The Torelli geometry and its applications

RESEARCH ANNOUNCEMENT

Benson Farb ^{∗} and Nikolai V. Ivanov ^{†}

November 7, 2003

Let S be a closed orientable surface of genus g. The mapping class group Mod(S) of S is defined as the group of isotopy classes of orientationpreserving diffeomorphisms S → S. We will need also the extended mapping class group Mod^{±}(S) of S which is defined as the group of isotopy classes of all diffeomorphisms S → S. Let us fix an orientation of S. Then the algebraic intersection number provides a nondegenerate, skew-symmetric, bilinear form on H = H_{1}(S, Z), called the intersection form. The natural action of Mod(S) on H preserves the intersection form. If we fix a symplectic basis in H, then we can identify the group of symplectic automorphisms of H with the integral symplectic group Sp(2g, Z) and the action of Mod_{S} on H leads to a natural surjective homomorphism

Mod(S) ⟶ Sp(2g, Z).

The Torelli group of S, denoted by I_{S}, is defined as the kernel of this homomorphism; that is, I_{S} is the subgroup of Mod(S) consisting of elements which act trivially on H_{1}(S, Z). In particular, we have the following wellknown exact sequence

1 ⟶ I_{S} ⟶ Mod(S) ⟶ Sp(2g, Z) ⟶ 1.  (1)

∗The first author was supported in part by NSF grants DMS-9704640 and DMS-0244542. He would also like to thank the Institut de Mathématiques de Bourgogne for its hospitality and support during the writing of this paper.

†This work was done during the visit of the second author to the University of Chicago.

He would like to thank the Department of Mathematics of the University of Chicago and the first author for the hospitality.

The group I_{S} plays an important role both in algebraic geometry and in low-dimensional topology. At the same time most of the basic questions about I_{S} remain open. See, e.g., [Jo1] and [Hain] for surveys. For example, while I_{S} is finitely generated if the genus g ≥ 3 by a famous theorem of D. Johnson [Jo2], it is not known whether or not I_{S} has a finite presentation if g ≥ 3.

In this note we introduce a new geometric object related to the Torelli group I_{S}, which we call the Torelli geometry T G(S) of S and announce several results about T G(S) and I_{S}. Our main result about T G(S) gives a complete description of its automorphisms (they are all induced by diffeomorphisms of S). We also give a purely algebraic characterization of some geometrically defined elements of I_{S}, namely, of the (powers of the) so-called bounding twists and bounding pairs, as also of some geometrically defined collections of such elements. When combined with the description of the automorphisms of T G(S), these characterizations lead to several results about the Torelli group I_{S}, among which are the following.

• The automorphism group of I(S) is isomorphic to the extended mapping class group Mod^{±}(S); the outer automorphism group Out(I(S)) contains the integral symplectic group Sp(2g, Z) as a subgroup of index 2.

• The abstract commensurator Comm(I(S)) is isomorphic to the extended mapping class group Mod^{±}(S).

• The Torelli group is not arithmetic.

The last result is known. Actually, any normal subgroup of Mod(S) was proved to be non-arithmetic in [Iv3]; for more general results, proving a conjecture of the second author, see [FM]. However, we feel that our proof sheds a new light on this result.

The details of the proofs will appear in [FI].

Torelli Geometry. A recurring theme in the theories of both finite and infinite groups is the investigation of a group by using its action on an appropriate geometry. We will use the term geometry in a narrow sense of a simplicial complex with some additional structure, namely, with some vertices and simplices marked by some colors. For example, the main construction of Bruhat-Tits theory applied to SL_{n}(Q_{p}), n > 2 gives an (n − 1)-dimensional simplicial complex X_{n,p} (with marked vertices) whose group of simplicial automorphisms preserving the marking is isomorphic to SL_{n}(Q_{p}). Similar constructions occur in the theory of finite simple groups.

For the mapping class group Mod(S) of a closed surface S, a useful geometry is the complex of curves, denoted CC(S). This simplicial complex, introduced by W. Harvey [Harv], is defined in the following way: CC(S) has one vertex for each isotopy class of simple (i.e. embedded) closed curve on S; further, k +1 distinct vertices of CC(S) form a k-simplex if the corresponding isotopy classes can be represented by disjoint curves. It is easy to see that CC(S) has dimension 3g − 4; in contrast with the classical situations it is not locally finite.

The complex CC(S) is a fundamental tool in the low-dimensional topology. The following theorem, which strongly supports the claim of CC(S) for being the right geometry for Mod(S), is of special importance for us.

Theorem 1 (Ivanov [Iv5]). Let S be closed and let genus(S) ≥ 2. Then the group of simplicial automorphisms of CC(S) is isomorphic to the extended mapping class group Mod^{±}(S).

