Modular groups, Hurwitz classes and
dynamic portraits of NET maps
An orientation-preserving branched covering is a nearly Euclidean Thurston (NET) map if each critical point is simple and its postcritical set has exactly four points. Inspired by classical, non-dynamical notions such as Hurwitz equivalence of branched covers of surfaces, we develop invariants for such maps. We then apply these notions to the classification and enumeration of NET maps. As an application, we obtain a complete classification of the dynamic critical orbit portraits of NET maps.
Key words and phrases:Thurston map, virtual endomorphism, branched covering
2010 Mathematics Subject Classification:Primary: 36F10; Secondary: 57M12
This paper is part of our program to investigate nearly Euclidean Thurston (NET) maps. A Thurston map is NET if each critical point is simple and it has exactly four postcritical points.
The program began with [cfpp], written with Jim Cannon. There, we showed that each NET map admits a factorization where is a Euclidean Thurston map which is affine in suitable coordinates and is a homeomorphism. This property motivated the term nearly Euclidean. Thus NET maps are among the simplest Thurston maps, and their associated Teichmüller spaces have complex dimension 1. In [cfpp], we exploited the connection to affine geometry to give effective algorithms for the computation of fundamental dynamic invariants, such as the pullback function on homotopy classes of simple closed curves.
Next, [fpp1] shows that every NET map has a presentation which allows it to be described by a simple diagram, such as that shown in Figures LABEL:fig:PrenDgm, LABEL:fig:firstprendgm and LABEL:fig:degfourprendgm, and gives examples showing how to find such diagrams in concrete cases. The data in the diagram can then be translated into input for a computer program, NETmap (see [NETmap]), which implements the algorithms to compute the invariants. The NET map website [NET] contains an overview, papers, many examples, and executable files for the computer program NETmap.
The survey [fkklpps], written jointly with G. Kelsey, S. Koch, R. Lodge, and E. Saenz, reports on findings and new phenomena from a study of the data generated and on connections to other areas.
This paper develops other invariants of NET maps, focusing on non-dynamical invariants.
We now list these invariants and outline our first main results, in roughly decreasing order of sensitivity. Suppose are Thurston maps with postcritical sets . In the first four items below, are orientation-preserving homeomorphisms of the sphere to itself.
We recall that are Thurston equivalent if there exist such , with , , , and are isotopic relative to . Put another way, and are conjugate up to isotopy relative to their postcritical sets.
We say are Hurwitz equivalent for the pure modular group if there exist such with , , and . The motivation for the terminology: pre- and/or post-composing with a homeomorphism fixing pointwise yields a map which is equivalent to . In suitable Euclidean coordinates, the pure modular group is .
We say are Hurwitz equivalent for the modular group if there exist such with and . Theorem LABEL:thm:hurwitz gives a complete algebraic invariant of the modular group Hurwitz type. In suitable Euclidean coordinates, the modular group is .
We say are topologically equivalent if there exist such with .
Every NET map is topologically equivalent to a Euclidean NET map induced by a linear map , where is a diagonal matrix with entries and . The pair are the elementary divisors of . Theorem LABEL:thm:eleydivrs shows that the elementary divisors form a complete invariant of the topological equivalence class of a NET map; see also Theorem LABEL:thm:hurwitz_special. Hence all NET maps in a given modular group Hurwitz class have the same elementary divisors.
Most NET maps have four critical values. There are two atypical cases: elementary divisors and . Maps with these elementary divisors have two and three critical values, respectively. In atypical cases, modular group Hurwitz classes may contain both NET and non-NET maps: the postcritical sets of some maps may have fewer than four points. In typical cases, that is, all other cases, modular group Hurwitz classes consist entirely of NET maps.
Given , the collection of local degrees defines a partition of . The branch data associated to is the set of such partitions as varies in the set of branch values of . Section LABEL:sec:bd gives a classification of the branch data associated to NET maps. There are three basic types. In even degrees, in addition to the obvious Riemann-Hurwitz condition, there is an exceptional condition: for branch data of type 3, the degree must be divisible by 4 [pp, Thm. 3.8].
