Geometries expanding into a three-dimensional Euclidean space

The paper considers a general approach to defining and studying geometries expanding into 3-dimensional Euclidean space Е3 through the operation of 'gluing' space Е3 using uniformly discontinuous groups of its movements. As an example, there is provided construction of space Е31 obtained as a result of 'gluing' the space with the help of the group G1 = \T-a }. Affine and some metric properties of this space are considered, the group of its movements is studied.