Invent. math. 119, 267 295 (1995) Inventiones mathematicae Springer-Verlag 1995 Subgroup growth and congruence subgroups Alexander Lubotzky Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel Oblatum 2-VII-1993 & 29-IV-1994 Summary. Let k be a global field, O its ring of integers, G an almost simple, simply connected, connected algebraic subgroups of GLm, defined over k and F = G(O) which is assumed to be infinite. Let a,(F) (resp. y,(F)) be the number of all (resp. congruence) subgroups of index at most n in F. We show: (a) If char(k) = 0 then: log 2 n log 2 n (i) C1 l ~n < log7,(F) < C 2 ~ for suitable constants C1 and C2. Under some mild assumptions we also have: (ii) F has the congruence subgroup property if and only if log~,(F) = o(log z n). (iii) If F is boundedly generated then F has the congruence subgroup property. (This confirms a conjecture of Rapinchuk I-R1] which was also proved by Platonov and Rapinchuk [PR3].) (b) If char(k) > 0 (and under somewhat stronger conditions on G) then for suitable constants C3 and C4, C3 log/n =< logT,(Y) < C410g 3 n. Introduction Let k be a global field, O its ring of integers, S a finite set of valuations containing all the archimedean ones S~, Os = {x ~ k lv(x) > O, Vvr G an almost simple, simply-connected, connected algebraic group defined over k with a fixed embedding into GL.,, and F -- G(Os). We assume throughout the paper that G(Os) is infinite, or equivalently that lqws G(kv) is not compact. For example, k = Q, S = { ~ }, G = SLn and F = SLa(7I). A subgroup H of /' is called a congruence subgroup if there exists a non-zero ideal ~2[ <10s such that H contains the kernel of the natural projection F(~I) = Ker(G(Os) -~ G(Os/9.1)). Let y,(F) (resp. a,(F)) denotes the number of congruence