In his book, D.Calegari proves the equivalence of the algebraic and geometric definitions of stable commutator length (Proposition 2.10, p. 15). I actually have some difficulties in understanding the proof. For example, when he says that the inequality in one direction is "obvious," he has the following argument:
$cl(a^n) \leq g$ if and only if there is an admissible map $f : S \rightarrow X$, where S has exactly one boundary component and satisfies $n(S) = n$ and $2g − 1 = -\chi^-(S)$. Hence $\lim_n cl(a^n)/n \geq \inf_S -\chi^- (S)/2n(S)$. [However, I do not see, why it is so.] We must have $$ \lim_n cl(a^n)/n \geq \inf_S (2g-1)/2n, $$
but why should this inequality hold?
In the second part of the proof, he also says that adding a $1$-handle to a surface increases genus by $1$ and reduces the number of boundary components by $1$. It is clear with genus, but why should it reduce the number of boundary components?