THE HOWE-MOORE PROPERTY FOR REAL AND p-ADIC GROUPS
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Proof. If G is Howe-Moore, by Proposition 3.2, G is quasi-finite. Let K be the locally finite (= locally elliptic) radical of G, i.e. the subgroup generated by all finite normal subgroups of G. If K is finite, then it is clear that G/K is finitely generated and still Howe-Moore, hence simple. Otherwise, G is locally finite and by a classical result of Hall and Kulatilaka [HK], G is isomorphic to C p∞
i=1 v i w i the standard scalar product on K n . Let M be
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