On the character variety of the four–holed sphere
We study the (relative) character varieties of the four-holed sphere and the action of the mapping class group on it. We describe a domain of discontinuity for this action, and, in the case of real characters, show that this domain of discontinuity may be non-empty on the components where the relative euler class is non-maximal.
In his PhD thesis , McShane established the following remarkable identity for lengths of simple closed geodesics on a once-punctured torus with a complete, finite area hyperbolic structure:
where varies over all simple closed geodesics on , and is the hyperbolic length of under the given hyperbolic structure on . This result was later generalized to (general) hyperbolic surfaces with cusps by McShane himself , to hyperbolic surfaces with cusps and/or geodesic boundary components by Mirzakhani , and to hyperbolic surfaces with cusps, geodesic boundary and/or conical singularities, as well as to classical Schottky groups by Tan, Wong and Zhang in , .
On the other hand, Bowditch in  gave an alternative proof of (1) via Markoff maps, and extended it in  to type-preserving representations of the once-punctured torus group into satisfying certain conditions which we call here the BQ–conditions (Bowditch’s Q–conditions). He also obtained in  a variation of (1) which applies to hyperbolic once-punctured torus bundles. Subsequently, Sakuma , Akiyoshi, Miyachi and Sakuma ,  and recently Sakuma and Lee  refined Bowditch’s results and generalized them to those which apply to hyperbolic punctured surface bundles. In  Tan, Wong and Zhang also further extended Bowditch’s results to representations of the once-punctured torus group into which are not type-preserving, that is, where the commutator is not parabolic, and also to representations which are fixed by an Anosov element of the mapping class group and which satisfy a relative version of the Bowditch’s Q–conditions. They also showed that the BQ-conditions defined an open subset of the character variety on which the mapping class group of the punctured torus acted properly discontinuously.
In a different direction, Labourie and McShane in  showed that the identity above has a natural formulation in terms of (generalised) cross ratios, and then, using this formulation, studied identities arising from the cross ratios constructed by Labourie for representations from fundamental groups of surfaces to .
The above papers provided much of the motivation for this paper, in particular, the identities obtained were in many cases valid for the moduli spaces of hyperbolic structures, so invariant under the action of the mapping class group, and in the case of cone structures, they could be interpreted as identities valid for certain subsets of the character variety which were invariant under the action of the mapping class group, even though the representations in the subset may be non-discrete or non-faithful. This leads naturally to the question of whether there were interesting subsets of the character varieties on which the mapping class group acts properly discontinuously, but which consists of more than just discrete, faithful representations, as explored in the punctured torus case in .
In this paper we will consider representations of the free group on three generators into . We adopt the viewpoint that is the fundamental group of the four-holed sphere , with identified with , and study the natural action of , the mapping class group of on the character variety
where we take the quotient in the sense of geometric invariant theory. If and , this action is given by
where is the map associated to in homotopy. We are interested in the dynamics of this action, in particular, on the relative character varieties , which is the set of representations for which the traces of the boundary curves are fixed.
There exists a domain of discontinuity for the action of on , that is, an open subset on which acts properly discontinuously.
This set is described by two conditions, much in the spirit of  and  given as follows. If denotes the set of free homotopy classes of essential, non-peripheral simple closed curves on , then the conditions for to be in are
for all ; and
for only finitely many , where is a fixed constant that depends only on .
Furthermore, the set of satisfying condition (ii) above satisfy a quasi-convexity property, equivalently, is connected when represented as the subset of the complementary regions of a properly embedded binary tree (see Theorem 3.1). This property is particularly important when writing a computer program to draw slices of the domain of discontinuity.
As already observed by several other authors in related situations (see Goldman , Tan–Wong–Zhang  and Minsky ), our domain of discontinuity contains the discrete faithful characters, but also characters which may not be discrete or faithful.
Of particular interest is the set of real characters, which consists of representations in or . In the latter case, Goldman  proved ergodicity of the mapping class group action for all orientable hyperbolizable surfaces. (This was generalized by the second author in the non-orientable case in ). On the other hand, in the case the dynamics is much richer and less understood. For example, when is a closed surface of genus , Goldman conjectured that the action of on the components of with non-maximal Euler class is ergodic. An approach towards a proof of this would be to use a cut-and-paste argument involving pieces homeomorphic to one-holed tori and four-holed spheres. While the case of the one-holed torus was completely described by Goldman in , we obtain partial results in the four-holed sphere case here. In fact, an important corollary of our analysis is the following (see Theorem 5.3):
The components of the relative character variety of with non-maximal Euler class may contain non-empty domains of discontinuity for the action.
