Properties of isometrically homogeneous curves

This paper is devoted to the study of isometrically homogeneous spaces from the view point of metric geometry. Mainly we focus on those spaces that are homeomorphic to lines. One can reduce the study to those distances on $\mathbb{R}$ that are translation invariant. We study possible values of various metric dimensions of such spaces. One of the main results is the equivalence of two properties: the first one is linear connectedness and the second one is 1-dimensionality, with respect to Nagata dimension. Several concrete pathological examples are provided.