The translation subgroup
The modular group of the four-times marked sphere contains a distinguished subgroup isomorphic to . In suitable Euclidean coordinates, its elements are induced by translations. This group acts trivially on the Teichmüller space modelled on . Suppose represents an element of this group and . Then many fundamental invariants of and coincide, though their dynamic portraits may be different. It may also happen that , i.e. is a deck transformation of the covering induced by . This creates some subtleties; §3 gives a thorough analysis.
Our first application is to the enumeration of representatives of NET map modular group Hurwitz classes. The website [NET] contains data organized first by degree, then by elementary divisors, and finally by modular group Hurwitz class, giving one representative for every modular group Hurwitz class through degree 30. Table 1 gives the numbers of modular group Hurwitz classes of NET maps through degree 9. For every choice of elementary divisors, one modular group Hurwitz class of NET maps consists of all Euclidean NET maps with elementary divisors .
We now turn to dynamical applications. Suppose is a Thurston map with set of critical points and postcritical set . The most rudimentary dynamical invariant of is its dynamic portrait. This invariant records the restriction and the local degree for . The website [NET] contains data organized by degree and dynamic portrait, giving one representative for every dynamic portrait through degree 40. Table LABEL:tab:portraitnum gives a precise count of the number of such NET map dynamic portraits in each degree. Theorem LABEL:thm:portrait characterizes when a given pair consisting of elementary divisors and a dynamic portrait is realizable by a NET map. We also present algorithms for computing the correspondence between dynamic portraits of NET maps and NET map presentations. We describe these algorithms in Sections LABEL:sec:maptopic and LABEL:sec:pictomap.
As an example of how non-dynamical invariants influence dynamical properties, we present the following. Below, a Thurston map is Böttcher expanding if away from periodic critical points, it is uniformly expanding with respect to a natural orbifold metric on the orbifold associated to . Any rational Thurston map is Böttcher expanding.
Suppose is a non-Euclidean NET map with elementary divisors and postcritical set . If then is isotopic relative to to a Böttcher expanding map.
The assumption that is not Euclidean is necessary: if is the map induced by the matrix , then while has a positive real eigenvalue which is strictly less than one; this is an obstruction to expansion in the isotopy class. There are plenty of Böttcher expanding NET maps with ; among them are many rational maps too, e.g. the Douady rabbit polynomial. So the condition is far from sharp.
Recall that is Levy-free if there is no simple closed curve which is essential, not peripheral and having a lift homotopic to which maps to with degree 1. Using Theorem 4.1 of [cfpp], we can see that the degree with which maps to is divisible by . So is Levy-free. By [bd0, Theorem C], is isotopic to a Böttcher expanding map. ∎
If a modular group Hurwitz class contains a non-Euclidean NET map, then every map in is non-Euclidean and we call non-Euclidean. Any two maps in have the same elementary divisors.
Suppose is a non-Euclidean modular group Hurwitz class of NET maps and the elementary divisor is strictly larger than one. Then every isotopy class in contains a Böttcher expanding representative.
As another application, we give an effective algorithm for calculating a basic dynamical invariant. Suppose is a NET map with postcritical set . If a homeomorphism represents an element of and lifts under to a map which again represents an element of , the assignment induces the virtual endomorphism on the pure mapping class group associated to .
For generic Thurston maps, this is difficult to compute, even for maps with . One method is the following. As shown by S. Koch [Ko], a Thurston map has an associated algebraic correspondence on the moduli space of conformal configurations of embeddings of into the Riemann sphere. Here, is a finite unramified covering map, and is holomorphic. If , in suitable coordinates, , is identified with the fundamental group of , is a compact Riemann surface minus a finite set of points and is given as a locus , and the maps are the coordinate projections. In the special case when is rational (for ease of exposition here), there is a distinguished “fixed” point with , and the virtual endomorphism is then the induced map on fundamental group . In cases with few critical points, such as those considered in [BN] and [L], the defining equation for the correspondence can be found explicitly, and from this the virtual endomorphism may be computed. For maps with many critical points, this quickly becomes computationally intractable.
