Bounded defect in partial Euler characteristics

By Alexander Schmidt at Heidelberg

Let G be a pro-p group and suppose that the numbers h^{i}(G) = dim_{ℤ/pℤ} H^{i}(G, ℤ/pℤ) are finite for i = 0, …, n. Defining the partial Euler characteristic by χ_{n}(G) = ∑_{i=0}^{n}(−1)^{i}h^{i}(G) it is well known that for every open subgroup U ⊂ G the following inequality holds

χ_{n}(U)(−1)^{n} ≤ (G : U)χ_{n}(G)(−1)^{n}  (1)

and the cohomological dimension of G is less or equal to n if and only if equality holds in (1) for all U (See [2] Satz 5.5). In this paper we show that in principal this fact remains true if the defect in (1) remains bounded independent of the open subgroup U.

Proposition 1 Let G be a pro-p group such that H^{i}(G, ℤ/pℤ) is finite for 0 ≤ i ≤ n and assume that there is a constant N such that

χ_{n}(U)(−1)^{n} + N ≥ (G : U)χ_{n}(G)(−1)^{n}

for all open subgroups U ⊂ G. Then G is finite or cd(G) ≤ n.

Proof: Assume G to be infinite and let U ⊂ G be an open subgroup. Defining k ≥ 0 by

χ_{n}(U)(−1)^{n} + k = (G : U)χ_{n}(G)(−1)^{n},

we will show that k = 0. For an open subgroup V ⊂ U we have by assumption

(G : V)χ_{n}(G)(−1)^{n} − N ≤ χ_{n}(V)(−1)^{n}
≤ (U : V)χ_{n}(U)(−1)^{n}
= (U : V)((G : U)χ_{n}(G)(−1)^{n} − k)
= (G : V)χ_{n}(G)(−1)^{n} − (U : V)k.

Hence k(U : V) ≤ N and therefore k = 0 since we can choose V ⊂ U sufficiently small. □ The following corollary is the pro-p version of a well known result for discrete groups (see [3]).

Corollary 1 Every non-trivial finitely generated closed normal subgroup in a free pro-p group is open. In particular a free pro-p group of infinite rank has no finitely generated closed normal subgroups.

Proof of the corollary: Let F be a free pro-p group of finite rank r and let K be a finitely generated closed normal subgroup in F. We denote the quotient by G. For any open subgroup U ⊂ G we denote its preimage in F by F_{U}. Then (G : U) = (F : F_{U}) and we have the following commutative diagram with exact columns:

1 ⟶ K ⟶ F ⟶ G ⟶ 1
[figure: ∥; ∪; ∪]
1 ⟶ K ⟶ F_{U} ⟶ U ⟶ 1 .

By the Hochschild-Serre sequence for the lower column we have that h^{1}(F_{U})−h^{1}(U) ≤ h^{1}(K). Computing r(F_{U}) by the Nielsen-Schreier formula we get that G fulfills the conditions of the proposition with n = 1 and N = h^{1}(K). If G is finite then K is open. Otherwise G is free pro-p of positive rank. By the Nielsen-Schreier formula for G and F and by the above inequality the following holds for every open subgroup U in G:

(G : U)(h^{1}(F) − h^{1}(G)) = h^{1}(F_{U}) − h^{1}(U) ≤ h^{1}(K).

This implies that F and G have the same rank and therefore K is trivial in this case. This proves the corollary in the case of finitely generated free pro-p groups. If F is not finitely generated then one easily finds a finitely generated closed subgroup F_{0} ⊂ F containing K such that K is of infinite index in F_{0}. By the first part of the proof this is impossible. □

Corollary 2 Let G be a finitely generated, infinite pro-p group with h^{2}(G) = c − 1. If h^{1}(U) − c = (G : U)(h^{1}(G) − c) for every open subgroup U ⊂ G then cd(G) ≤ 2.

Proof: We have χ_{2}(U) = h^{2}(U) − h^{2}(G) + (G : U)χ_{2}(G) by assumption. Hence G fulfills the conditions of the proposition with n = 2 and N = h^{2}(G). □ Remarks: 1. Corollary 2 was also proved by U.Jannsen (unpublished). 2.(Jannsen) The only finite groups satisfying the assumptions of corollary 2 are G = 1 (with c = 1) and G = ℤ/2ℤ (with c = 2).

3. In the case c = 2 it is not difficult to prove that G is a Demushkin group (see [4], Lemma 4). However there exists a much stronger result in this case (see [1]).

References

[1] D.Dummit, J.Labute On a new characterization of Demushkin groups Invent. Math. 732(1983) 413-418

[2] H.Koch Galoissche Theorie der p-Erweiterungen Dt. Verl. d. Wiss. Berlin 1970

[3] O.Schreier Die Untergruppen der freien Gruppen Abh. Math. Sem. d. Univ. Hamburg 5(1928) 161-183 [4] K.Wingberg Positiv-zerlegte p-Erweiterungen algebraischer Zahlkörper

J. reine u. ang. Math. 357 (1985), 193-204

Alexander Schmidt Mathematisches Institut der Universität Im Neuenheimer Feld 288 D-69120 Heidelberg Germany e-mail: schmidt@harmless.mathi.uni-heidelberg.de