Controlled coarse homology and isoperimetric inequalities
Controlled coarse homology and isoperimetric inequalities
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受控粗同源性和等周不等式
DOI：
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发表时间：
2008
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通讯作者：
Ján Špakula
中科院分区：
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作者：
Piotr W. Nowak;Ján Špakula
我们研究了一个具有特定增长条件的粗同调理论。对于具有字长度量的有限生成群G，这一同调理论与G的可修饰性有关。我们用G上的一个等周不等式刻画了同调中某一基本类的消失，并证明了在任何群上，这类基本类的消失至多需要线性控制。后者是无限群的经典Burnside问题的同调版本，具有正解。作为应用，我们刻画了具有指定增长的体积形式的基元的存在性，并证明了粗同调类阻碍了加权Poincaré不等式。
We study a coarse homology theory with prescribed growth conditions. For a finitely generated group G with the word length metric this homology theory turns out to be related to amenability of G. We characterize vanishing of a certain fundamental class in our homology in terms of an isoperimetric inequality on G and show that on any group at most linear control is needed for this class to vanish. The latter is a homological version of the classical Burnside problem for infinite groups, with a positive solution. As applications we characterize the existence of primitives of volume forms with prescribed growth and show that coarse homology classes obstruct weighted Poincaré inequalities.