Iterated monodromy groups

Iterated monodromy groups (IMGs) are a novel and promising connection between geometric group theory and dynamical systems (developed by Nekrashevych who wrote his fundamental monograph about 10 years ago here in Bremen). Every branched cover  f: X \to X of a topological space X has an associated iterated monodromy group \text{IMG}(f). If f is a postcritically finite rational map over \mathbb{C}, then \text{IMG}(f) encodes in a computationally efficient way combinatorial information about f and how it acts topologically on the Julia set of f.

IMGs relate a very active branch of geometric group theory (automata groups and groups acting on rooted trees) with symbolic and holomorphic dynamics. At this time, it is already evident that the IMGs are very important for both mathematical areas. They have been used to solve open questions in holomorphic dynamics like the Hubbard Twisted Rabbit Problem (see "Algorithmic aspects" below) and help to construct important examples which lead to development of new machinery for answering group theoretical questions (addressed in the "Algebraic properties of IMGs" section below).

Characterisation of the IMGs

One of Nekrashevych’s achievements is that he managed to find a complete description of groups that arise as IMGs of postcritically finite polynomials in terms of a specific class of finite automata, called kneading automata. There is no known characterization of automata generating IMGs of post-critically finite rational maps. The difference between the polynomial and the rational case is the fact that polynomials admit a concise combinatorial description (e.g., in terms of spiders), while no such description is known for general rational functions.

However, our group managed to extend the latter result to the class of expanding Thurston maps introduced by Bonk and Meyer (and also by Haïssinsky and Pilgrim). Examples of expanding Thurston maps are given by postcritically finite rational maps whose Julia sets are the whole Riemann sphere. Hlushchanka (partly joined with Meyer) showed that for every sufficiently large iterate F=f^n of an expanding Thurston map f there exists an F-invariant star T (i.e., a tree with a single branch point), s.t. P_f are leaves of T. This result was recently extended to the case of rational maps with Sierpiński carpet Julia set. This essentially gives a description of the map F and its dynamics in terms of finite combinatorial data (a cellular Markov partition). Moreover, it provides sufficient information and the easiest way for computing the action of iterated monodromy groups.

Algebraic properties of IMGs

Before IMGs were introduced, groups generated by finite automata served mainly as isolated exotic (counter)examples; now they appear naturally by iteration of postcritically finite rational functions. For instance, \text{IMG}(z^2+i) is a group of intermediate growth and \text{IMG}(z^2-1) is an amenable group of exponential growth. Unfortunately, we still lack of general theory which would unify and explain these nice examples. The only known general result is due to Nekrashevych and is the following: if two bounded Fatou components of a polynomial have intersecting closures then the IMG of this polynomial has exponential growth. In that way, the result relates the geometry (topology) of the Julia sets with algebraic properties of IMGs. It is conjectured that the IMGs of postcritically finite polynomials with a dendrite Julia set, so that the postcritical points do no separate it, have intermediate growth (a motivating example is the IMG of z^2+i).

"What is the growth of IMG of postcritically finite polynomial/rational/Thurston maps? What is the relation between the topological structure of the limit space of a contracting self-similar group and the growth of the group?" —  those are some of the questions that our group attempts to answer. Recently, we managed to show that the IMGs of two new families of rational maps have exponential growth. These maps have either empty Fatou set or Sierpinski carpet Julia set (i.e., the statement does not follow from the Nekrashevych result). The proof uses the geometry of maps coming from the invariant trees/curves for their dynamics (we talked about this in "Characterisation of the IMGs" section). One of our current goals is to adapt our method at least to the class of (rational) expanding Thurston maps.

At the same time, we study the amenabilty properties of the IMGs. Nekrashevych (together with Bartholdi and Kaimanovich) showed that a large class of self-similar groups satisfy an amenability condition. In particular, the iterated monodromy groups of all postcritically finite polynomials are amenable (for this he showed that IMGs of polynomials are generated by bounded automata). Amenability of IMGs of rational maps still remains open. As Nekrashevych's result shows, the description of IMGs may be very important for this matter. This description we already established for expanding Thurston maps.

Algorithmic aspects

Hubbard famously posed the "twisted rabbit problem", which asks for the combinatorial class of the rabbit polynomial f_R postcomposed by powers of a Dehn twist; one could ask the same question when the Dehn twist is replaced by any homeomorphism h that fixes the postcritical set.  The problem (and its defiance of a solution for over a decade) highlighted the lack of invariants for combinatorial equivalence; its celebrated solution came in work of Bartholdi and Nekrashevych (2006).  A certain biset associated to f_R was shown to be subhyperbolic, which means that there is a finite set of representatives equivalent to h\circ f_R for all h.  Finally it is shown that the members of this finite set can be distinguished in terms of the nucleus of their iterated monodromy groups.

The methods described here have no clear generalization to the case when f is a postcritically finite rational map or a post-singularly finite transcendental map, and our group wishes to investigate this.  For instance, is the f-biset always subhyperbolic?  What is an algorithmically efficient way to distinguish bisets?  A promising invariant comes in work of Lodge who solves the twisting problem for f(z)=\frac{2z^3}{3z^2+1} in terms of a new biset invariant.  The invariant is essentially the dynamics of multicurves under f-preimage or alternatively the dynamics of the extension (shown to exist by Selinger) of Thurston's pullback map to augmented Teichmuller space.

Dudko managed to use spiders to compute the IMG (or, more precisely, the bisetassociated with a polynomial; he uses it to compute the external angles describing (in Poirier's formulation) a topological polynomial. This can be also used for many combinatorial operations with polynomials, such as a composition, renormalization, tuning. Moreover, Bartholdi managed to implement the Thurston iteration scheme into a GAP program package that allows to compute the rational map out of its biset. The latter generalizes Thurston-Hubbard-Schleicher spider algorithm.

Our group together with Laurent Bartholdi (University of Göttingen) addresses many other algorithmic aspects of the IMG theory. To list just a couple of them: "Can one check whether two contracting groups are isomorphic more easily by checking whether the associated Julia sets are the same?", "Can one check whether two Fatou components of a rational map have intersecting closures?"

Some of our publications:

  • Volodymyr Nekrashevych: Self-similar groups. Mathematical Surveys and Monographs, vol. 117, American Mathematical Society (2005).
  • Russell Lodge: Boundary values of the Thurston pullback map, Conform. Geom. 17 (2013), 77-118. (arxiv)
  • Laurent Bartholdi, Dzmitry Dudko: Algorithmic aspects of branched coverings. (arxiv)

Experts within our team: Dzmitry Dudko, Mikhail Hlushchanka, Russell Lodge