On Turing dynamical systems and the Atiyah problem
On Turing dynamical systems and the Atiyah problem
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DOI：
10.1007/s00222-013-0497-5
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发表时间：
2010-04
影响因子：
3.1
通讯作者：
Łukasz Grabowski
中科院分区：
文献类型：
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作者：
Łukasz Grabowski
本文的主要定理是关于betti数可能值的Atiyah问题。证明了所有非负实数都是betti数，而“多”（例如所有非负代数）实数是关于自由紧致作用的单连通流形的betti数。并构造了一个关于点灯群的三重直接积作用的具有超越betti数的单连通流形的显式例子。主要的新思想是将图灵机嵌入到整群环中。开发的主要工具将已知的某些随机游走算子的谱计算技术推广到离散测量群类群环中的任意算子。
Main theorems of the article concern the problem of Atiyah on possible values of-Betti numbers. It is shown that all non-negative real numbers are-Betti numbers, and that “many” (for example all non-negative algebraic) real numbers are-Betti numbers of simply connected manifolds with respect to a free cocompact action. Also an explicit example is constructed which leads to a simply connected manifold with a transcendental-Betti number with respect to an action of the threefold direct product of the lamplighter group. The main new idea is embedding Turing machines into integral group rings. The main tool developed generalizes known techniques of spectral computations for certain random walk operators to arbitrary operators in groupoid rings of discrete measured groupoids.