  We obtain a slow exponential growth estimate for the spherical principal
series representation rho_s of Lie group Sp(n, 1) at the edge (Re(s)=1) of
Cowling's strip (|Re(s)|<1) on the Sobolev space H^alpha(G/P) when alpha is the
critical value Q/2=2n+1. As a corollary, we obtain a slow exponential growth
estimate for the homotopy rho_s (s in [0, 1]) of the spherical principal series
which is required for the first author's program for proving the Baum--Connes
conjecture with coefficients for Sp(n,1).
