On the Higson corona of uniformly contractible spaces
On the Higson corona of uniformly contractible spaces
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关于均匀可收缩空间的希格森日冕
DOI：
10.1016/s0040-9383(97)00048-7
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发表时间：
1998
期刊：
影响因子：
--
通讯作者：
V. Uspenskij
中科院分区：
文献类型：
--
作者：
A. Dranishnikov;J. Keesling;V. Uspenskij
设X是固有度规空间νX是它的希格森电晕。证明了νX的覆盖维数不超过M. Gromov引入的X的渐近维数asdimX。特别地，它意味着对于Rn上的欧几里得度量和双曲度量，dim νRn= n。我们证明了对于具有词度量的有限生成群Γ ‘、Γ，不等式dimνΓ ’≤dimνΓ成立。我们还证明了几何有限群Γ在无穷远处对泛覆盖空间X = EΓ的紧化X ‘的一个小作用，可以将希格森紧化映射到X ’上。在这种情况下，X '的有理不周期性意味着S. Weinberger关于X的猜想，它是Γ的诺维科夫猜想的一种形式。
Let X be a proper metric space and let νX be its Higson corona. We prove that the covering dimension of νX does not exceed the asymptotic dimension asdimX of X introduced by M. Gromov. In particular, it implies that dim νRn= n for euclidean and hyperbolic metrics on Rn. We prove that for finitely generated groups Γ′ ⊃ Γ with word metrics the inequality dimνΓ′ ⩽ dim νΓ holds. Also we prove that a small action at infinity of a geometrically finite group Γ on some compactification X′ of the universal covering space X = EΓ enables one to map the Higson compactification onto X′. In that case the rational acyclicity of X′ implies the conjecture by S. Weinberger for X which is a form of the Novikov Conjecture for Γ.