So let $X$ be a finite CW complex which is connected.

Q1: Is $\pi_1(X)$ necssarily a finitely presented group?

If the answer is yes, then how does prove it. I've tried to prove it using an induction argument but I'm stuck... So every time one glues a cell then one needs to show that this only throws in finitely many new generators and finitely many new relations... If the argument is too involved then I'd like to have a reference.

For every finitely presented group $G=\langle g_1,\ldots g_n| R_1,\ldots R_m\rangle$ one may construct a connected finite CW complex $X$ having only cells of dimensions $\leq 2$ such that $\pi_1(X)\simeq G$. You have one circle for every generator and one 2-cell for every relation.

Q2: So did topologists try to prove results going in the other direction namely, say that $X$ is a finite connected CW complex with $n_i$ cells of dimension $i$ for $i\leq k$. Suppose that we know nothing about the incidence relations of these cells (except that $X$ is connected) then what can we say about the fundamental group of $X$ (outside the fact that it is finitely presented)?

One might ask similar questions where one imposes some incidence relations on the various cells etc.

randompresentations but it doesn't tell you much about any specific presentation. – Ryan Budney Mar 6 '11 at 21:13