Counting subgroups of the fundamental groups of certain 3-manifolds

The count of subgroups of finitely generated groups originates from famous M.Hall's formula for subgroups of the free groups. Motivated by topological applications (manifold coverings), this talk is devoted to a further development of our previous results and techniques for the fundamental groups of surfaces. We consider a well-known class of 3-manifolds: orientable circle bundles over compact surfaces. We find the number of subgroups of index n in their fundamental groups by expressing it as a linear combination of the numbers of surface subgroups of indices m|n. The respective coefficients are equal to (n/m)^{2-\chi m}, or vanish. In all cases, the number of subgroups turns out to be independent of the orientability of the base surface, closed or bordered.

We rely upon the known bijection between subgroups and transitive permutational representations of a group. In our case, the commutativity or "anti-commutativity" condition makes it possible to manipulate within the centralizers of regular permutations and to reduce the question to the count of solutions of certain non-homogeneous systems of linear congruences. Similar formulas have also been obtained for some non-orientable Seifert spaces without exceptional fibres. However, the whole class of Seifert fibre spaces still remains unsettled.

No preliminary knowledge of circle bundles is assumed. This is joint work with A.Mednykh (Novosibirsk).