Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds
Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds
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DOI：
10.1155/s1073792804131619
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发表时间：
2003-05
影响因子：
1
通讯作者：
S. Francaviglia
中科院分区：
文献类型：
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作者：
S. Francaviglia
令 W 为紧流形，并令 ρ 为其基本群在 PSL(2, C) 中的表示。然后，通过采用从通用覆盖 W~ 到 ℍ 3 的任何 ρ 等变映射，然后在基本域上积分双曲体积形式的回拉，来定义 ρ 的体积。事实证明，这样的体积并不取决于等变图的选择。邓菲尔德将这种构造扩展到非紧（尖点）流形 M 的情况，但他没有证明在所有情况下体积都被明确定义。我们在这里证明，表示的体积总是明确定义的，并且仅取决于表示。此外，我们表明，可以通过拉直 M 的任何理想三角剖分来轻松计算该体积。我们表明，表示的体积从上方受到 M 的相对单纯体积的限制。最后，我们证明了双曲流形基本群表示的刚性定理。也就是说，我们证明如果 M 是双曲线且 vol(ρ) = vol(M)，则 ρ 是离散且忠实的。
Let W be a compact manifold and let ρ be a representation of its fundamental group into PSL(2, C). Then the volume of ρ is defined by taking any ρ-equivariant map from the universal cover W~ to ℍ 3 and then by integrating the pullback of the hyperbolic volume form on a fundamental domain. It turns out that such a volume does not depend on the choice of the equivariant map. Dunfield extended this construction to the case of a noncompact (cusped) manifold M, but he did not prove that the volume is well defined in all cases. We prove here that the volume of a representation is always well defined and depends only on the representation. Moreover, we show that this volume can be easily computed by straightening any ideal triangulation of M. We show that the volume of a representation is bounded from above by the relative simplicial volume of M. Finally, we prove a rigidity theorem for representations of the fundamental group of a hyperbolic manifold. Namely, we prove that if M is hyperbolic and vol(ρ) = vol(M), then ρ is discrete and faithful.