This implies that there are representations in these components for which all essential simple closed curves on have hyperbolic representatives, even though these representations may not be discrete and faithful. There are also some surprises here, in particular, certain slices of the real character variety satisfying some general condition always have non-empty intersection with the domain of discontinuity.
The mapping class group consisting of equivalence classes of diffeomorphisms of fixing the boundary is isomorphic to the index two normal subgroup of the triangle group generated by reflections on an ideal triangle and it acts on the character variety as given in the earlier part of the introduction. For a representation , we can look at its character
The character variety is exactly the set of characters, by results of , see for example , each character is in turn determined entirely by its value on the seven elements as follows, satisfying a single equation:
Hence, we identify with the variety consisting of points satisfying the equation
The action extends to an action of on , which is generated by
Elements of the mapping class group correspond to words of even length in and , in particular, , and correspond to Dehn twists about essential simple closed curves on and generate the action on .
It is somewhat remarkable that many of the results of  and  generalize to the problem we study here, although the analysis is necessarily more technical and complicated, but in some sense, also more interesting. There are however some results which do not generalize, see Example 3.11. Finally, we note that recent work of Hu, Tan and Zhang  on Coxeter group actions on quartic varieties indicate that in fact, there should be a deeper underlying theory for analyzing the domains of discontinuity for group actions of this type.
The rest of this paper is organized as follows. In §2 we set the notation and give some basic definitions. In §3 we prove the generalizations of the basic lemmas (in terms of Markoff maps) required to analyse and understand the orbit of a character under the action of the mapping class group. In particular, generalizations of the “fork” lemma (Lemma 3.3) and the quasi-convexity result (Theorem 3.1) from  and , as well as an analysis of the values taken by the neighbors around a region are covered. In §4 we give a proof of our main theorem (Theorem A) which describes the domain of discontinuity for the action in terms of the BQ-conditions. In §5, we consider the real case and show that domains of discontinuity can occur in the components of the relative character variety which do not have maximal relative Euler class (Theorem B). Finally, in §6 we give some concluding remarks.
Acknowledgements. This project was initiated when the authors were participating in the trimester program on “Geometry and Analysis of surface group representations” at the Institut Henri Poincaré (Jan-Mar 2012). The authors are grateful to the organizers of the program for the invitations to participate in the program, and to the IHP and its staff for their hospitality and generous support. In particular, we would like to thank Bill Goldman for helpful conversations.
In this section we set the notation which we will use in the paper and give some important definitions. Since many of the results included in this article are influenced by, and are generalisations of the results of Tan, Wong and Zhang’s article , we will try to follow closely the notation and structure of that paper, so that the interested reader will find it easier to compare our results with theirs. We should also note that their paper was a generalisation of Bowditch’s results .
2.1. The four-holed sphere group
Let be a (topological) four-holed sphere, that is, a sphere with four open disks removed, and let be its fundamental group. The group admits the following presentation
where , , and correspond to homotopy classes of the four boundary components, one for each removed disk. Note that is isomorphic to the free group on three generators .
We define an equivalence relation on by: if and only if is conjugate to or . Note that can be identified with the set of free homotopy classes of unoriented closed curves on .
2.2. Simple closed curves on the sphere
Let be the set of free homotopy classes of essential (that is, non-trivial and non-peripheral) simple closed curves on and let be the subset corresponding to . Note that can be identified with by considering the ‘slope’ of , see, among others, Proposition 2.1 of Keen and Series . For example, we can identify with , with , with , and so on.
We also observe that inherits a cyclic ordering from the cyclic ordering of induced from the standard embedding into .
2.3. Relative character variety of
The character variety is the space of equivalence classes of representations , where the equivalence classes are obtained by taking the closure of the orbit under the conjugation action by .
A representation is said to be a –representation, or –character, where , if for some fixed generators corresponding to the boundary components of , , , , . The space of equivalence classes of -representations is denoted by and is called the –relative character variety. These representations correspond to representations of the four-holed sphere where we fix the conjugacy classes of the four boundary components.
