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.
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 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…
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.
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…
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.
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 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 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.
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…
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.
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…
Arboreal Galois Groups of a PCF Map with Strictly Pre-periodic Critical Points
Koc University（科奇大学）；Adam Mickiewicz University（亚当·密茨凯维奇大学）；Bogazici University（博阿齐奇大学）；Johann Wolfgang Goethe-Universität（约翰·沃尔夫冈·歌德大学）；Academia Sinica, Institute of Mathematics（中央研究院数学研究所）
We show that if a group $G$ acting faithfully on a rooted tree $T$ has a free subgroup, then either there exists a point $w$ of the boundary $\partial T$ and a free subgroup of $G$ with trivial stabilizer of $w$, or there exists $w\in\partial T$ and a free subgroup of $G$ fixing $w$ and acting faithfully on arbitrarily small neighborhoods of $w$. This can be used to prove absence of free subgroups for different known classes of groups. For instance, we prove that iterated monodromy groups of expanding coverings have no free subgroups and give another proof of a theorem by S. Sidki.
To every automorphism w of an infinite rooted regular binary tree we associate a two variable generating function Φ_w that encodes information on the orbit structure of w. We prove that this is a rational function if w can be described by finitely many recursion relations of a particular form. We show that this condition is satisfied for all elements of the discrete iterated monodromy group Γassociated to a postcritically finite quadratic polynomial over C. For such Γwe also prove that there are only finitely many possibilities for the denominator of Φ_w, and we describe a procedure to determi…
In symmetric cryptography, the round functions used as building blocks for iterated block ciphers are often obtained as the composition of different layers providing confusion and diffusion. The study of the conditions on such layers which make the group generated by the round functions of a block cipher a primitive group has been addressed in the past years, both in the case of Substitution Permutation Networks and Feistel Networks, giving to block cipher designers the receipt to avoid the imprimitivity attack. In this paper a similar study is proposed on the subject of the Lai-Massey scheme,…
This paper is a survey, with few proofs, of ideas and notions related to self-similarity of groups, semi-groups and their actions. It attempts to relate these concepts to more familiar ones, such as fractals, self-similar sets, and renormalizable dynamical systems. In particular, it presents a plausible definition of what a "fractal group" should be, and gives many examples of such groups. A particularly interesting class of examples, derived from monodromy groups of iterated branch coverings, or equivalently from Galois groups of iterated polynomials, is presented. This class contains interes…
