17.99 (2010)

Consider the group $B = \langle a, b \mid (bab^{-1})a(bab^{-1})^{-1} = a^2 \rangle$ introduced by Baumslag in 1969. The same relation is satisfied by the functions $f(x) = 2x$ and $g(x) = 2^x$ under the operation of composition in the group of germs of monotonically increasing to $\infty$ continuous functions on $(0, \infty)$, where two functions are identified if they coincide for all sufficiently large arguments. Is the representation $a \to f, b \to g$ of the group $B$ faithful?