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270 U. Bunke, T. Schick, M. Spitzweck, and A. ThomDefinition 4.2. A locally compact abelian topological group G 2 S lc is called admissibleon S lc if Ext i Sh Ab S lc.G; T/ Š 0 for i D 1; 2 (and correspondingly for S lc-acyc ).Lemma 4.3. Let f W S lc ! S be the inclusion. If F 2 Sh Ab S lc and f F is admissible,then F is admissible on S lc .Proof. This is an application of Corollary 3.33.Corollary 4.4. If G 2 S lc is admissible, then it is admissible on S lc .This is an application of Corollary 3.34.Lemma 4.5. The class of admissible sheaves is closed under finite products and extensions.Proof. For finite products the assertions follows from the fact that Ext ShAb S .:::;T/commutes with finite products. Given an extension of sheaves0 ! F ! G ! H ! 0such that F and H are admissible, then also G is admissible. This follows immediatelyfrom the long exact sequence obtained by applying Ext Sh Ab S .:::;T/.4.1.2 In this paragraph we formulate one of the main theorems of the present paper.Let G be a locally compact topological abelian group. We first recall some technicalconditions.Definition 4.6. We say that G satisfies the two-three condition, if1. it does not admit Q n2NZ=2Z as a subquotient,2. the multiplication by 3 on the component G 0 of the identity has finite cokernel.Definition 4.7. We say that G is locally topologically divisible if for all primes p 2 Nthe multiplication map p W G ! G has a continuous local section.Using that the class of admissible groups is closed under finite products and extensionswe get the following general theorem.Theorem 4.8. 1. If G is a locally compact abelian group which satisfies the two-threecondition, then it is admissible over S lc-acyc .2. Assume that(a) G satisfies the two-three condition,(b) G has an open subgroup of the form C R n with C compact such that G=C R nis finitely generated(c) the connected component of the identity of G is locally topologically divisible.Then G is admissible over S lc .