Commensurated Subgroups and Ends of Groups

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 CommG(A), is {g ∈ G|(gAg−1) ∩A has finite index in both A and gAg−1}. It is straightforward to check that CommG(A) is a subgroup of G. A subgroup A is commensurated in G if CommG(A) = G. The centralizer of A in G is a subgroup of the normalizer of A in G which is a subgroup of CommG(A). We develop geometric versions of commensurators in finitely generated groups. In particular, g ∈ CommG(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.