For , we let , and . A classical result on the character varieties (see (9) in p. 298 of Fricke and Klein ) states that is identified with the set
The mapping class group of , acts on , see . For concreteness, we adopt the convention here that consists of orientation-preserving homeomorphisms fixing the boundary, there will be no essential difference to the ensuing discussion. The mapping class group is generated by Dehn twists along the simple closed curves corresponding to , and . The action of each Dehn twist can be read easily on the trace coordinates of . For example the Dehn twist about the separating curve is the map given by:
where are defined as above. This corresponds to the action of given in the Introduction, see also Remark 3.6.
2.4. The BQ-conditions
For a fixed , let be a constant depending only on , that we will define later in Definition 3.9. A -representation (or ) is said to satisfy the BQ-conditions (Bowditch’s Q-conditions) if
for all ; and
for only finitely many (possibly no) .
We also call such a representation a BQ-representation, or Bowditch representation, and the space of equivalence classes of such representations the Bowditch representation space, denoted by .
Note that if (since is conjugate to or its inverse by definition); so the conditions (BQ1) and (BQ2) make sense.
2.5. The binary tree
Let be a countably infinite simplicial tree properly embedded in the plane all of whose vertices have degree . As an example, we can consider, as , the binary tree dual to the Farey triangulation of the hyperbolic plane (also called an infinite trivalent tree). See  for the definition of .
2.6. Complementary regions
A complementary region of is the closure of a connected component of the complement.
We denote by the set of complementary regions of . Similarly, we use , for the set of vertices and edges of respectively.
We use the letters to denote the elements of . For , we also use the notation to indicate that and and are the endpoints of ; see Figure 1.
2.7. A tri-coloring of the tree
We choose a coloring of the regions and edges, namely a map such that for any edge we have and such that , , are all different. The coloring is completely determined by a coloring of the three regions around any specific vertex, and hence is unique up to a permutation of the set . We denote by the set of complementary regions with color , and by the set of edges with color .
As a convention, in the following, when are complementary regions around a vertex, we will have , and .
2.8. –Markoff triples
For a complex quadruple , a -Markoff triple is an ordered triple of complex numbers satisfying the –Markoff equation:
Note that, if is a –Markoff triple in the sense of Tan-Zhang-Wong, then is a –Markoff triple in our sense.
It is easily verified that, if is a –Markoff triple, so are the triples
It is important to note that permutations triples are not –Markoff triples, contrary to the situation with –Markoff triples.
2.9. Relation with -representations
Let be the map defined by:
This map is defined and studied by Goldman and Toledo in , where they show, among many other results, that the map is onto and proper. (Note that Goldman and Toledo denote this map .)
Given , a representation is in if and only if is a –Markoff triple with .
The elementary operations defined in (3) are intimately related with the action of the mapping class group on the character variety, as we will see later.
2.10. –Markoff map
A -Markoff map is a function such that
for every vertex , the triple is a –Markoff triple, where are the three regions meeting such that , and ;
For any and for every edge we have:
If and , then
If and , then
If and , then
where , , and
We shall use to denote the set of all –Markoff maps and lower case letters to denote the values of the regions. For example, we have .
There exists a bijective correspondence between –Markoff maps and –Markoff triples. Hence, using Remark 2.1, there exists a bijective correspondence between the set of –Markoff maps and the –relative character variety , where .
In fact, as in the case of Markoff maps and –Markoff maps, if the edge relations (4), (5) and (6) are satisfied along all edges, then it suffices that the vertex relation (2) is satisfied at a single vertex. So one may establish a bijective correspondence between –Markoff maps and –Markoff triples, by fixing three regions which meet at some vertex . This process may be inverted by constructing a tree of –Markoff triples as Bowditch did in  for Markoff triples and as Tan, Wong and Zhang did in  for the –Markoff triples: given a triple , set , and extend over as dictated by the edge relations. In this way one obtains an identification of with the algebraic variety in given by the –Markoff equation. In particular, gets an induced topology as a subset of .
2.11. The subsets
Given and , the set is defined by
These sets will be crucial in the proof of our main results.
We can now state the BQ-conditions in terms of Markoff maps.
For a fixed , let be a constant depending only on , that we will define later in Definition 3.9. A -Markoff map is said to satisfy the BQ-conditions if
We denote by the set of all -Markoff maps which satisfy the BQ-conditions, and call the set of all such maps the Bowditch -Markoff maps.
We will see that it is sufficient to give a large enough constant to define the BQ-conditions. Indeed, taking a bigger constant in the definition will give rise to the same subset of Markoff maps.
3. Estimates on Markoff maps
The main aim of this section is to prove the following result, which is a generalisation of Theorem 3.1 of Tan–Wong–Zhang .
