Iterated Monodromy Groups Research Articles

Amenability of bounded automata groups on infinite alphabets

AbstractWe study the action of groups generated by bounded activity automata with infinite alphabets on their orbital Schreier graphs. We introduce an amenability criterion for such groups based on the recurrence of the first‐level action. This criterion is a natural extension of the result that all groups generated by bounded activity automata with finite alphabets are amenable. Our motivation comes from the investigation of iterated monodromy groups of entire transcendental functions in holomorphic dynamics.

Maximal subgroups of a family of iterated monodromy groups

Abstract The Basilica group is a well-known 2-generated weakly branch, but not branch, group acting on the binary rooted tree. Recently, a more general form of the Basilica group has been investigated by Petschick and Rajeev, which is an $s$ -generated weakly branch, but not branch, group that acts on the $m$ -adic tree, for $s,m\ge 2$ . A larger family of groups, which contains these generalised Basilica groups, is the family of iterated monodromy groups. With the new developments by Francoeur, the study of the existence of maximal subgroups of infinite index has been extended from branch groups to weakly branch groups. Here we show that a subfamily of iterated monodromy groups, which more closely resemble the generalised Basilica groups, have maximal subgroups only of finite index.

Iterated monodromy groups of exponential maps

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.

Iterated monodromy groups of rational functions and periodic points over finite fields

Iterated monodromy groups of rational functions and periodic points over finite fields

Iterated monodromy groups of Chebyshev-like maps on $\mathbb{C}^n$

Every affine Weyl group appears as the iterated monodromy group of a Chebyshev-like polynomial self-map of \mathbb{C}^n .

Geometry and Algebra of the Deltoid Map

The geometry of the deltoid curve gives rise to a self-map of that is expressed in coordinates by . This is one in a family of maps that generalize Chebyshev polynomials to several variables. We use this example to illustrate two important objects in complex dynamics: the Julia set and the iterated monodromy group.

Galois groups and Cantor actions

In this paper, we study the actions of profinite groups on Cantor sets which arise from representations of Galois groups of certain fields of rational functions. Such representations are associated to polynomials, and they are called profinite iterated monodromy groups. We are interested in a topological invariant of such actions called the asymptotic discriminant. In particular, we give a complete classification by whether the asymptotic discriminant is stable or wild in the case when the polynomial generating the representation is quadratic. We also study different ways in which a wild asymptotic discriminant can arise.

Maximal subgroups of groups of intermediate growth

Finding the number of maximal subgroups of infinite index of a finitely generated group is a natural problem that has been solved for several classes of “geometric” groups (linear groups, hyperbolic groups, mapping class groups, etc). Here we provide a solution for a family of groups with a different geometric origin: groups of intermediate growth that act on rooted binary trees. In particular, we show that the non-torsion iterated monodromy groups of the tent map (a special case of some groups first introduced by Šunić in [32] as “siblings of the Grigorchuk group”) have exactly countably many maximal subgroups of infinite index, and describe them up to conjugacy. This is in contrast to the torsion case (e.g. Grigorchuk group) where there are no maximal subgroups of infinite index. It is also in contrast to the above-mentioned geometric groups, where there are either none or uncountably many such subgroups.Along the way we show that all the groups defined by Šunić have the congruence subgroup property and are just infinite.

Exponential growth of some iterated monodromy groups

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 Sierpi\'{n}ski carpet Julia set, and obstructed Thurston maps. Furthermore, we construct the first example of a non-renormalizable polynomial with a dendrite Julia set whose $\operatorname{IMG}$ has exponential growth.

Fixed-point-free elements of iterated monodromy groups

The iterated monodromy group of a post-critically finite complex polynomial of degreed≥2d \geq 2acts naturally on the completedd-ary rooted treeTTof preimages of a generic point. This group, as well as its profinite completion, acts on the boundary ofTT, which is given by extending the branches to their “ends” at infinity. We show that, in most cases, elements that have fixed points on the boundary are rare, in that they belong to a set of Haar measure00. The exceptions are those polynomials linearly conjugate to multiples of Chebyshev polynomials and a case that remains unresolved, where the polynomial has a non-critical fixed point with many critical preimages. The proof involves a study of the finite automaton giving generators of the iterated monodromy group, and an application of a martingale convergence theorem. Our result is motivated in part by applications to arithmetic dynamics, where iterated monodromy groups furnish the “geometric part” of certain Galois extensions encoding information about densities of dynamically interesting sets of prime ideals.

Hyperbolic polynomial diffeomorphisms of [formula omitted]. III: Iterated monodromy groups

This paper is a sequel to Part I [13] and Part II [14,15]. In the current article we relate several combinatorial descriptions of the Julia sets for hyperbolic polynomial diffeomorphisms of C2: quotients of solenoids [3], automata [22] and Hubbard trees [14,15]. The notion of iterated monodromy groups are defined for such diffeomorphisms and are used to construct automata from Hubbard trees.