We consider a slightly more general notion, and calculate it using different methods. We relax the condition that the homeomorphisms fix pointwise to merely fixing setwise, and we obtain an associated virtual multi-endomorphism: the lift might not be well-defined, due to the possible presence of deck transformations. Theorem LABEL:thm:viredmp gives a method for calculating from knowledge of the slope function ; see §2. This method is then implemented in the NETmap program. Our method relies on two ingredients. First, there is a simple finite linear-algebraic condition for an element to be liftable under , i.e. to lie in the domain of the virtual multi-endomorphism (Lemma LABEL:lemma:when_liftable). To calculate its image , we exploit two facts: (i) the linear part of an element of is uniquely determined by its value at a pair of distinct extended rationals , and (ii) the slope function is algorithmically computable [cfpp]. Section LABEL:sec:exvme illustrates this in a complicated example.
As a further application, we obtain information about the geometry of the correspondence . The NETmap program uses classical geometric methods to find explicit matrix generators for the domain of the virtual endomorphism (equivalently, for the Fuchsian group that uniformizes ), and their images under the virtual endomorphism. The lifting of complex structures under defines an analytic self-map on the Teichmüller space modelled on that covers the correspondence . The NETmap program translates algebraic information about the virtual endomorphism into a rather complete geometric description about ; cf. [cfpp]. The data on the website tabulates this geometric information.
Section 2 collects concepts and facts used throughout the paper. Sections 3 and 4 discuss various modular groups and related actions. Sections 5 and 6 introduce Hurwitz classes, Hurwitz invariants and elementary divisors. Section 7 illustrates the calculation of the virtual endomorphism in a concrete complicated example. Section 8 introduces branch data, used in section 9 to classify dynamic portraits. Sections 10 and 11 give algorithms for constructing a portrait from a map and for constructing maps with a given portrait, respectively.
The authors gratefully acknowledge support from the American Institute for Mathematics. Kevin Pilgrim was also supported by Simons grant #245269.
In this section we summarize some background material from [cfpp] which will be used repeatedly. Let be a NET map. There exist a lattice in and branched covering maps , and for so that , as in Figure 1 in Section 1 of [cfpp]. We may always, and usually do, assume . The postcritical set of is . Statement 2 of Lemma 1.3 of [cfpp] states that contains exactly four points which are not critical points of . This set of four points is . The covering is normal with group of deck transformations equal to the group of all Euclidean isometries of the form for and . We will often use the finite Abelian group .
We recall Lemmas 2.1 and 2.2 stated in [fpp1].
Let and be lattices in . Let , respectively , be the groups of Euclidean isometries of the form for some , respectively . Also let and be the canonical quotient maps. Let be an affine isomorphism such that . Then induces a branched covering map such that . The map preserves orientation if and only if preserves orientation. The set is a coset of a sublattice of , and the degree of equals the index .
Let and be lattices in . Let , respectively , be the group of Euclidean isometries of the form for some , respectively . Also let and be the canonical quotient maps. Let be a branched covering map such that . Then we have the following three statements.
There exists a homeomorphism such that the restriction of to is affine and . If , then is a sublattice of .
There exists an affine isomorphism such that the branched map of Lemma 2.1 which induces from to is up to isotopy rel . If , then is a sublattice of .
The maps and are unique up to precomposing with an element of . They are also unique up to postcomposing with an element of .
We conclude this section by quickly recalling some dynamical invariants defined by pullback.
