At Wikipedia's Erlangen program I read that "quite often, it appears there are two or more distinct geometries with isomorphic automorphism groups". Some examples are given.
But what are examples of geometries nevertheless being uniquely defined (up to isomorphism) by the isomorphism class of their automorphism group?
Are there examples of non-Euclidean geometries with an automorphism group isomorphic to the Euclidean group?