Our definition of a geometry for the Torelli group is guided by the definition of the complex of curves CC(S). In order to arrive at a complex intimately related to the Torelli group, it is natural to use as vertices the isotopy classes of separating curves and of bounding pairs, since they correspond to the natural generators of the Torelli group. By a separating curve γ ⊂ S we mean a homologically trivial curve, or equivalently, a curve which bounds a subsurface in S. A bounding pair in S is a pair of disjoint, non-isotopic, nonseparating curves in S whose union bounds a subsurface in S. It turns out that it is useful also to introduce a marking.

Definition (Torelli geometry). The Torelli geometry of S, denoted T G(S), is the following simplicial complex with additional structure:

Vertices of T G(S): The vertices of T G(S) consist of:

1. Isotopy classes of separating curves in S, and

2. Isotopy classes of bounding pairs in S.

Simplices of T G(S): A collection of k ≥ 2 vertices in T G(S) forms a (k − 1)-simplex if these vertices have representatives which are mutually non-isotopic and disjoint.

We also endow T G(S) with a marking, which consists of the following two pieces of data:

Marking for T G(S):

1. Each vertex in T G(S) is marked by its type: separating curve or bounding pair.

2. A 2-simplex Δ in T G(S) is marked if there are 3 non-isotopic, disjoint, nonseparating curves γ_{1}, γ_{2}, γ_{3} so that the vertices of Δ are bounding pairs γ_{i} ∪ γ_{i+1} for i = 1, 2, 3 (mod 3).

It is easy to see that the group Mod(S) acts on T G(S) by automorphisms, by which we mean simplicial automorphisms preserving the marking. Our main result about T G(S) is the following.

Theorem 2 (Automorphisms of T G(S)). Suppose genus(S) ≥ 5. Then every automorphism of the Torelli geometry T (S) is induced by a diffeomorphism of S. Further, the natural map

Mod^{±}(S) ⟶ Aut(T G(S))

is an isomorphism.

The fact that Aut(T G(S)) is the extended mapping class group and not the Torelli group itself reflects intrinsic extra symmetries of the Torelli group; indeed, exactly by this reason Theorem 2 will allow us to compute the automorphisms group of T G(S) and the abstract commensurator of T G(S).

Connectedness theorems. A key ingredient in the proof of Theorem 2 is the following result, which we believe is of independent interest.

Theorem 3 (T G(S) is connected). For a closed orientable surface S of genus(S) ≥ 3 the complex T G(S) is connected.

There is a basic and useful subcomplex of the complex of curves, which is in fact a subcomplex of T G(S) for closed S.

Definition (C_{sep}(R)). Let R be any compact orientable surface, perhaps with nonempty boundary. We define C_{sep}(R) to be the simplicial complex whose vertices consist of isotopy classes of curves in R which bound a subsurface of positive genus with one boundary component, and with k + 1 distinct vertices forming a k-simplex if they can be represented by disjoint curves on S.

Note that, for a closed surface S, the complex C_{sep}(S) is just the subcomplex of CC(S) spanned by the (isotopy classes of) separating curves.

Theorem 4 (C_{sep}(R) is connected). Suppose that R is a compact surface with genus(R) ≥ 3. Then C_{sep}(R) is connected.

Note that if γ ⊂ R is a separating curve, and genus(R) ≥ 3, then R \ γ has one component of genus at least two; in particular there is a separating curve β ∈ R disjoint from γ and bounding a subsurface of genus one. We will call such separating curves the genus one separating curves. Hence Theorem 4 follows from the following more strong result which we prove:

For genus(R) ≥ 3,the full subcomplex of C_{sep}(R) spanned by the (isotopy classes of) genus one separating curves is connected.

Notice that both connectedness theorems do not hold if we omit the restriction on the genus; in fact, if the genus is equal to 2, then both complexes have an infinite number of vertices and no edges.

The idea of proof of Theorem 2. The main step in the proof of Theorem 2 is to show that every automorphism of the Torelli geometry T G(S) canonically induces an automorphism of the complex of curves CC(S) when genus(S) ≥ 5. Once this is proved, we can quote Ivanov’s Theorem (Theorem 1 above) and complete the proof.

The problem is to encode a nonseparating curve in CC(S) in terms of homologically trivial curves. The main difficulty results from the “infiniteness” of this problem; for example, any nonseparating curve can be completed to a bounding pair in infinitely many different ways, none of which is more natural than any other. The key idea is to consider all possible completions at once.

