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Let $G$ be the fundamental group of a closed surface of genus $g$. We know $G$ has a presentation $$\langle a_1,b_1,a_2,b_2,\dots,a_g,b_g \mid [a_1,b_1][a_2,b_2]\dots[a_g,b_g]=1 \rangle.$$ The deficiency of this presentation is $2g-1$.

The question is: Does there exist a presentation of $G$ like $\langle x_1,\dots,x_{2g} \mid r_1,\dots,r_m \rangle$ with $2g$ generators and $m>1$ relators such that each $r_i$ is not in the normal closure of $r_j$ for $i\neq j$? The deficiency of the presentation is less than $2g-1$.

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You can replace any relator $r$ in any group presentation by $k$ equivalent relators: just choose positive integers $n_1,\ldots,n_k$ with ${\rm gcd}(n_1,\ldots,n_k)=1$, for which any $k-1$ of the $n_i$ have a common factor, and replace $r$ by $r^{n_1},\ldots,r^{n_k}$. In most cases, including for the presentation of a surface group, the normal closure of any proper subset of the $r_i$ will not contain $r$.

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