Kai-Uwe Bux, Rodrigo Perez
Nekrashevych conjectured that the iterated monodromy groups of quadratic polynomials with preperiodic critical orbit have intermediate growth. We illustrate some of the difficulties that arise in attacking this conjecture and prove subexponential growth for the iterated monodromy group of $z^2+i$. This is the first non-trivial example supporting the conjecture.
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This paper discusses iterated monodromy groups for transcendental functions. We show that for every post-singularly finite entire transcendental function, the iterated monodromy action can be described by bounded activity automata of a special form, called "dendroid automata". In particular, we conclude that the iterated monodromy group of a post-singularly finite entire function is amenable if and only if the monodromy group is.
We describe the iterated monodromy groups associated with post-critically finite quadratic polynomials, and explicit their connection to the `kneading sequence' of the polynomial. We then give recursive presentations by generators and relations for these groups, and study some of their properties, like torsion and `branchness'.
This paper introduces iterated monodromy groups for transcendental functions and discusses them in the simplest setting, for post-singularly finite exponential functions. These groups are self-similar groups in a natural way, based on an explicit construction in terms of kneading sequences. We investigate the group theoretic properties of these groups, and show in particular that they are amenable, but they are not elementary subexponentially amenable.
We associate a group $IMG(f)$ to every covering $f$ of a topological space $M$ by its open subset. It is the quotient of the fundamental group $π_1(M)$ by the intersection of the kernels of its monodromy action for the iterates $f^n$. Every iterated monodromy group comes together with a naturally defined action on a rooted tree. We present an effective method to compute this action and show how the dynamics of $f$ is related to the group. In particular, the Julia set of $f$ can be reconstructed from $\img(f)$ (from its action on the tree), if $f$ is expanding.
A just-infinite iterated monodromy group without the congruence subgroup property
本研究证明多项式z²+i的迭代单值群具备刚无限、正则分支且无同余子群性质，给出了此类多项式迭代单值群的首个实例，并补充了该群相关核的信息。
Iterated monodromy groups of postcritically-finite rational maps form a rich class of self-similar groups with interesting properties. There are examples of such groups that have intermediate growth, as well as examples that have exponential growth. These groups arise from polynomials. We show exponential growth of the $\operatorname{IMG}$ of several non-polynomial maps. These include rational maps whose Julia set is the whole sphere, rational maps with Sierpiński carpet Julia set, and obstructed Thurston maps. Furthermore, we construct the first example of a non-renormalizable polynomial with…
We provide a self-similar measure for the self-similar group $G$ acting faithfully on the binary rooted tree, defined as the iterated monodromy group of the quadratic polynomial $z^2+i$. We also provide an $L$-presentation for $G$ and calculations related to the spectrum of the Markov operator on the Schreier graph of the action of $G$ on the orbit of a point on the boundary of the binary rooted tree.
Let $π:X\to Y$ be an $n$-dimensional iterated Kodaira fibration with fiber of genus $g$ and injective monodromy. Llosa Isenrich and Py proved that we can pass to a finite index subgroup of $π_1(X)$ to get the base space of an n+1-dimensional iterated Kodaira fibration with injective monodromy and they asked about bounding the index of such a group. We provide a bound on this index.
Profinite geometric iterated monodromy groups of postcritically finite polynomials in degree 3
本文研究数域上三次后临界有限多项式的投射有限几何迭代单值群，证明其在特定条件下有限不变生成，并由分歧画像的同构类决定，同时刻画了其正则分支结构与挠元素性质。
Maximal subgroups of a family of iterated monodromy groups
针对迭代单值群中与广义Basilica群相似的子群族，结合分支群到弱分支群的极大子群研究进展，证明该子群族的极大子群仅具有限指数。
We study in detail the profinite group G arising as geometric étale iterated monodromy group of an arbitrary quadratic morphism f with an infinite postcritical orbit over a field of characteristic different from two. This is a self-similar closed subgroup of the group of automorphisms of a regular rooted binary tree. In many cases it is equal to the automorphism group of the tree, but there remain some interesting cases where it is not. In these cases we prove that the conjugacy class of G depends only on the combinatorial type of the postcritical orbit of f. We also determine the Hausdorff di…
We study in detail the profinite group G arising as geometric étale iterated monodromy group of an arbitrary quadratic polynomial over a field of characteristic different from two. This is a self-similar closed subgroup of the group of automorphisms of a regular rooted binary tree. (When the base field is \C it is the closure of the finitely generated iterated monodromy group for the usual topology which is also often studied.) Among other things we prove that the conjugacy class and hence the isomorphism class of G depends only on the combinatorial type of the post-critical orbit of the polyn…
We give two new examples of groups of intermediate growth, by a method that was first used by Bux and Pérez. Our examples are the groups generated by the automata with the kneading sequences 11(0) and 0(011). By results of Nekrashevych, both of these groups are iterated monodromy groups of complex post-critically finite quadratic polynomials. We include a complete, systematic description of the Bux-Pérez method. We also prove, as a sample application of the method, that the groups determined by the automata with kneading sequence 1(0^k) (k >= 2) have intermediate growth, although this result i…
We define iterated monodromy groups of more general structures than partial self-covering. This generalization makes it possible to define a natural notion of a combinatorial model of an expanding dynamical system. We prove that a naturally defined "Julia set" of the generalized dynamical systems depends only on the associated iterated monodromy group. We show then that the Julia set of every expanding dynamical system is an inverse limit of simplicial complexes constructed by inductive cut-and-paste rules.
