On the Basilica operation

IF 0.6 3区数学 Q3 MATHEMATICS Groups Geometry and Dynamics Pub Date : 2021-03-09 DOI:10.4171/ggd/702

Jan Moritz Petschick, Karthik Rajeev

引用次数: 4

Abstract

Inspired by the Basilica group B, we describe a general construction which allows us to associate to any group of automorphisms G ď AutpT q of a rooted tree T a family of Basilica groups BasspGq, s P N`. For the dyadic odometer O2, one has B “ Bas2pO2q. We study which properties of groups acting on rooted trees are preserved under this operation. Introducing some techniques for handling BasspGq, in case G fulfills some branching conditions, we are able to calculate the Hausdorff dimension of the Basilica groups associated to certain GGS-groups and of generalisations of the odometer, O m. Furthermore, we study the structure of groups of type BasspO mq and prove an analogue of the congruence subgroup property in the case m “ p, a prime.

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受Basilica群B的启发，我们描述了一个一般的结构，它允许我们将一个Basilica群族BasspGq, s P N '与根树T的任意自同构群G G G T T关联起来。对于二元里程计O2，有B " Bas2pO2q。我们研究了作用于根树的群在这种操作下保留了哪些性质。引入一些处理BasspGq的技术，在G满足某些分支条件的情况下，我们能够计算出与某些ggs群相关的Basilica群的Hausdorff维数，以及omom的推广。此外，我们研究了BasspGq型群的结构，并证明了在m ' p, a素数的情况下的同余子群性质的类似。

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来源期刊

Groups Geometry and Dynamics 数学-数学

CiteScore

1.10

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Groups, Geometry, and Dynamics is devoted to publication of research articles that focus on groups or group actions as well as articles in other areas of mathematics in which groups or group actions are used as a main tool. The journal covers all topics of modern group theory with preference given to geometric, asymptotic and combinatorial group theory, dynamics of group actions, probabilistic and analytical methods, interaction with ergodic theory and operator algebras, and other related fields. Topics covered include: geometric group theory; asymptotic group theory; combinatorial group theory; probabilities on groups; computational aspects and complexity; harmonic and functional analysis on groups, free probability; ergodic theory of group actions; cohomology of groups and exotic cohomologies; groups and low-dimensional topology; group actions on trees, buildings, rooted trees.

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