Consider a vertex γ of T G(S) represented by a bounding pair of nonseparating curves C_{0}, C_{1}. We would like to single out one of these curves using only the information available in T G(S). This can be done by using a separating curve D which has i(D, C_{0}) ≠ 0 and i(D, C_{1}) = 0. The isotopy class ⟨C_{0}⟩ obviously can be recovered from the pair (γ, δ), where δ = ⟨D⟩. Therefore the isotopy class ⟨C_{0}⟩ can be encoded by (γ, δ). For technical reasons, we use only pairs which we will call admissible (see below). Of course, we need to be able to tell when two such pairs encode the same isotopy class.

Definition (Admissible pair). Let γ be a bounding pair and let δ be a separating curve. We call the pair (γ, δ) an admissible pair if both:

1. γ is a vertex of some marked triangle {γ, γ′, β} of T (S), and

2. δ is connected by an edge of T (S) with β and is not connected by an edge with either γ or γ′.

We will say that the triangle {γ, γ′, β} certifies the admissibility of the pair (γ, δ).

Let the marked triangle {γ, γ′, β} be determined by three curves C_{0}, C_{1}, C_{2} (as in the definition of the marked triangles), and suppose that β is the isotopy class of the pair C_{1}, C_{2} (and, hence, γ, γ′ are the isotopy classes of two pairs including C_{0}). Then δ can be represented by a curve D disjoint from both C_{1} and C_{2}. Since δ is not connected by an edge with γ, we have i(D, C_{0}) ≠ 0. It follows, in particular, that the isotopy class ⟨C_{0}⟩ of the nonseparating curve C_{0} is uniquely determined by the pair (γ, δ). We will say that the isotopy class ⟨C_{0}⟩ is encoded by (γ, δ). Obviously, such an encoding of ⟨C_{0}⟩ is far from being unique. We will account for this non-uniqueness by using the following moves.

Two moves on admissible pairs: We consider the following two types of moves for admissible pairs (γ, δ).

Type I move: If a marked triangle {γ, γ′, β} certifies the admissibility of the pair (γ, δ), then replace (γ, δ) by (γ′, δ).

Type II move: If a marked triangle {γ, γ′, β} certifies the admissibility of the pair (γ, δ) and simultaneously certifies the admissibility of the pair (γ, δ′), then replace (γ, δ) by (γ, δ′).

The key result about the encodings of nonseparating curves by homologically trivial ones is the following.

Theorem 5 (Equivalence of encodings). Two admissible pairs encode the same nonseparating curve if and only if they are connected by a sequence of moves of types I and II.

This theorem is deduced from the above connectedness theorems, namely from Theorems 3, 4.

Once this is proven, the proof that an automorphism of T G(S) induces an automorphism of CC(S) follows easily, noting that the definition of admissible pairs and of moves of types I and II are given entirely in terms of the Torelli geometry.

Applications to the Torelli group. In order to state the main results about the Torelli groups, we need a counterpart of the extended mapping class group Mod^{±}(S) for the symplectic groups. Namely, we need the extended symplectic group Sp^{±}(2g, Z), which is defined as the group of automorphisms of Z^{2g} preserving the standard symplectic form up to an overall sign. Clearly, Sp^{±}(2g, Z) contains Sp(2g, Z) as a subgroup of index 2.

Let S be a closed surface of genus g. The conjugation action of Mod^{±}(S) on the normal subgroup I(S) induces homomorphisms

Mod^{±}(S) ⟶ Aut(I(S)) and Sp^{±}(2g, Z) ⟶ Out(I(S))

where Aut(I(S)) and Out(I(S)) are the groups of automorphisms and outer automorphisms, respectively, of I(S).

Our first application of Theorem 2 is the following.

Theorem 6 (Automorphisms of I(S)). Let S be a compact surface with genus(S) ≥ 5. Then the natural homomorphisms

Mod^{±}(S) ⟶ Aut(I(S)) and Sp^{±}(2g, Z) ⟶ Out(I(S))

are isomorphisms. In fact there is an isomorphism of exact sequences

1 ⟶  I(S)  ⟶ Mod^{±}(S) ⟶ Sp^{±}(2g, Z) ⟶ 1
↓ ≈  ↓ ≈  ↓ ≈
1 ⟶ Inn(I(S))) ⟶ Aut(I(S)) ⟶ Out(I(S)) ⟶ 1,

where the first exact sequence is an obvious version of the exact sequence (1) and the second exact sequence is the usual one for a centerless group, with Inn denoting the group of inner automorphisms induced by conjugation.

Note that the conclusion of Theorem 1 is false when genus(S) = 2, as Mess [Me] has shown that in this case I(S) is a countably generated free group. Theorem 6 was inspired by the theorem of Ivanov [Iv1], [Iv2] (see also [McC]) to the effect that Out(Mod^{±}(S)) = 1 for genus(S) ≥ 3.