Motivated by a problem in complex dynamics, we examine the block structure of the natural action of monodromy groups on the tree of preimages of a generic point. We show that in many cases, including when the polynomial has prime power degree, there are no large blocks other than those arising naturally from the tree structure. However, using a method of construction based on real graphs of polynomials, we exhibit a non-trivial example of a degree 6 polynomial failing to have this property. This example settles a problem raised in a recent paper of the second author regarding constant weighted…
Liouville property for groups and conformal dimension
本文证明：若迭代单值群的底层空间共形维数小于2，则该群上所有有限二阶矩对称随机游走具有Liouville性质，从而群可均，并应用于次双曲有理函数的Julia集。
Given a fibration over the circle, we relate the eigenspace decomposition of the algebraic monodromy, the homological finiteness properties of the fiber, and the formality properties of the total space. In the process, we prove a more general result about iterated group extensions. As an application, we obtain new criteria for formality of spaces, and 1-formality of groups, illustrated by bundle constructions and various examples from low-dimensional topology and singularity theory.
On subgroup growth of iterated wreath products in product action
本文研究了通过迭代wreath产物在积作用下得到的hereditarily just infinite profinite群的子群增长类型，证明了存在任意子群增长类型的群。
Branch iterated Galois groups with positive fixed-point proportion and positive Hausdorff dimension
本文研究了后临界无限unicritical多项式的算术 profinite 迭代 monodromy 群，证明其具有正豪斯多夫维度和正固定点比例，扩展了正分支群的研究范围。
Arboreal Galois Groups of a PCF Map with Strictly Pre-periodic Critical Points
本文研究了定义在数域k上的后临界有限二次有理函数f(x)=2/(x-1)^2的算术和几何迭代monodromy群，其临界点均为严格预周期。研究给出了几何迭代monodromy群的拓扑生成元的显式递归描述，并证明算术迭代monodromy群的Hausdorff维数为零。还给出了确定关联的树状Galois群达到最大可能大小的显式判据，并证明最大性可以在第四层验证，具有计算可行性。最后确定了算术迭代monodromy群的常数域与k(μ_{2^{\infty}})的交集，首次全面研究了具有非阿贝尔常数域的PCF二次映射。
On polynomial free-by-cyclic groups
研究 free-by-cyclic 群中自由群自同构单值性的动力学性质在拟等距下的不变性，利用循环分裂和细长分裂证明并刻画了单值性增长类型作为几何不变量的特性。
Given a sequence of $(G_i)_{i \in \N}$ of finite transitive groups of degree $n_i$, let $W_\infty$ be the inverse limit of the iterated permutational wreath products of the first m groups. We prove that $W_\infty$ is (topologically) finitely generated if and only if $\prod_{i=1}^{\infty} (G_i/G_i')$ is finitely generated and the growth of the minimal number of generators of $G_i$ is bounded by $ d...n_1...n_{i-1}$ for a constant $d$. Moreover we give a criterion to decide whether $W_\infty$ is positively finitely generated.
Torsion homology growth of polynomially growing free-by-cyclic groups
本研究证明多项式增长单值群的自由循环群同调挠增长各维度均为零且与Farber链选取无关，推导出整挠等于ℓ²-挠，验证了Lück的相关猜想。
Virtual retracts in groups acting on rooted trees
本文研究群作用于根树上的虚拟 retract，证明有限生成分支群不具有 LR 性质，并揭示 LR 性质在后临界有限二次复多项式中与欧几里得 orbifold 的关系，同时提供新的 pro-2 群例子。
Maximal subgroups of groups of intermediate growth
本文解决了作用于有根二叉树的中间增长群的无穷指数最大子群数量问题，证明帐篷映射的非挠迭代单项群恰有可数无穷多个此类子群，并证实Šunić定义的所有群均具有同余子群性质且恰好无穷。
Settled Elements in Arboreal Galois Groups of Quadratic PCF Polynomials
研究二次PCF多项式迭代单子群的定居性质，证明其在数域中存在无限多个参数使对应的树形伽罗华表示密集定居。
Quasi-isometric rigidity for lamplighters with lamps of polynomial growth
本文证明多项式增长灯泡群的拟等距具有测度缩放刚性，缩放因子由增长次数决定，并应用于构造非双Lipschitz等价的群对及迭代圈积分类，首次给出灯泡刚性群例子。
Galois theory of quadratic rational functions with periodic critical points
针对二次有理函数的伽罗瓦树表示，本文通过研究迭代单值群，展示了临界点周期的二次有理函数的树伽罗瓦群非稠密稳定，否定相关问题并证明特定常数字段猜想。
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