Iterated Monodromy Groups

迭代单群{第一个摘要。该项目通过自覆盖的迭代单群的概念探索几何群论与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（自动机群和作用于有根树的群）与符号和全纯动力学联系起来。在引入迭代单群之前，有限自动机生成的群主要是孤立的反例。现在很清楚，这种群与拓扑空间的自覆盖的迭代（特别是后临界有限有理函数的迭代）自然地出现在一起。这为全纯动力学在群论中的应用开辟了新的前景。另一方面，自动机群的计算效率有助于深入研究自覆盖的组合学和符号动力学。例如，如果自覆盖是扩展的，那么它的Julia集是唯一由迭代一元群确定的。该项目建议进一步研究动力系统和自动机群之间的联系。我们特别希望在有根树作用的群理论、自动机产生的群理论、群的增长理论中得到新的结果，并找到几何群论在有理函数和多项式的迭代组合中的新应用。{第二抽象。该项目将数学的两个分支结合在一起，这两个分支以前并不那么接近：几何群论和全纯动力学。几何群理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函数和多项式函数的迭代，是复杂而美丽的分形结构（如Mandelbrot和Julia集）的来源。研究者引入的迭代单群与理性迭代自然相关，并且是用紧凑符号形式编码迭代的所有复杂拓扑行为及其Julia集合的几何形状的群。迭代单群的构造是很自然的，但从经典群论的角度来看，这样得到的群是很奇特的。经典群和迭代单群之间的区别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分形“野性”形状之间的区别。迭代单群的代数性质仍然是相当神秘的，并且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系统之间的这种新联系将对数学的两个部分都有成果，我们将能够利用群论在全纯动力学中获得新的结果，并将全纯动力学应用于研究新的群类。

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.