Introduction to Iterated Monodromy Groups

The theory of iterated monodromy groups was developed by Nekrashevych [9]. It is a wonderful example of application of group theory in dynamical systems and, in particular, in holomorphic dynamics. Iterated monodromy groups encode in a computationally efficient way combinatorial information about any dynamical system induced by a post-critically finite branched covering. Their power was illustrated by a solution of the Hubbard Twisted Rabbit Problem given by Bartholdi and Nekrashevych [2].

Free subgroups in groups acting on rooted trees

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 ∂T and a free subgroup of G with trivial stabilizer of w , or there exists w \in ∂T and a free subgroup of G fixing w and acting faithfully on arbitrarily small neighborhoods of w . This can be used to prove the 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.

Blocks of monodromy groups in complex dynamics

Motivated by a problem in complex dynamics, we examine the block structure of the natural action of iterated 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 sums of polynomials in the complex plane. We also show that degree 6 is exceptional in another regard, as it is the lowest degree for which the monodromy group of a polynomial is not determined by the combinatorics of the post-critical set. These results give new applications of iterated monodromy groups to complex dynamics.

A group of non-uniform exponential growth locally isomorphic to $IMG(z^2+i)$

We prove that a sequence of marked three-generated groups isomorphic to the iterated monodromy group of z 2 + i converges to a group of non-uniform exponential growth, which is an extension of the infinite direct sum of cyclic groups of order 4 by a Grigorchuk group.

Iterated monodromy groups of quadratic polynomials, I

We describe the iterated monodromy groups associated with post-critically finite quadratic polynomials, and make 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’.

Self-similar groups and their geometry

This is an overview of results concerning applications of self-similar groups generated by automata to fractal geometry and dynamical systems. Few proofs are given, interested reader can find the rest of the proofs in the monograph [Nek05]. We associate to every contracting self-similar action a topological space JG called limit space together with a surjective continuous map s : JG −→ JG. On the other hand if we have an expanding self-covering f : M1 −→ M of a topological space by its open subset, then we construct the iterated monodromy group (denoted IMG(f)) of f , which is a contracting self-similar group. These two constructions (dynamical system (JG, s) from a self-similar group and self-similar group IMG(f) from a dynamical system) are inverse to each other. The action of f on its Julia set is topologically conjugate to the action of s on the limit space JIMG(f) (see Theorem 6.1). We get in this way on one hand interesting examples of groups from dynamical systems (like the “basilica group” IMG ( z − 1 ) , which is a first example of an amenable group not belonging to the class of the sub-exponentially amenable groups). On the other hand, iterated monodromy groups are algebraic tools giving full information about combinatorics of self-coverings. The paper has the following structure. Section “Self-similar actions and automata” provides the basic notions from automata theory and theory of groups acting on rooted trees. It also gives some classical examples of self-similar groups. Section “Permutational bimodules” develops algebraic tools which are used in the study of self-similar groups. We define the notion of a permutational bimodule, which gives a convenient algebraic interpretation of automata. A closely related notion is virtual endomorphism, which can be used to construct explicit self-similar actions. We describe at the end of the section self-similar actions of free abelian groups and show how they are related to numeration systems on Z. Section 4 defines iterated monodromy groups. We show how to compute them (their standard actions) as groups generated by automata. Section 5 studies contracting self-similar actions and defines their limit spaces JG. We also prove some basic properties of the limit spaces, limit G-spaces and tiles. The last section shows connections of the obtained results with other topics of Mathematics. Subsection 6.2 shows that Julia sets of post-critically finite rational functions are limit spaces of their iterated monodromy groups. Next two subsections show a connection between topology of the limit spaces and a notion of bounded automata from [Sid00] and construct an iterative algorithm finding approximations of the limit space of actions by bounded automata. In Subsection 6.5 automata generating iterated monodromy groups of complex polynomials are described. We will see in particular, that iterated monodromy groups of complex polynomials are generated by bounded automata, so that the algorithm of the previous subsection can be used to draw topological approximations of the Julia sets of polynomials. We study in the last subsection the limit spaces of free Abelian groups and fit the theory of self-affine “digit” tilings in the framework of self-similar groups and their limit spaces.

Finitely ramified iterated extensions

Let K be a number field, t a parameter, F = K(t), and '(x) ∈ K(x) a polyno- mial of degree d ≥ 2. The polynomialn(x, t) = ' ◦n (x)−t ∈ F(x), where ' ◦n = '◦'◦� � �◦ ' is the n-fold iterate of ', is absolutely irreducible over F; we compute a recursion for its discriminant. Let F' be the field obtained by adjoining to F all roots (in a fixed F) of �n(x, t) for all n ≥ 1; its Galois group Gal(F'/F) is the iterated monodromy group of '. The iterated extension F' is finitely ramified over F if and only if ' is post-critically finite. We show that, moreover, for post-critically finite ', every specialization of F'/F at t = t0 ∈ K is finitely ramified over K, pointing to the possibility of studying Galois groups of number fields with restricted ramification via tree representations associated to iterated monodromy groups of post-critically finite polynomials. We discuss the wildness of rami- fication in some of these representations, describe prime decomposition in terms of certain finite graphs, and also give some examples of monogene number fields that arise from the construction.