On groups that can be covered by conjugates of finitely many cyclic or procyclic subgroups

Abstract

Given a discrete (resp. profinite) group $G$, we define $NCC(G)$ to be the smallest number of cyclic (resp. procyclic) subgroups of $G$ whose conjugates cover $G$. In this paper we determine all residually finite discrete groups with finite NCC and give an almost complete characterization of profinite groups with finite NCC.

v3: 30 pages, to appear in Math. Annalen. Major revision based on the referee's report. Sections 6 and 7 have been completely rewritten. We have also removed the results from the original paper dealing with NCC for families of finite p-groups or relied on computations in the group PGL_1(D) where D is the quaternion division algebra over Q_p. We plan to include those results in a follow-up paper