Show, using covering space theory, that a finitely presented group \(G\) has only finitely many distinct subgroups \(H\le G\) of index \(4\).
Solution
Choose a finite presentation \(G=\langle g_1,\ldots,g_m\mid r_1,\ldots,r_s\rangle\), and let \(X_G\) be its presentation complex: one \(0\)-cell \(v\), one oriented \(1\)-cell for each \(g_i\), and one \(2\)-cell for each \(r_j\), attached along the loop spelling \(r_j\). By the Seifert–van Kampen theorem, \(\pi_1(X_G,v)\cong G\).
1. Subgroups \(H\le G\) of index \(4\) correspond bijectively to isomorphism classes of connected \(4\)-sheeted covering spaces \(\widetilde X\to X_G\) with a chosen point \(\tilde v\) over \(v\).
\(X_G\) is a connected CW complex, so the classification of covering spaces applies: the based covering \((\widetilde X,\tilde v)\) corresponds to the subgroup \(p_*\pi_1(\widetilde X,\tilde v)\), whose index equals the number of sheets.
2. A connected \(4\)-sheeted covering with the fiber over \(v\) labeled \(\{1,2,3,4\}\) determines permutations \(\rho(g_1),\ldots,\rho(g_m)\in S_4\) satisfying \(r_j(\rho(g_1),\ldots,\rho(g_m))=1\) for every \(j\), and it is determined by them up to isomorphism.
Lifting the loop \(g_i\) from each point of the fiber ends at a point of the fiber; this defines the monodromy permutation \(\rho(g_i)\). The attaching loop of each \(2\)-cell lifts to closed loops, because the \(2\)-cell lifts, so each relator acts trivially. Conversely, the permutations determine the covering of the \(1\)-skeleton, which extends uniquely over the \(2\)-cells when the relators act trivially.
3. Q.E.D.
There are at most \(24^m\) choices of \((\rho(g_1),\ldots,\rho(g_m))\in S_4^m\), and for each, at most four choices of \(\tilde v\) in the fiber. By steps 1 and 2, \(G\) has at most \(4\cdot24^m\) subgroups of index \(4\).