Suppose is a NET map with postcritical set . The set of (unoriented, simple, essential, non-peripheral, closed, up to free homotopy) curves on is naturally identified with via slopes of representing geodesics with respect to the Euclidean metric induced via pushforward under . Pulling back via induces the slope function , where is a symbol representing the set of inessential and peripheral curves. [cfpp, §5] gives an algorithm for calculating , implemented in the program NETmap.
Let be a simple closed curve in with slope . The multiplier of both and is defined as a fraction . The numerator is the number of connected components of which are neither inessential nor peripheral in . Theorem 4.1 of [cfpp] shows that the local degrees of on these connected components are equal. This common local degree is .
The set of marked complex structures on is naturally identified with the upper half-plane . We briefly recall this identification. To , one associates the lattice , the group , and the quotient marked at the four points and equipped with special generators for its orbifold fundamental group in the obvious way. Conversely, such a quotient equipped with such generators yields a lattice in equipped with an ordered basis with , and this yields an element .
Pulling back complex structures on under induces an analytic self-map . The map extends continuously to the Weil-Petersson boundary so that if and only if .
3. Modular group actions
This section deals with actions of modular groups on isotopy classes of NET maps. We formally present results only for the modular group. At the end of the section we discuss the analogs for the pure modular group and for the extended modular group, which allows for reversal of orientation. Let be a NET map. This will be fixed for all of this section. As in Section 2, let be branched covering maps so that . The set of branch values of in is the postcritical set of . Let be the lattice of branch points of in for .
It is always possible to take the lattice to be . We do so for simplicity. We define the special affine group to be the group of all orientation-preserving affine isomorphisms such that . So is the set of all maps , where , is a column vector in and is a column vector in . As usual, let be the group of all Euclidean isometries of the form for some . Let be the modular group of the pair . Proposition 2.7 of Farb and Margalit’s book [fm] shows that is a semidirect product with normal subgroup isomorphic to and quotient group . This is essentially the content of the following proposition.
The branched map induces a surjective group homomorphism with kernel . Hence
We define a group homomorphism as follows. Let . Lemma 2.1 with obtains an orientation-preserving homeomorphism such that . We define to be the isotopy class of in . Using Lemma 2.2, we see that is surjective, that it is a group homomorphism and that its kernel is . This proves Proposition 3.1. ∎
Because consists of maps of the form with and , Proposition 3.1 implies that every element of has a -term and a translation term. The translation term is an element of modulo . We say that an element of is elliptic, parabolic or hyperbolic according to whether its -term is elliptic, parabolic or hyperbolic. (We define reflections and glide reflections similarly when dealing with the extended modular group.) We say that an element of is a translation if its -term is trivial. In this way every element of the extended modular group has a type, that is, it is either elliptic, parabolic, hyperbolic, a reflection, a glide reflection or a translation.
Let denote the subgroup of consisting of those isotopy classes represented by homeomorphisms for which there exists a homeomorphism such that . Hence the bottom half of the diagram in Figure LABEL:fig:lift is commutative. We call the subgroup of liftables in . Let . We define the special affine group of to be the group of all orientation-preserving affine isomorphisms such that , and . The group actually depends on , and in addition to , but it is well defined up to a conjugation isomorphism. Its elements are affine planar homeorphisms that first descend under the projection and then further descend under , as in Figure LABEL:fig:lift.
The group homomorphism of Proposition 3.1 restricts to a surjective group homomorphism . In particular, every pair of homeomorphisms and stabilizing as in Figure LABEL:fig:lift can be modified by isotopies rel so that they lift to an affine isomorphism as in Figure LABEL:fig:lift. The kernel of this group homomorphism is . So .
Let . Lemma 2.1 obtains homeomorphisms and as in Figure LABEL:fig:lift. Hence the group homomorphism of Proposition 3.1 maps to . Statement 2 of Lemma 2.2 proves the second statement of Proposition 3.2, giving surjectivity. The kernel of this restriction group homomorphism is clearly . This proves Proposition 3.2.