A counter example for the Homogeneity Conjecture

TL;DR

This paper constructs a specific counterexample on the Lie group Sp(2) demonstrating that the Homogeneity Conjecture, which relates Clifford-Wolf translations to homogeneity, is false.

Contribution

It provides the first explicit counterexample to the Homogeneity Conjecture, showing that certain Clifford-Wolf translations do not imply homogeneity of the quotient space.

Findings

Counterexample on Sp(2) disproves the conjecture

Existence of a cyclic subgroup with special translation properties

Riemannian quotient is not homogeneous despite Clifford-Wolf translations

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.