentations for automata groups generated by Cayley machines of finite groups of nilpotency class two and prove that these automata groups are all cross-wired lamplighter groups.

Infinitely presented permutation stable groups and invariant random subgroups of metabelian groups

AbstractWe prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

Spectra of Cayley graphs of the lamplighter group and random Schrödinger operators

We show that the lamplighter group L has a system of generators for which the spectrum of the discrete Laplacian on the Cayley graph is a union of an interval and a countable set of isolated points accumulating to a point outside this interval. This is the first example of a group with infinitely many gaps in the spectrum of its Cayley graph. The result is obtained by a careful study of spectral properties of a one-parametric family of convolution operators on L. Our results show that the spectrum is a pure point spectrum for each value of the parameter, the eigenvalues are solutions of algebraic equations involving Chebyshev polynomials of the second kind, and the topological structure of the spectrum makes a bifurcation when the parameter passes the points 1 and -1.

Poisson boundaries of lamplighter groups: proof of the Kaimanovich–Vershik conjecture

We answer positively a question of Kaimanovich and Vershik from 1979, showing that the final configuration of lamps for simple random walk on the lamplighter group over \mathbb Z^d (d \geq 3) is the Poisson boundary. For d \geq 5 , this had been shown earlier by Erschler (2011). We extend this to walks of more general types on more general groups.

Solvable groups of interval exchange transformations

We prove that any finitely generated torsion free solvable subgroup of the group ${\rm IET}$ of all Interval Exchange Transformations is virtually abelian. In contrast, the lamplighter groups $A\wr \mathbb{Z}^k$ embed in ${\rm IET}$ for every finite abelian group $A$, and we construct uncountably many non pairwise isomorphic 3-step solvable subgroups of ${\rm IET}$ as semi-direct products of a lamplighter group with an abelian group. We also prove that for every non-abelian finite group $F$, the group $F\wr \mathbb{Z}^k$ does not embed in ${\rm IET}$.

Recoding Lie algebraic subshifts

<p style='text-indent:20px;'>We study internal Lie algebras in the category of subshifts on a fixed group – or Lie algebraic subshifts for short. We show that if the acting group is virtually polycyclic and the underlying vector space has dense homoclinic points, such subshifts can be recoded to have a cellwise Lie bracket. On the other hand there exist Lie algebraic subshifts (on any finitely-generated non-torsion group) with cellwise vector space operations whose bracket cannot be recoded to be cellwise. We also show that one-dimensional full vector shifts with cellwise vector space operations can support infinitely many compatible Lie brackets even up to automorphisms of the underlying vector shift, and we state the classification problem of such brackets. <p style='text-indent:20px;'>From attempts to generalize these results to other acting groups, the following questions arise: Does every f.g. group admit a linear cellular automaton of infinite order? Which groups admit abelian group shifts whose homoclinic group is not generated by finitely many orbits? For the first question, we show that the Grigorchuk group admits such a CA, and for the second we show that the lamplighter group admits such group shifts.

Lamplighter groups, bireversible automata, and rational series over finite rings

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.

A property of the lamplighter group

AbstractWe show that the inert subgroups of the lamplighter group fall into exactly five commensurability classes. The result is then connected with the theory of totally disconnected locally compact groups and with algebraic entropy.

A note on torsion subgroups of groups acting on finite-dimensional CAT(0) cube complexes

In this article, we state and prove a general criterion which prevents some groups from acting properly on finite-dimensional CAT(0) cube complexes. As an application, we show that, for every non-trivial finite group F, the lamplighter group F≀F2 over a free group does not act properly on a finite-dimensional CAT(0) cube complex (although it acts properly on an infinite-dimensional CAT(0) cube complex). We also deduce from this general criterion that, given a group G acting on a CAT(0) cube complex of finite dimension and an infinite torsion subgroup L≤G, either the normaliser NG(L) is virtually (locally finite)-by-(free abelian) or, for every k≥1, NG(L) contains a non-abelian free subgroup commuting with a subgroup of L of size ≥k.

A characterization of superreflexivity through embeddings of lamplighter groups

We prove that finite lamplighter groups { Z 2 ≀ Z n } n ≥ 2 \{\mathbb {Z}_2\wr \mathbb {Z}_n\}_{n\ge 2} with a standard set of generators embed with uniformly bounded distortions into any non-superreflexive Banach space and therefore form a set of test spaces for superreflexivity. Our proof is inspired by the well-known identification of Cayley graphs of infinite lamplighter groups with the horocyclic product of trees. We cover Z 2 ≀ Z n \mathbb {Z}_2\wr \mathbb {Z}_n by three sets with a structure similar to a horocyclic product of trees, which enables us to construct well-controlled embeddings.

𝐾-theory for generalized Lamplighter groups

We compute K K -theory for the reduced group C*-algebras of generalized Lamplighter groups.

