Use covering space theory to prove that if \(H\) is a finite index subgroup of a finitely presented group, then \(H\) is finitely presented.
Solution
Let \[ d=[G:H]<\infty, \] and choose a finite presentation \[ G\cong\langle g_1,\ldots,g_n\mid r_1,\ldots,r_m\rangle. \]
1. There is a connected finite \(2\)-dimensional CW complex \(X\) with basepoint \(x_0\) such that\[ \pi_1(X,x_0)\cong G. \]
Form the presentation complex: take one \(0\)-cell, attach \(n\) oriented \(1\)-cells corresponding to the generators, and attach \(m\) \(2\)-cells along loops spelling the relators. The Seifert–van Kampen theorem gives the stated fundamental group. There are finitely many cells because the presentation is finite.
2. The subgroup \(H\le\pi_1(X,x_0)\) corresponds to a connected \(d\)-sheeted covering\[ p:(\widetilde X,\widetilde x_0)\longrightarrow(X,x_0) \]with\[ p_*\pi_1(\widetilde X,\widetilde x_0)=H. \]
The connected CW complex \(X\) is locally path-connected and semilocally simply connected, so the classification theorem for connected based covering spaces applies. It associates to \(H\) a connected based cover whose induced subgroup is \(H\).
For a connected covering, the number of sheets equals the index of its associated subgroup. Thus the fiber over \(x_0\) has \[ d=[G:H] \] points, and the cover has \(d\) sheets.
3. The CW structure of \(X\) lifts to a finite CW structure on \(\widetilde X\) with exactly \(d\) lifts of every cell of \(X\).
For an open cell \(e^k\subseteq X\), the restriction \[ p:p^{-1}(e^k)\longrightarrow e^k \] is a \(d\)-sheeted covering. Since the open cell \(e^k\) is homeomorphic to \(\mathbb R^k\) and hence simply connected, this restricted covering is trivial: its inverse image is the disjoint union of \(d\) components, each mapped homeomorphically onto \(e^k\).
Equivalently, a characteristic map \[ \chi:D^k\to X \] for the cell has one lift for each choice of a point over an interior basepoint, and the lifted boundary maps attach these \(d\) disks to the lifted lower skeleton. Doing this for every cell gives a CW structure upstairs in which each base cell has exactly \(d\) lifts.
The base \(X\) has finitely many cells by 1 and \(d<\infty\), so \(\widetilde X\) has finitely many cells.
4. The fundamental group of a connected finite CW complex is finitely presented.
Let \(Y\) be a connected finite CW complex. Choose a maximal tree \(T\) in its finite connected \(1\)-skeleton. After collapsing \(T\), the remaining finitely many \(1\)-cells form a wedge of finitely many circles and hence give finitely many generators.
Each of the finitely many \(2\)-cells has an attaching loop in the \(1\)-skeleton; after the tree collapse this loop gives one word in those generators. The Seifert–van Kampen theorem says that attaching the \(2\)-cells imposes the normal closures of these finitely many words. Attaching cells of dimension at least \(3\) does not change the fundamental group, because their attaching spheres \(S^{k-1}\) are simply connected for \(k\ge3\). Thus \(\pi_1(Y)\) has finitely many generators and finitely many relators.
5. The subgroup \(H\) is finitely presented.
By 3, \(\widetilde X\) is a connected finite CW complex, so 4 implies that \[ \pi_1(\widetilde X,\widetilde x_0) \] is finitely presented. For a covering map, \(p_*\) is injective, and by 2 its image is \(H\). Hence \[ H\cong\pi_1(\widetilde X,\widetilde x_0), \] so \(H\) is finitely presented.