A counter example for the Homogeneity Conjecture

Abstract

We construct a counter example to show that the Homogeneity Conjecture, first proposed by J.A. Wolf in 1962, is not true. To be precise, we prove that on the Lie group Sp(2), there exists a left invariant Riemannian metric and a cyclic subgroup {\Gamma} of order (2n+1), such that the left translation of each element of {\Gamma} on Sp(2) is a Clifford-Wolf translation, but the Riemannian quotient {\Gamma}\Sp(2) is not homogeneous.