Numerical studies of Thompson’s group F and related groups

We have developed polynomial-time algorithms to generate terms of the cogrowth series for groups [Formula: see text], the lamplighter group, [Formula: see text] and the Brin–Navas group [Formula: see text]. We have also given an improved algorithm for the coefficients of Thompson’s group [Formula: see text], giving 32 terms of the cogrowth series. We develop numerical techniques to extract the asymptotics of these various cogrowth series. We present improved rigorous lower bounds on the growth-rate of the cogrowth series for Thompson’s group F using the method from [S. Haagerup, U. Haagerup and M. Ramirez-Solano, A computational approach to the Thompson group F, Int. J. Alg. Comp. 25 (2015) 381–432] applied to our extended series. We also generalise their method by showing that it applies to loops on any locally finite graph. Unfortunately, lower bounds less than 16 do not help in determining amenability. Again for Thompson’s group F we prove that, if the group is amenable, there cannot be a sub-dominant stretched exponential term in the asymptotics. Yet the numerical data provides compelling evidence for the presence of such a term. This observation suggests a potential path to a proof of non-amenability: If the universality class of the cogrowth sequence can be determined rigorously, it will likely prove non-amenability. We estimate the asymptotics of the cogrowth coefficients of F to be [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. The growth constant [Formula: see text] must be 16 for amenability. These two approaches, plus a third based on extrapolating lower bounds, support the conjecture [M. Elder, A. Rechnitzer and E. J. Janse van Rensburg, Random sampling of trivial words in finitely presented groups, Expr. Math. 24 (2015) 391–409, S. Haagerup, U. Haagerup and M. Ramirez-Solano, A computational approach to the Thompson group F, Int. J. Alg. Comp. 25 (2015) 381–432] that the group is not amenable.

The Conjugacy Ratio of Groups

AbstractIn this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is 0 for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups and the lamplighter group.

The lamplighter group of rank two generated by a bireversible automaton

We construct a 4-state 2-letter bireversible automaton generating the lamplighter group of rank two. The action of the generators on the boundary of the tree can be induced by the affine transformations on the ring of formal power series over .

On self-similar Lie algebras and virtual endomorphisms

We introduce the notion of virtual endomorphisms of Lie algebras and use it as an approach for constructing self-similarity of Lie algebras. This is done in particular for a class of metabelian Lie algebras having homological type $$FP_n$$ , which are Lie algebra analogues of lamplighter groups. We establish several criteria when the existence of virtual endomorphism implies a self-similar Lie structure. Furthermore, we prove that the classical Lie algebra $$sl_n(k)$$ , where char(k) does not divide n affords non-trivial faithful self-similarity.

The degree of commutativity and lamplighter groups

The degree of commutativity of a group [Formula: see text] measures the probability of choosing two elements in [Formula: see text] which commute. There are many results studying this for finite groups. In [Y. Antolín, A. Martino and E. Ventura, Degree of commutativity of infinite groups, Proc. Amer. Math. Soc. 145 (2017) 479–485, MR 3577854], this was generalized to infinite groups. In this note, we compute the degree of commutativity for wreath products of the form [Formula: see text] and [Formula: see text], where [Formula: see text] is any finite group.

Furstenberg entropy of intersectional invariant random subgroups

We study the Furstenberg-entropy realization problem for stationary actions. It is shown that for finitely supported probability measures on free groups, any a priori possible entropy value can be realized as the entropy of an ergodic stationary action. This generalizes results of Bowen. The stationary actions we construct arise via invariant random subgroups (IRSs), based on ideas of Bowen and Kaimanovich. We provide a general framework for constructing a continuum of ergodic IRSs for a discrete group under some algebraic conditions, which gives a continuum of entropy values. Our tools apply, for example, for certain extensions of the group of finitely supported permutations and lamplighter groups, hence establishing full realization results for these groups. For the free group, we construct the IRSs via a geometric construction of subgroups, by describing their Schreier graphs. The analysis of the entropy of these spaces is obtained by studying the random walk on the appropriate Schreier graphs.

On NP-completeness of subset sum problem for Lamplighter group

In the paper “Knapsack problem for groups” A. Myasnikov, A. Nikolaev, and A. Ushakov stated a group version of well known Subset sum and Knapsack problems and many others. Actually they created a new direction of research as intersection of group theory and discrete optimization. They called the new direction as non-commututative optimization. Our work belongs to this new direction. In present paper we show that Subset sum problem is NP-complete for lamplighter group. Our result was obtained by polynomial reduction of well known Exact set cover problem which is strongly NP-complete to Subset sum problem for lamplighter group. As we expected, this result provides a convenient approach to prove NP-completeness of Subset sum problem for a wide class of groups.

Generators of split extensions of Abelian groups by cyclic groups

Let G \simeq M \rtimes C be an n -generator group which is a split extension of an Abelian group M by a cyclic group C . We study the Nielsen equivalence classes and T-systems of generating n -tuples of G . The subgroup M can be turned into a finitely generated faithful module over a suitable quotient R of the integral group ring of C . When C is infinite, we show that the Nielsen equivalence classes of the generating n -tuples of G correspond bijectively to the orbits of unimodular rows in M^{n -1} under the action of a subgroup of GL _{n - 1}(R) . Making no assumption on the cardinality of C , we exhibit a complete invariant of Nielsen equivalence in the case M \simeq R . As an application, we classify Nielsen equivalence classes and T-systems of soluble Baumslag–Solitar groups, split metacyclic groups and lamplighter groups.