The commensurator. Recall that the (abstract) commensurator group Comm(Γ) of a group Γ is defined to be the set of equivalence classes of isomorphisms ϕ : H → N between finite index subgroups H, N of Γ, where ϕ_{1} : H_{1} → N_{1} is equivalent to ϕ_{2} : H_{2} → N_{2} if ϕ_{1} = ϕ_{2} on H_{1} ∩H_{2}. The composition of homomorphisms induces a natural multiplication on Comm(Γ), which turns Comm(Γ) into a group.

The group Comm(Γ) is in general much larger than Aut(Γ); for example Aut(Z^{n}) = GL(n, Z) whereas Comm(Z^{n}) = GL(n, Q). The group Comm(Γ) was computed for mapping class groups of surfaces by Ivanov [Iv5]. For the Torelli group we have the following:

Theorem 7 (Commensurator Theorem). Let S be a closed surface of genus(S) ≥ 5. Then the natural injection

Mod^{±}(S) ⟶ Comm(I_{S})

is an isomorphism.

The non-arithmeticity. Our proof of the non-arithmeticity of I_{S} is inspired by the proof of the non-arithmeticity of Mod(S) given in [Iv5]. Theorem 7 implies that I_{S} is normal in its abstract commensurator. Using this fact, it is easy to see that I_{S} cannot be arithmetic. As an illustrating example, notice that the arithmetic group GL(n, Z) is not normal in its abstract commensurator Comm(GL(n, Z)) = GL(n, Q).

The idea of proof of Theorems 6 and 7. The main step in the proof of Theorems 6 and 7 is to prove that the given isomorphism Φ : Γ_{1} ⟶ Γ_{2} between two subgroups of finite index in I_{S} induces an automorphism of the Torelli Geometry T G(S); Theorem 2 can then be applied to produce a mapping class inducing Φ. First of all, we note that the isotopy class of a curve is uniquely determined by (any non-zero power of) the Dehn twist about that curve. If one has a purely algebraic characterization of (the powers of) Dehn twists in the Torelli group, this can be used to produce an automorphism of T G(S)

We will denote the Dehn twist about a curve γ by T_{γ}, and define the Dehn twist about a bounding pair {a, b} as T_{ab^{−1}} = T_{a}T_{b^{−1}}. Dehn twist about a separating curve or bounding pair will be called a simple twist. For a group Γ and element f ∈ Γ, let C_{Γ}(f) denote the subgroup of elements g ∈ Γ commuting with f, and let Z(Γ) denote the center of Γ. The following proposition gives us an algebraic characterization of (powers of) simple twists.

Proposition 8 (Characterizing simple twists in I(S)). Let S be a closed surface with genus(S) ≥ 3, and let f ∈ I(S) be nontrivial. Then f is a power of a simple twist if and only if both of the following hold:

1. Z(C_{I(S)}(f)) = Z, and

2. The maximum of ranks of abelian subgroups of I(S) that contain f is 2g − 3.

From Proposition 8 one can easily deduce that any isomorphism Φ : Γ_{1} ⟶ Γ_{2} between two subgroups Γ_{1}, Γ_{2} of finite index in I_{S} takes powers of simple twists to powers of simple twists. The main tool in proving Proposition 8 is the Thurston normal form theory for mapping classes extended to abelian subgroups of Mod(S), for example, in [Iv4].

Next, we need further to distinguish purely algebraically between the two types of simple twists.

Proposition 9 (Characterizing bounding pairs). Let g ≥ 4, and let R be the set of powers of simple twists in I(S). Then the following are equivalent.

1. f ∈ R is a power of a Dehn twist about a bounding pair.

2. There exist g, h ∈ R, distinct from f and from each other, so that the group generated by f, g, h is isomorphic to Z^{2}.

Finally, we need to give a purely algebraic characterization of marked triangles in T G(S).

Proposition 10 (Characterizing marked triangles). Let g ≥ 4, and let R be the set of powers of simple twists in I(S). Then the following are equivalent.

1. f, g, h ∈ R are powers of a Dehn twists about a bounding pairs γ_{1}, γ_{2}, γ_{3} such that {γ_{1}, γ_{2}, γ_{3}} is a marked triangle.

2. The groups generated by elements f, g, h and by any two of them are all isomorphic to Z^{2}.

The proofs of the last two Propositions are similar, and are based on an analysis of all possible configurations of three separating circles and bounding pairs on a surface.

References

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Benson Farb: Dept. of Mathematics, University of Chicago 5734 University Ave. Chicago, Il 60637 E-mail: farb@math.uchicago.edu

Nikolai V. Ivanov: Michigan State University Dept. of Math, Wells Hall East Lansing, MI 48824-1027 E-mail: ivanov@math.msu.edu