According to a branch of rigidity theory, the “virtual” form of a very strong relation between these structures is implied by their quasi-isometry. One can define a left-invariant finitely additive measure on one of these objects, meaning it is amenable, (15[1])if and only if it cannot be paradoxically decomposed. Borel measures on these structures that are finite on compact sets are named for Alfréd Haar. The word metric on these structures is defined using the distance between vertices of a (*) Cayley graph. (-5[1])Reduced words define “free” examples of these structures. An exponential map from algebras into these structures is defined in a theory that treats them as differentiable manifolds. The equivalence classes of all loops under homotopy form the “fundamental” one of these structures. (10[2])For 10 points, "topological" examples of what structures, like GL(n) (“G L N”), were studied by Sophus Lie? (-5[1])■END■