Use covering space theory to show that if \(H\) is a subgroup of index \(3\) in a finitely presented group \(G\), then \(H\) is finitely presented.
Solution
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\[ \pi_1(X,x_0)\cong G. \]
Start with one \(0\)-cell and attach \(n\) oriented \(1\)-cells, one for each generator \(g_i\). Then attach \(m\) \(2\)-cells along loops spelling the relators \(r_j\). The Seifert–van Kampen theorem gives exactly the chosen presentation for the fundamental group. Because the presentation has finitely many generators and relators, this presentation complex has finitely many cells.
2. The subgroup \(H\le G\cong\pi_1(X,x_0)\) determines a connected \(3\)-sheeted covering\[ p:(\widetilde X,\widetilde x_0)\longrightarrow(X,x_0) \]such that\[ p_*\pi_1(\widetilde X,\widetilde x_0)=H. \]
Every CW complex is path-connected componentwise, locally path-connected, and semilocally simply connected. Since \(X\) is connected, the covering-space classification theorem identifies subgroups of \(\pi_1(X,x_0)\) with connected based covering spaces of \(X\). Apply it to \(H\).
For a connected covering, the cardinality of the fiber over the basepoint equals the index of the associated subgroup. Thus \[ |p^{-1}(x_0)|=[G:H]=3, \] so the covering has three sheets.
3. The covering space \(\widetilde X\) admits a finite CW structure for which \(p\) maps every open cell homeomorphically onto an open cell of \(X\).
The CW structure of a base space lifts through a covering map. For completeness, proceed inductively over the cells of \(X\).
Over the unique \(0\)-cell there are exactly three points, giving three \(0\)-cells upstairs. Suppose the lift has been constructed over the \((k-1)\)-skeleton. For each open \(k\)-cell \(e^k\subseteq X\), choose a characteristic map \[ \chi:D^k\to X. \] Its restriction to the open disk is a homeomorphism onto \(e^k\). Because the open cell is simply connected and \(p\) is a three-sheeted covering, the inverse image \[ p^{-1}(e^k) \] is a disjoint union of three components, each mapped homeomorphically onto \(e^k\); equivalently, the characteristic map has three lifts after a choice of a point over its center. The lifted boundary maps attach these three disks to the already lifted \((k-1)\)-skeleton.
Thus every cell of \(X\) has exactly three lifted cells in \(\widetilde X\). Since \(X\) has finitely many cells by 1, \(\widetilde X\) has finitely many cells as well.
4. The fundamental group of a connected finite CW complex is finitely presented.
Let \(Y\) be a connected finite CW complex. Its finite connected \(1\)-skeleton \(Y^{(1)}\) contains a maximal tree \(T\). Collapse \(T\) to a point. The quotient of \(Y^{(1)}\) is a wedge of finitely many circles, one for each \(1\)-cell not in \(T\); hence these circles give finitely many generators of the fundamental group.
Each \(2\)-cell of \(Y\) has an attaching loop in the \(1\)-skeleton. After collapsing \(T\), that loop gives one word in the finite set of generators. By Seifert–van Kampen, attaching the finitely many \(2\)-cells imposes precisely the normal closures of these finitely many words. Cells of dimension at least \(3\) do not change the fundamental group: their attaching spheres \(S^{k-1}\) are simply connected for \(k\ge3\), so van Kampen adds no new generators or relations to \(\pi_1\). Consequently \(\pi_1(Y)\) has a presentation with finitely many generators and finitely many relators.
5. The subgroup \(H\) is finitely presented.
By 3, \(\widetilde X\) is a connected finite CW complex. Hence 4 shows that \[ \pi_1(\widetilde X,\widetilde x_0) \] is finitely presented. The covering homomorphism \(p_*\) is injective and, by 2, has image \(H\). Therefore \[ \pi_1(\widetilde X,\widetilde x_0)\cong H, \] so \(H\) is finitely presented.