Finite-dimensional representations constructed from random walks
Finite-dimensional representations constructed from random walks
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DOI：
10.4171/cmh/444
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发表时间：
2018-01-01
影响因子：
0.9
通讯作者：
Ozawa, Narutaka
中科院分区：
文献类型：
--
作者：
Erschler, Anna;Ozawa, Narutaka
给出一个系数为正交表示的1-上循环b，我们证明了b的每个有限维求和是上同调平凡的当且仅当||b(X-n)||(2)/n在概率上趋于常数，其中X-n是随机游动(G，Mu)的轨迹。作为推论，我们得到了G满足Shalom性质HFD的充分条件。另一个应用是对于没有无限虚拟阿贝尔商的任何有限生成的无限服从群，收敛到由其关于Mu*m的平均值归一化的Mu*(N)(E)-Mu*(N)(G)，n>；m的概率常数。最后，我们证明了G到Hilbert空间的调和等变映射作为归一化Mu*n-g Mu*n的U-超极限可以依赖于某些群的超滤子U。
Given a 1-cocycle b with coefficients in an orthogonal representation, we show that every finite dimensional summand of b is cohomologically trivial if and only if ||b(X-n)||(2)/n tends to a constant in probability, where X-n is the trajectory of the random walk (G, mu). As a corollary, we obtain sufficient conditions for G to satisfy Shalom's property HFD. Another application is a convergence to a constant in probability of mu*(n)(e)-mu*(n)(g), n >> m normalized by its average with respect to mu*m, for any finitely generated infinite amenable group without infinite virtually abelian quotients. Finally, we show that the harmonic equivariant mapping ofG to a Hilbert space obtained as an U-ultralimit of normalized mu*n - g mu*n can depend on the ultrafilter U for some groups.