There, $\gamma_{n}(G)=G_{n}$ it's just a description notation, I mean, this is like when you take a sequence and write $a_{n}=(whatever)$ but in this case, as the lower central series of a group is unique we can think in an "operator" family ($\gamma_{n}$) that sends every group to the $n-$th term of his lower central series ($G_{n}$).
Are there known conditions under which a Markov Chain is also a Martingale? I know only that the only Random Walk that is a Martingale is the symmetric one, i.e., p= 1-p =1/2.