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 (15[1])objects, meaning it is amenable, (15[1])if (15[1])and only if it cannot be paradoxically decomposed. Borel measures on these structures that are finite on compact sets are named for Alfréd Haar. (15[3])The (-5[1])word metric (15[1])on these structures is defined using (-5[1])the distance between vertices of a (*) Cayley graph. (-5[1])Reduced (10[1])words define “free” examples of these (10[2])structures. (10[3])An exponential map (10[1])from algebras into these structures is defined in a theory (10[1])that treats them as differentiable manifolds. The equivalence classes of all loops under homotopy form the “fundamental” one of these structures. (10[4])For 10 points, "topological" examples of what structures, like GL(n) (“G L N”), were studied by Sophus Lie? (10[1]-5[1])■END■