Let $S = S_g$ be a closed surface.
An author of a paper writes:
We say $\langle a_1, b_1, \cdots, a_{2g}, b_{2g} \ | \ R \rangle$ is a geometric presentation of the fundamental group $\pi_1(S)$ if the corresponding one vertex 2-complex is homeomorphic to S
Could somebody clarify exactly what this means?
I am familiar with the canonical presentation of $\pi_1(S)$, $$\pi_1(S) = \langle a_1, b_1, \cdots, a_{2g}, b_{2g} \ | [a_1,b_1] \cdots [a_{2g}, b_{2g}] = 1 \rangle $$ and how it stems from idenitifying the sides of a hyperbolic 4g-gon, which labelled so that they spell out the product of the commutators $[a_1,b_1] \cdots [a_{2g}, b_{2g}]$, however I'm not sure how that translates to the definition above.
The definition of Cayley complex helps.
Briefly, given a presentation $\langle x_1,...,x_n|R_1,...,R_k\rangle $ of a group, you construct a 2-dim complex as folows:
i) take a single vertex $v$
ii) attach to $v$ a loop $\gamma_i$ for any generator $x_i$
iii) for any relation of the form $x_{i_1}\cdots x_{i_s}=1$ attach a polygon with $s$ egdes the complex, by labeling edges $e_1,...,e_s$ and attaching $e_r$ to $x_{i_r}$
The result is the Cayley complex. Now, given $S$ and a presentation of $\pi_1(S)$ you wonder if the Cayley complex is homeomorphic to $S$ or not.
If yes, the author of the paper you are reading call such a presentation geometric.
An example of a non-geometric presentation could be constructed for instance by artificially adding generators and relations that kills them.