Normal Subgroup Growth of Linear Groups: the (G2, F4, E8)-Theorem

Let Γ be a finitely generated residually finite group. Denote by sn(Γ) (resp. tn(Γ)) the number of subgroups (resp. normal subgroups) of Γ of index at most n. In the last two decades the study of the connection between the algebraic structure of Γ and the growth rate of the sequence {sn(Γ)}n=1 has become a very active area of research under the rubric “subgroup growth” (see [L1], [LS] and the references therein). The subgroup growth rate of a finitely generated group is bounded above by e , which is the growth rate for a finitely generated nonabelian free group. On the other end of the spectrum, the groups with polynomial subgroup growth (PSG-groups for short), i.e., those satisfying sn(Γ) ≤ n, were characterized ([LMS]) as the virtually solvable groups of finite rank. This was originally proved for linear groups ([LM]). The linear case was then used to prove the theorem for general residually finite groups. In recent years, interest has also developed in the normal subgroup growth {tn(Γ)}n=1. In [L3] it was shown that the normal subgroup growth of a nonabelian free group is of type n, just a bit faster than polynomial growth. One cannot, therefore, expect that the condition on Γ of being of “polynomial normal subgroup growth” (PNSG, for short) will have the same strong structural implications as that of polynomial subgroup growth. In particular, PNSG-groups (unlike PSG-groups) need not be virtually solvable. In fact, the examples produced in [S, Py] (which, incidentally, show that essentially every rate of subgroup growth between polynomial and factorial can occur) all have sublinear normal subgroup growth and are very far from being solvable. For linear groups, however, the situation is quite different. First fix some notations: Let F be an (algebraically closed) field and Γ a finitely generated subgroup of GLn(F ). Let G be the Zariski closure of Γ, R(G) the solvable radical of G, and G◦-the connected component of G. Write G =