Alexander Mednykh
Sobolev Institute of Mathematics, Russia Counting conjugacy classes of subgroups in finitely generated group
Abstract: General formula for the number of conjugacy classes of subgroups of a given index in an arbitrary finitely generated group is obtained. As application we give complete solutions of the following three problems.
Problem 1. (Hurwitz enumeration problem) Define the number of coverings of a given multiplicity over closed and bordered Riemann surfaces.
Problem 2. (Tutte problem) Define the number of maps (=Dessins d'enfants) on a given Riemann surface up to homeomorphism.
Problem 3. (V.A. Liskovets problem) Define the number of orientable coverings of a given multiplicity over non-orientable manifold with a finitely generated fundamental group.