On Parabolic Subgroups and Hecke Algebras of Some Fractal Groups
On Parabolic Subgroups and Hecke Algebras of Some Fractal Groups
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关于一些分形群的抛物线子群和赫克代数
DOI：
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发表时间：
1999
期刊：
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通讯作者：
R. Grigorchuk
中科院分区：
文献类型：
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作者：
L. Bartholdi;R. Grigorchuk
我们研究了一些分支型分形群的子群结构、Hecke代数、拟正则表示和渐近性质。引入抛物子群，证明了抛物子群是弱极大的，并且相应的拟正则表示是不可约的。这些(无限维)表示用有限维准正则表示来逼近。与这些抛物子群相关的Hecke代数是可交换的，因此有限拟正则表示的不可约分量的分解由抛物子群的双陪集给出。由于我们的结果源于对有限指标子群的考虑，它们也适用于群G的无限完备化。所涉及的表示具有在数学中研究的有趣的谱性质。GR/9910102。本文是上述研究的群论对立面。我们更仔细地研究了几个分形群的例子，在这样做的过程中展示了第一个无挠分支-无限群的例子。我们还给出了中间增长的分枝刚无限群的一个新的例子，并给出了生成元和相关者的L式的表示。
We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are weakly maximal, and that the corresponding quasi-regular representations are irreducible. These (infinite-dimensional) representations are approximated by finite-dimensional quasi-regular representations. The Hecke algebras associated to these parabolic subgroups are commutative, so the decomposition in irreducible components of the finite quasi-regular representations is given by the double cosets of the parabolic subgroup. Since our results derive from considerations on finite-index subgroups, they also hold for the profinite completions $hat G$ of the groups G. The representations involved have interesting spectral properties investigated in math.GR/9910102. This paper serves as a group-theoretic counterpart to the studies in the mentionned paper. We study more carefully a few examples of fractal groups, and in doing so exhibit the first example of a torsion-free branch just-infinite group. We also produce a new example of branch just-infinite group of intermediate growth, and provide for it an L-type presentation by generators and relators.