多项式增长群具有性质A的初等证明方法
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摘要
在粗几何的研究中,顺从性可以推出性质A;具有性质A的离散度量空间可以粗嵌入到Hilbert空间;粗Baum-Connes猜想和Novikov猜想可以由粗嵌入性质推出.因此,顺从性和性质A在粗几何研究中占有重要地位.主要关注一种特殊的群,称为多项式增长群.仅从多项式增长群和性质A的基本定义出发,证明了多项式增长群具有顺从性和性质A.
Amenability and Property A are of importance in coarse geometry, since amenability implies property A, then, property A metric spaces coarsely embed into Hilbert spaces.Thus, coarse Baum-Connes conjecture and coarse Novikov conjecture can be implied from such coarse embeddability. This note focuses on a special kind of groups, called polynomial growth type groups. It proves that a polynomial growth type group has both amenability and property A only using the basic definitions of polynomial growth type group and property A.
Amenability and Property A are of importance in coarse geometry, since amenability implies property A, then, property A metric spaces coarsely embed into Hilbert spaces.Thus, coarse Baum-Connes conjecture and coarse Novikov conjecture can be implied from such coarse embeddability. This note focuses on a special kind of groups, called polynomial growth type groups. It proves that a polynomial growth type group has both amenability and property A only using the basic definitions of polynomial growth type group and property A.
引文
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[7]Nowak P W, Yu G. Large Scale Geometry[M]. Zurich:European Mathematical Society, 2012.
[2]Atiyah M F, Singer I M. The index of elliptic operators on compact manifolds[J]. Bulletin of the American Mathematical Society, 1963, 69:29-38.
[3]Roe J. Coarse cohomology and index theory on complete Riemannian manifolds[J]. Memoirs of the American Mathematical Society, 1993, 104(497):1-90.
[4]Gromov M. In Geometric Group Theory Vol.2[M]. Cambridge:Cambridge University Press, 1993.
[5]Yu G. The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space[J]. Inventiones Mathematicae, 2000, 139(1):201-240.
[6]Grigorchuk R. Milnor's problem on the growth of groups and its consequences[J]. Mathematics,2013, 5:1-61.
[7]Nowak P W, Yu G. Large Scale Geometry[M]. Zurich:European Mathematical Society, 2012.