PRO-p GROUPS OF POSITIVE DEFICIENCY

Let Γ be a finitely presentable pro-p group such that def(Γ) > 0. If Γ has a nontrivial finitely generated closed normal subgroup N of infinite index then def(Γ) = 1, N is a free pro-p group and Γ/N is virtually free. If π is a finitely presentable group with β (2) 1 (π) = 0 then def(π) ≤ 1, with equality only if c.d.π = 2 or π ∼= Z. (See Theorem 2.5 of [1].) The L-Betti number condition holds if π has a finitely generated infinite normal subgroup of infinite index, or if π has an infinite amenable normal subgroup. We are interested in finding analogous results for pro-p groups. There is at present no good p-adic analogue of the von Neumann algebra, providing an invariant with the formal properties of the L-Betti numbers in the discrete case. However Schmidt has shown that if a finitely presentable pro-p group Γ has a finitely generated infinite closed normal subgroup N of infinite index then def(Γ) ≤ 1 [7]. We shall modify his argument to show that if def(Γ) = 1 then v.c.d.Γ ≤ 2. We shall also show that if a finitely presentable pro-p group has a nontrivial finite normal subgroup then it has deficiency ≤ 0. (I do not know whether the corresponding result holds in the discrete case.) If G is a pro-p group let h(G) = dimFpH (G;Fp). Then def(G) = h(G)− h(G). The partial Euler characteristic χ2(G) = 1− def(G) = 1− h(G) + h(G) is submultiplicative: if U is an open subgroup of G then χ2(U) ≤ χ2(G)[G : U ]. (See Chapter 3.3 of [5].) Theorem 1. Let Γ be a finitely presentable pro-p group with a nontrivial finite normal subgroup N . Then def(Γ) ≤ 0. Proof. Let A be a nontrivial abelian subgroup of N and let Γ = CΓ(A) and N ′ = N ∩ Γ. Then N ′ is central in Γ, which is a closed subgroup of finite index in Γ, and hence is also open. Since N ′ is finite the extension is classified by an element of H(Γ;N ) = lim −→ H(Γ/U ;N ). Thus on passing to an open subgroup U in Γ, if necessary, we may assume that the extension splits: U ∼= N ′ × U/N . But it is then easy 1991 Mathematics Subject Classification. 20E18.