The Lamplighter Group as a Group Generated by a 2-state Automaton, and its Spectrum
The Lamplighter Group as a Group Generated by a 2-state Automaton, and its Spectrum
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作为由 2 状态自动机生成的群的点灯者群及其频谱
DOI：
10.1023/a:1012061801279
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发表时间：
2001
影响因子：
0.5
通讯作者：
A. Zuk
中科院分区：
文献类型：
--
作者：
R. Grigorchuk;A. Zuk
摘要我们认识到，
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/2
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≀
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一个由两态自动机定义的群。我们研究了这个群在二叉树上和它的边界上的相应作用。最终的目标是计算一个特殊的生成元系统的马尔可夫（或随机游走）算子的谱，在这种情况下是[-1，1]，谱测度是一个离散测度，集中在[-1，1]中的一个稠密可数点集上（这是一个以前从未见过的马尔可夫算子在群上的新效应，它导致了强阿蒂亚猜想的反例）。这是通过计算频谱的有限维近似的运营商和使用的想法分形以类似的方式它是由巴塞洛迪和Grigorchuk用于计算频谱的一些分支集团。我们还得到了谱测度在1邻域内的e−1/1−x型渐近性，并证明了Følner集是指数增长的。
AbstractWe realize the lamplighter group
$$\mathbb{Z}$$
/2
$$\mathbb{Z}$$
≀
$$\mathbb{Z}$$
as a group defined by a 2-state automaton. We study the corresponding action of this group on a binary tree and on its boundary. The final goal is the computation for a special system of generators of the spectrum of the Markov (or the random walk) operator which is [−1,1] in this case and of the spectral measure which is a discrete measure concentrated on a dense countable set of points in [−1,1] (a new effect unseen before for Markovian operators on groups which leads to a counterexample to the Strong Atiyah Conjecture). This is done by the computation of spectra of finite-dimensional approximations of the operator and uses an idea of fractalness in a similar way it was used by Bartholdi and Grigorchuk for the computation of the spectra of some branch groups. We also obtain the asymptotic of type e−1/1−x of the spectral measure in the neighborhood of 1 and show that Følner sets grow exponentially.