Iterated Monodromy Groups

迭代单峰群

• 批准号: 0605019
• 负责人: Volodymyr Nekrashevych
• 金额: $ 9.62万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-09-01 至 2009-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0605019&HistoricalAwards=false
• 关键词: Iterated; Monodromy; Groups

Iterated monodromy groups{First abstract.} The project explores new connectionsbetween geometric group theory and dynamical systems via thenotion of the iterated monodromy group of a self-covering. Itrelates a very active branch of geometric group theory (automatagroups and groups acting on rooted trees) with symbolic andholomorphic dynamics. Before introduction of iterated monodromygroups, groups generated by finite automata were mainly isolatedexotic (counter) examples. It is clear now that such groups appearnaturally in connection with iterations of self-coverings oftopological spaces (in particular iterations of post-criticallyfinite rational functions). This opens new perspectives ofapplication of holomorphic dynamics to group theory. On the otherhand, computational effectiveness of automata groups helps tostudy deeper the combinatorics and symbolic dynamics ofself-coverings. For example, if the self-covering is expanding,then its Julia set is uniquely determined by the iteratedmonodromy group. The project suggests further investigation of theconnections between dynamical systems and automata groups. Wehope, in particular, to get new results in the theory of groupsacting on rooted trees, groups generated by automata, theory ofgrowth of groups, find new applications of geometric group theoryto combinatorics of iterations of rational functions andpolynomials.{Second abstract.} The project brings together two branchesof Mathematics, which where not so close before: geometric grouptheory and holomorphic dynamics. Geometric groups theory is a partof algebra which studies large-scale properties of groups ofsymmetries. Holomorphic dynamics studies iterations of rationaland polynomial functions and is a source of complex and beautifulfractal structures such as Mandelbrot and Julia sets. Iteratedmonodromy groups, introduced by the investigator, are naturallyassociated with rational iterations and are groups encoding in acompact symbolic form all the complex topological behavior of theiterations and geometry of their Julia sets. The construction ofthe iterated monodromy group is very natural, but groups obtainedin this way are very exotic from the point of view of classicalgroup theory. Difference between classical groups and iteratedmonodromy groups resembles in some way the difference between the``smooth'' geometric shapes of the classical geometry and fractal``wild'' shapes of holomorphic dynamics. Algebraic properties ofiterated monodromy groups remain to be rather mysterious and apart of the project is to understand them better. We are alsohoping that this new connection between group theory and dynamicalsystems will be fruitful for both parts of mathematics, that wewill be able to use group theory to obtain new results inholomorphic dynamics and to apply holomorphic dynamics to studynew classes of groups.

迭代单元群{第一摘要。}本项目通过自覆盖的迭代单元群的运动，探索几何群论和动力系统之间的新联系。它将几何群论的一个非常活跃的分支(自动机群和作用在有根树上的群)与符号动力学和全纯动力学联系起来。在引入迭代单元群之前，有限自动机生成的群主要是孤立的(反)例子。现在很清楚，这样的群与拓扑空间的自覆盖的迭代(特别是临界后有限有理函数的迭代)一起自然地出现。这为全纯动力学在群论中的应用开辟了新的视角。另一方面，自动机群的计算效率有助于更深入地研究自覆盖的组合学和符号动力学。例如，如果自覆盖是扩张的，那么它的Julia集是由迭代Mondromy群唯一确定的。该项目建议进一步研究动力系统和自动机群之间的联系。特别是，我们希望在作用于根树上的群论、自动机生成的群、群的增长论方面取得新的结果，找到几何群论在有理函数和多项式迭代的组合学中的新应用。{第二摘要}该项目汇集了数学的两个分支：几何群论和全纯动力学，这两个分支以前并不是那么密切。几何群论是研究对称群的大规模性质的代数的一部分。全纯动力学研究有理和多项式函数的迭代，是复杂和美丽的分形结构的来源，如Mandelbrot集和Julia集。由研究者引入的迭代单元群自然地与有理迭代联系在一起，是以紧符号形式编码迭代的所有复杂拓扑行为及其Julia集的几何的群。迭代单向群的构造是非常自然的，但从经典群论的观点来看，以这种方式得到的群是非常奇特的。经典群和迭代单调群之间的差异在某种程度上类似于经典几何的“光滑”几何形状和全纯动力学的“狂野”形状之间的差异。迭代单元群的代数性质仍然是相当神秘的，该项目的一部分是更好地理解它们。我们也希望群论和动力学系统之间的这种新的联系将对数学的两个部分都是有益的，我们将能够利用群论来获得全纯动力学的新结果，并将全纯动力学应用于研究新的群类。