Commensurated subgroups and ends of groups

Abstract

Abstract. If G is a group, then subgroups A and B are commensurable if A \ B has finite index in both A and B. The commensurator of A in G, denoted by Comm G .A/, is ¹g 2 G j .gAg 1 / \ A has finite index in both A and gAgIt is straightforward to check thatWe develop geometric versions of commensurators in finitely generated groups. In particular, g 2 Comm G .A/ iff the Hausdorff distance between A and gA is finite. We show a commensurated subgroup of a group is the kernel of a certain map, and a subgroup of a finitely generated group is commensurated iff a Schreier (left) coset graph is locally finite. The ends of this coset graph correspond to the filtered ends of the pair .G; A/. This last equivalence is particularly useful for deriving asymptotic results for finitely generated groups. Our primary goals in this paper are to develop and compare the basic theory of commensurated subgroups to that of normal subgroups, and to initiate the development of the asymptotic theory of commensurated subgroups.

Continuous cocycle superrigidity for coinduced actions and relative ends

“…• Let K be a finitely generated normal subgroup of G such that G/K has one end. Then e(G, K) = e(G/K) = 1 by Proposition 2.5 (8).…”

“…One main property of commensurated subgroups we use is the following, see [8,Section 2]. Proposition 2.1.…”

“…Using these examples, one can build more examples using basic properties of commensurators, see e.g. [8,26].…”

“…Note that Geoghegan gave a topological description of e(G, K), which is called the number of filtered ends of (G, K) in [12, Section 14.5] and [8,Section 5].…”

A property of the lamplighter group

“…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.…”