A property of the lamplighter group

Abstract

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.A subgroup H of a group G is said to be inert if H and g −1 Hg are commensurate for all g ∈ G, meaning that H ∩ H g always has finite index in both H and H g . The terminology was introduced by Kegel and has been explored in many contexts (see, for example, the recent survey [8]). In abstract group theory, Robinson's investigation [13] focusses on soluble groups. Here we study a particular special family of soluble groups: the lamplighter groups and our interest is in the connection with the theory of totally disconnected locally compact groups. In that context, inert subgroups are particularly important in the light of van Dantzig's theorem that every totally disconnected locally compact group has a compact open subgroup and of course all such subgroups are commensurate with one another and therefore inert. It should be noted that in recent literature it is common to use the term commensurated in place of inert, see for example [5,6,7,10]. In §2 we remark a dynamical aspect of the property investigated here and relate it to the concept of algebraic entropy.The relation of commensurability is an equivalence relation amongst the subgroups of a group. By a class we shall here mean an equivalence class of subgroups under this relation. For a prime p, the corresponding lamplighter group is the standard restricted wreath product F p wr Z, i.e., the standard restricted wreath product of a group of order p by an infinite cyclic group. These are the simplest of soluble groups that fall outside the classes considered by Robinson [13]. Our main observation is as follows.Theorem. The inert subgroups of the lamplighter group F p wr Z fall into exactly five classes.

On uniformly fully inert subgroups of abelian groups

“…(a) In the language of non-commutative groups, the argument of Example 2.14 proves exactly that in the wreath product Z(p ) Z G ϕ a subgroup may be commensurable with a normal subgroup even if it is not commensurable neither with its normal closure nor with its normal core. Moreover, the subgroup G − is inert in G (in Belyaev's terminology) but not uniformly inert (for a similar example see also[5], where a quotient of this group is considered).…”