Let M be an oriented complete hyperbolic n-manifold of finite volume. Using the definition of volume of a representation previously given by the authors in [3] we show that the volume of a representation \(\rho :\pi _1(M)\rightarrow \mathrm {Isom}^+({{\mathbb {H}}}^n)\), properly normalized, takes integer values if n is even and \(\ge 4\). If M is not compact and 3-dimensional, it is known that the volume is not locally constant. In this case we give explicit examples of representations with volume as arbitrary as the volume of hyperbolic manifolds obtained from M via Dehn fillings.

中文翻译：

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陈述量的完整性

令M为有限体积的定向完全双曲n流形。使用作者先前在[3]中给出的表示量的定义，我们表明表示量\（\ rho：\ pi _1（M）\ rightarrow \ mathrm {Isom} ^ +（{{\ mathbb如果n为偶数和\（\ ge 4 \），则经过正确归一化的{H}}} ^ n）\）将采用整数值。如果M不是紧凑的3维的，则已知体积不是局部恒定的。在这种情况下，我们给出了具有任意体积表示形式的显式示例，该体积与通过Dehn填充从M获得的双曲流形的体积一样。