The algebraic solvability of the novel lamplighter puzzle

In this study, the group of finite cyclic lamplighter states is reinterpreted as the novel lamplighter puzzle. The rules of the puzzle are outlined and related back to properties of the lamplighter group with specific interest placed upon the discussion of which puzzle instances are solvable. The paper shows that, through the use of algebra, many puzzle instances can be identified as solvable without the use of an exhaustive search algorithm. Solvability depends upon the creation of irregular generating sets for subgroups of the finite cyclic lamplighter group and the cosets formed by these subgroups. Further possible generalizations of the lamplighter puzzle are also discussed in closing.

Law of large numbers for the drift of the two-dimensional wreath product

We prove the law of large numbers for the drift of random walks on the two-dimensional lamplighter group, under the assumption that the random walk has finite $$(2+\epsilon )$$ ( 2 + ϵ ) -moment. This result is in contrast with classical examples of abelian groups, where the displacement after n steps, normalised by its mean, does not concentrate, and the limiting distribution of the normalised n-step displacement admits a density whose support is $$[0,\infty )$$ [ 0 , ∞ ) . We study further examples of groups, some with random walks satisfying LLN for drift and other examples where such concentration phenomenon does not hold, and study relation of this property with asymptotic geometry of groups.

$$L^2$$-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 $$\ell ^2$$ ℓ 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 $$\ell ^2$$ ℓ 2 -Betti numbers arising from the lamplighter group algebra $${\mathbb Q}[{\mathbb Z}_2 \wr {\mathbb Z}]$$ Q [ Z 2 ≀ 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 $$\ell ^2$$ ℓ 2 -Betti numbers from the algebras $${\mathbb Q}[{\mathbb Z}_n \wr {\mathbb Z}]$$ Q [ Z n ≀ Z ] , where $$n \ge 2$$ n ≥ 2 is a natural number. We also apply the techniques developed to the generalized odometer algebra $${\mathcal {O}}({\overline{n}})$$ O ( n ¯ ) , where $${\overline{n}}$$ n ¯ is a supernatural number. We compute its $$*$$ ∗ -regular closure, and this allows us to fully characterize the set of $${\mathcal {O}}({\overline{n}})$$ O ( n ¯ ) -Betti numbers.

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

We 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.

Extensions of automorphisms of self-similar groups

In this work, we study automorphisms of synchronous self-similar groups and the existence of extensions to continuous automorphisms over the closure of these groups with respect to the depth metric. We obtain conditions for the continuity of such extensions, but we also construct examples of groups where such extensions do not exist. We study in detail the case of the lamplighter group L k = Z k ≀ Z \mathcal{L}_{k}=\mathbb{Z}_{k}\wr\mathbb{Z} .

Model theory and metric convergence II: Averages of unitary polynomial actions

We use model theory of metric structures to prove the pointwise convergence, with a uniform metastability rate, of averages of a polynomial sequence { T n } \{T_n\} (in Leibman’s sense) of unitary transformations of a Hilbert space. As a special case, this applies to unitary sequences { U p ( n ) } \{U^{p(n)}\} where p p is a polynomial Z → Z \mathbb {Z}\to \mathbb {Z} and U U a fixed unitary operator; however, our convergence results hold for arbitrary Leibman sequences. As a case study, we show that the non-nilpotent “lamplighter group” Z ≀ Z \mathbb {Z}\wr \mathbb {Z} is realized as the range of a suitable quadratic Leibman sequence. We also indicate how these convergence results generalize to arbitrary Følner averages of unitary polynomial actions of any abelian group G \mathbb {G} in place of Z \mathbb {Z} .

A property of the lamplighter group

We 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.

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

In 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.