  It is well known that when $G$ is the fundamental group of a closed surface
of negative Euler characteristic, it has the $R_{\infty}$ property. In this
work we compute the least integer $c$, {\it called the $R_{\infty}$-nilpotency
degree of $G$}, such that the group $G/ \gamma_{c+1}(G)$ has the $R_{\infty}$
property, where $\gamma_r(G)$ is the $r$-th term of the lower central series of
$G$. We show that $c=4$ for $G$ the fundamental group of any orientable closed
surface $S_g$ of genus $g>1$. For the fundamental group of the non-orientable
surface $N_g$ (the connected sum of $g$ projective planes) this number is
$2(g-1)$ (when $g>2$). A similar concept is introduced using the derived series
$G^{(r)}$ of a group $G$. Namely {\it the $R_{\infty}$-solvability degree of
$G$}, which is the least integer $c$ such that the group $G/G^{(c)}$ has the
$R_{\infty}$ property. We show that the fundamental group of an orientable
closed surface $S_g$ has $R_{\infty}$-solvability degree $2$.
