surface presentation Given the following group and presentation, how could I go about showing if there exists a compact surface with that fundamental group?
The group is $\big \langle a, b, c, d, e $ $\mid$ $e^2acbd^{-1}ba^{-1}dc^{-1} = 1\big \rangle$
I am aware of the various elementary transformations like cutting, pasting, rotation, reflection, etc., do not change the surface, but my question is: how and where do I begin?
I am also aware of the presentation for sphere, n-torus, and n-projective plane, and if the above is a surface, it would be one of them.
Thank you.
 A: To begin: compute the abelianization of this group, and compare it with the abelianizations of the $\pi_1$ of all the surfaces you know.  This will cut things down pretty drastically.
A: Take a decagon. Starting from one vertex and going around in counterclockwise order, label the 10 edges
$$e, e, a, c, b, d^{-1}, b, a^{-1}, d, c^{-1}
$$
Observe that, ignoring the $-1$ exponents, each label occurs exactly two times (this is the key observation). 
Thus, you can do pairwise identifications of the sides, making sure that for each letter, if the two occurrences of that letter have the same sign (namely $b$ and $e$) that the gluing takes the counterclockwise orientation to the counterclockwise orientation, and that if the two occurrences of that letter have opposite signs (namely $a,c,d$) that the gluing takes counterclockwise orientation to clockwise orientation.
You will now have constructed a surface $S$, and a 2-dimensional CW complex structure on $S$. IF you know one other piece of information, namely that all 10 of the vertices of the decagon are identified to a single vertex of the CW complex, then you'll be done: the presentation of $\pi_1 S$ coming from the CW complex is the presentation written down in your question.
So, is it true that all 10 of the vertices are identified to a single vertex?
