Local commensurability graphs of solvable groups

Abstract

The commensurability index between two subgroups A, B of a group G is [A :This gives a notion of distance amongst finite-index subgroups of G, which is encoded in the plocal commensurability graphs of G. We show that for any metabelian group, any component of the p-local commensurabilty graph of G has diameter bounded above by 4. However, no universal upper bound on diameters of components exists for the class of finite solvable groups. In the appendix we give a complete classification of components for upper triangular matrix groups in GL(2, F q ).