(Quasi-convexity) Let . Then:
There exists a constant such that , we have that is non-empty.
There exists a constant such that and , the set is connected.
For doing that, we need to do lots of estimates. In order to shorten a bit the formulae which will appear, we introduce the following notation.
Given a –Markoff map , let
For the rest of this section, let us fix and a –Markoff map .
3.1. Arrows assigned by a –Markoff map
As Bowditch  and Tan, Wong and Zhang  did, we may use to assign to each undirected edge, , a particular directed edge, , with underlying edge . Suppose . If , then the arrow on points towards ; in other words, . If , we put an arrow on pointing towards , that is, . If it happens that then we choose arbitrarily. Let be the set of oriented edges.
A vertex with all three arrows pointing towards it is called a sink, one where two arrows point towards it and one away is called a merge, and vertex with two (respectively three) arrows pointing away from it is called a fork (respectively source).
(Fork lemma) Suppose meet at a vertex , and the arrows on the edges and both point away from . Then at least one of the following is true:
We have similar results if the edges and (or and ) both point away from .
Let and be the arrows pointing away from . Let and be the regions such that and . The edge relation gives . Similarly . Adding both inequalities, one obtains
If and , then we have
A weaker statement will be sufficient for most of the paper.
Suppose meet at a vertex , and two arrows point away from , that is, is a fork or a source. Then
There is a constant such that if three regions meet at a sink, then
We show that, if are all sufficiently large, then the vertex cannot be a sink. We may assume and . We can rewrite (2) as:
There exists such that if , we have are all smaller than . It follows that
On the other hand, we have
So we infer that . Hence the arrow on the edge is directed away from which proves that is not a sink. ∎
3.2. Neighbors around a region
For each , its boundary is a bi-infinite path consisting of a sequence of edges of the form alternating with , where and are bi-infinite sequences of complementary regions in and . The edge relations (5), (6) are such that
The map is exactly the map defined by the Dehn twist along the curve defined in the introduction.
We can reformulate these equations in terms of matrices:
If , this can be rewritten as
Note: and are the coordinates of the center of the conic in coordinates defined by the vertex relation.
The matrix has determinant one. Hence its eigenvalues and are such that . Explicitly, if is a square root of in , then the eigenvalues are given by:
The matrix is
elliptic, if ;
parabolic, if ;
loxodromic, if .
Case 1: is elliptic, that is . Let such that . In this case, there exists an invertible matrix such that
So, the sequences , are given by:
Hence, the sequences and are bounded.
Case 2: is parabolic, that is . When , we get the exact formulae:
Similarly, when , we get the formulae:
Case 3: is loxodromic, that is . In this case and so is invertible. It follows that both and are non-zero. We have the following formula:
Using the definition of , we can see that is a square root of . And similarly is a square root of . Moreover we have . So we obtain the following closed formulae for and :
This means we can express these formulae as:
Hence, we see that both sequences have the same behaviour. We have . Hence, when and are both non-zero, the sequences and grow exponentially in and . To determine when at least one of , we have the following identity concerning the product :
The vertex relationship is
so it is a quadric and its type depends on the sign of the determinant
For detecting when the conic is degenerate we need the following determinant:
Note that . We infer that the conic is degenerate when is a solution of the equation AB=0. Following Benedetto and Goldman’s notation , we can rewrite as where are such that and .
Let and denote by the two solutions of the equation . Using this notation the solutions of the equation are .
We note that does not depend on and . The coefficients and are both non-zero unless . Note also that, in the case , we recover the formula of Tan-Wong-Zhang, namely . We can now conclude the discussion with the following lemma.
Suppose that has neighboring regions and , . Then:
If , then and remain bounded.
If , then and grow at most quadratically.
If and is not a solution of , then and grow exponentially as and as .
If and is a solution of , then:
, and and grows exponentially as ;or
, and and grows exponentially as ; or
and for all .
Given , let
Note that the case (4) of the previous Lemma can only happen for a Markoff map if one of the neighboring regions takes value in . In particular, the subcases of (4) correspond to the cases , and .
We can now give the definition of the constant used to define the BQ-conditions. First define as follows:
So is the maximum modulus of the coordinates of the center of the conic equation taken on the finite number of cases where the conic is degenerate.
Let , and, similarly, we define .
Suppose is an infinite ray in consisting of a sequence of edges of such that the arrow on each assigned by is directed towards . Then the ray meets at least one region X with and an infinite number of regions with . Moreover, if , then meets an infinite number of regions with