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.

Since A is G-almost invariant, there is a finite dimensional subspace F of B such that Ax and Ax −1 are both contained in A + F . Let n + and n − denote the maximum and minimum integers in the finite set {j; x j belongs to the support of some non-zero element of F }.
We now distinguish two cases each of which has two subcases.
Case 1 A has an element a with upper degree ≥ n + .
By enlarging F if necessary, we may assume a has upper degree n + . We will define a sequence (a j ) j≥0 of elements of A inductively, starting with a 0 = a, so that a j has upper degree n + + j and so that for almost all j, a j has lower degree ≥ n − .
For j > 0, we suppose that a j−1 ∈ A has been chosen with upper degree n + + j − 1. Then a j−1 x has upper degree n + + j and a j−1 x = a j + f for some a j in A and f in F . Since n + + j > n + , the upper degree of a j is the same as that of a j−1 x, namely n + + j. The lower degree of a j−1 x is 1 greater than the lower degree of a j−1 and so the lower degree of a j is either greater than that of a j−1 or is ≥ n − . As a consequence the terms of the sequence (a j ) eventually all have lower degree ≥ n − . Now the span of the a j is almost equal to B + .
Subcase 1a A has an element with lower degree ≤ n − .
If this happens then the same reasoning as above produces a sequence with lower degrees decreasing by one and the terms of the sequence span a subspace almost equal to B − . It follows that A is almost equal to B.
Subcase 1b All elements of A have lower degree > n − .
In this case, A is almost equal to B + .
Case 2 All elements of A have upper degree < n + . Similar reasoning shows that either A is finite dimensional or it is almost equal to B − . Remark 1.1. Essentially the same strategy can be used to prove a more general result: suppose R is a commutative noetherian ring, G = x is infinite cyclic, and B = RG = n∈Z R. If A is a G-almost invariant R-submodule of B then A is almost equal to n<0 I ⊕ n>0 J, for some right ideals I, J of R.
Proof of the Theorem. Let B denote the base of the lamplighter group, i.e., B is the infinite direct sum Z F p of countably many copies of F p . This can be identified with the Laurent polynomial ring F p [x, x −1 ]. If H is an inert subgroup of G then H ∩ B is an x -almost invariant F p -subspace of B and so is almost equal to one of 0, B + , B − , B by the above lemma. Commensuration and almost equality are the same thing here because the ground field is finite.
If H has no elements of infinite order then H ⊆ B and we are done. If H has an element of infinite order and also an element of finite order then H ∩ B contains Laurent polynomials of arbitrarily large positive and arbitrarily large negative degrees. In this case H ∩ B has finite index in B and H has finite index in G. If H has an element of infinite order and no elements of finite order then it is infinite cyclic and it is not commensurated.
We thank Pierre-Emmanuel Caprace for pointing out that one now has the following consequence. Corollary 1. If G is a totally disconnected locally compact group which has a dense subgroup isomorphic to a lamplighter group then G is isomorphic to one of the following. 1. A discrete lamplighter group.

A compact group.
3. The group F p ((t)) ⋊ t Z for some prime p.
4. The unrestricted wreath product F p wr Z for some prime p.
In order to justify Corollary 1 we need a few preliminaries. Recall that a locally compact group G is totally disconnected if the identity 1 G is its own connected component. For a totally disconnected locally compact group G, van Dantzig's theorem ensures that the family of all compact open subgroups of G forms a base of neighbourhoods of 1 G . Therefore, every totally disconnected locally compact group G has a distinguished class CO(G) of inert subgroups, namely its compact open subgroups.
For a discrete group H, let φ : H → G be a group homomorphism with dense image in G. Such a homomorphism is referred to as a totally disconnected locally compact completion of H (a general framework for totally disconnected locally compact completions can be found in [12]). Note that φ is not required to be injective. For every U ∈ CO(G), the preimage φ 2. for each class form the Belyaev completion and classify its quotients with compact kernels.
If we start with the lamplighter group, which is residually finite, F p wr Z densely embeds in its Belyaev completions; see [2, Theorem 7.1]. Therefore, to obtain the exhaustive list in Corollary 1 it suffices to form the Belyaev completion of each pair (H, I) where I represents one of the 5 classes of inert subgroups of F p wr Z.

Connection with algebraic entropy
The concept of inert subgroup tacitly involves inner automorphisms and, therefore, it is amenable to being extended to the case of a general endomorphism ϕ of a group G: a subgroup H of G is said to be ϕ-inert if H ϕ ∩ H has finite index in the image H ϕ (see [9]). Consequently, a subgroup H is inert in G if H is ϕ-inert for every inner automorphism ϕ of G. The family of all ϕ-inert subgroups of G is denoted by I ϕ (G).
This definition can be easily adapted to the context of vector spaces: for an endomorphism φ of a denote the family of all φ-inert linear subspaces of V . Notice that LI φ (V ) ⊆ I φ (V ) whenever K is a finite field.
The notion of φ-inert subspace allows to point out a dynamical aspect of the Lemma above. Indeed, let β K : be the two sided Bernoulli shift on K. Then we have the following reformulation of the Lemma.
Several different notions of algebraic entropy have been introduced in the past (see [1,11,17,14] and references there). In particular, the possibility to define φ-inert subobjects has recently turned out to be a very helpful tool for the study of the dynamical properties of the given endomorphism φ. The leading example is the so-called intrinsic entropy ent. It was introduced in [9] to obtain a dynamical invariant able to treat also endomorphisms of torsion-free abelian groups where other entropy functions vanish completely for the lack of non-trivial finite subgroups. Afterwards, the intrinsic valuation entropy ent v was introduced in [15] with the aim of extending ent to the context of modules over a non-discrete valuation domain and also the algebraic entropy for locally linearly compact vector spaces defined in [4] has the same "intrinsic" flavour. Therefore, going down the same path, one defines the intrinsic dimension entropy ent dim for linear endomorphisms by In this new context, Corollary 2 can be then used to compute the intrinsic dimension entropy of the two sided Bernoulli shift β K , which turns out to equal 1. Indeed, Corollary 2 and a limit-free formula as in [4,15] provide ent dim (β K ) = dim K (V − /β −1 K (V − )) = 1. Quite remarkably, φ-inert subspaces do not enrich the dynamics of linear flows like φ-inert subgroups do in the framework of abelian groups (see [9]). Indeed, one verifies that ent dim (φ) = ent dim (φ) for every φ : V → V , where ent dim (φ) := sup lim n→∞ dim K (T n (φ, F )) n | F ≤ V and dim K (F ) < ∞ , which is a classical entropy function for vector spaces and their endomorphisms (details about this entropy function can be found in [3]). Indeed, since every finite-dimensional subspace is φ-inert, one easily has ent dim (φ) ≥ ent dim (φ). Conversely, proceeding as in [4, Lemma 3.9], for every U ∈ I φ (V ) one can find a finite-dimensional subspace F U such that T n (φ, U ) = U + T n (φ, F U ). Consequently, dim K (T n (φ, U )/U ) ≤ dim K (T n (φ, F U )) and ent dim (φ) ≤ ent dim (φ).
In other words, there are always enough finite-dimensional linear subspaces.