Embeddings of biuniform spaces into topological groups

Given a completely regular space X with two uniformities Ul and Ur both generating the original topology of X, we consider the question whether there exists a Hausdorff topological group G containing X as a subspace such that ∗VX = Ul and V∗X = Ur, where ∗V and V∗ are respectively the left and right group uniformities of G. We show that in general the answer is in the negative and present certain conditions implying the existence of an embedding of X to a topological group with the above properties. This approach enables us to conclude that the difference between the left and right indices of boundedness for subsets of a topological group can be arbitrary large.