Isospectrality and 3-manifold groups (Q2719024)

Using the Chern-Simons invariant and the \(\eta\)-invariant, the author develops a new invariant which can show that some finite groups are not fundamental groups of any closed \(3\)-manifold. First, define \(S(H)\) to be the number of \(2\)-torsion summands of the abelianization of a finitely generated group \(H\). It is known that the difference between the \(\eta\)-invariant and the Chern-Simons invariant of a closed \(3\)-manifold \(M\) is measured by \(S(\pi_1(M))\). Second, consider a finite group \(G\) and a pair of subgroups \(H\) and \(K\) of \(G\) which are almost conjugate. This means that they meet every conjugacy class of \(G\) in the same number of elements. For a homomorphism \(\varphi\colon\pi\to G\), define \(\Delta S(\pi;\varphi)\in {\mathbb Z}/2\) to be \(S(\varphi^{-1}(H))-S(\varphi^{-1}(K))\). When \(\pi\) is \(\pi_1(M)\), the finite coverings of \(M\) corresponding to the subgroups \(\phi^{-1}(H)\) and \(\phi^{-1}(K)\) are isospectral, since \(H\) and \(K\) are almost conjugate, so have the same \(\eta\)-invariant. Since they have the same degree as covers of \(M\), they also have the same Chern-Simons invariant. Consequently, \(\Delta S(\pi_1(M);\varphi)\) must equal \(0\) for all choices of \(\varphi\), \(H\), and \(K\). By constructing \(\varphi\), \(H\), and \(K\) for which \(\Delta S(\pi;\varphi)\neq 0\), the author is able to give a new proof that the Mathieu group \(M_{23}\) and the symmetric groups \(S_n\) with \(n\geq 16\) are not \(3\)-manifold groups. For determining which finite \(G\) are \(3\)-manifold groups, the key open case is the generalized quaternion groups \(Q(8a,b,c)\). The author finds that the invariant vanishes for these groups, so it does not resolve those cases.