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In this note we present a very simple method of proving that some hyperbolic manifoldsM have finite sheeted covers with positive first Betti number.
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The method applies to the standard arithmetic subgroups ofSO(n,1) (a case which was proved previously by Millson [Mi]), to the non-arithmetic lattices inSO(n,1) constructed by Gromov and Piatetski-Shapiro [GPS] and to groups generated by reflections.
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(HereM3 denotes the vector space of 3×3 matrices over k andp>amp;gt;3.) The method of proof involves an induction, and is potentially of wide applicability.
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The proof is based on a variant of Moser's method using time-dependent vector fields.
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In the second part we apply this method to obtain pseudo-Riemannian homogeneous manifolds with real Killing spinors.
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