geometric group theory

Problem 9. Let (R2, dE) denote the usual Euclidean metric space. Let O denote the origin, and let ((x, y) denote the straight line through the points x and y in the plane, if x * y. Recall that the 'railway metric' dSNCF: R2 X R2 -> R is defined as follows: Vx, y E R2 dSNCF(x, y) = 0 x = y f (2) = dSNCF(x, y) = dE(x, y) OEl(x, y ) dSNCF(x, y) = dE(x, O) + dE(O, y) Of l(x, y) (1) Recall that in a metric space (X, d), the closed ball of radius r 2 0 with center v E X is the set Ba(v, r) = {x EX |d(v, x)