Commensurated subgroups and ends of groups. (Q2856532)

Let \(G\) be a group and let \(A,B\) be subgroups. \(A\) and \(B\) are \textit{commensurable} if \(A\cap B\) has finite index in both \(A\) and \(B\). The \textit{commensurator} of \(A\) in \(G\) is defined to be NEWLINE\[NEWLINE\{g\in G\mid (gAg^{-1})\cap A\text{ has finite index in both }A\text{ and }gAg^{-1}\}NEWLINE\]NEWLINE and denoted by \(\text{Comm}_G(A)\) which is a subgroup of \(G\). A subgroup \(A\) of \(G\) is \textit{commensurated} in \(G\) if \(\text{Comm}_G(A)=G\). The normalizer of \(A\) in \(G\) is a subgroup of \(\text{Comm}_G(A)\).NEWLINENEWLINE The authors show that a subgroup \(A\) of a finitely generated group \(G\) is commensurated if and only if the Hausdorff distance between \(A\) and \(gA\) is finite for every \(g\in G\). It is also shown that the intersection of two commensurated subgroups is commensurated. The union of a commensurated subgroup and a normal subgroup generates a commensurated subgroup. The image of a commensurated subgroup under an epimorphism is commensurated. The authors also study commensurated subgroups in amalgamated products and HNN extensions of groups and in word hyperbolic groups.NEWLINENEWLINE Commensurated subgroups of finitely generated groups are characterized as kernels of certain maps and in Schreier coset graphs. The authors show that if \(A\) is a commensurated subgroup of \(G\) then \(G\) acts transitively on a left coset graph and this graph has either 0, 1, 2 or an uncountable number of ends. A connection between the bounded packing ideas of Hruska and Wise and a left Schreier coset graph is examined.