топология / Farb, Margalit, A primer on mapping class groups

For fixed φ, different choices of γ give rise to maps φ that differ by conjugation.

If φ is a homeomorphism then it is invertible, and so φ is an automorphism. It follows that we have a well-defined homomorphism

Ψ : Mod±(S) → Out(π1(S))

which is injective by the correspondence given above. We have the following remarkable theorem.

THEOREM 4.1 (Dehn–Nielsen–Baer) Let g ≥ 1. The homomorphism

Ψ : Mod±(Sg ) −→ Out(π1(Sg ))

is an isomorphism.

As noted above, the proof of Theorem 4.1 reduces to the statement that Ψ is surjective. The original proof of this is due to Dehn, although Nielsen was the first to publish a proof [134]. Baer was the first to prove in jectivity.

Note that in the case g = 1 the Dehn–Nielsen–Baer Theorem recovers the fact that Mod±(T 2) ≈ GL(2, Z). Note too that the statement of the theorem does not hold when g = 0 since

Mod±(S2) ≈ Z/2Z 6≈1 ≈ Out(π1(S2)).

Action on the fundamental class. The action of Mod±(Sg ) on H2(Sg ; Z) ≈

Z and the action of Out(π1(Sg )) on H2(π1(Sg ); Z) ≈ Z are related by the Dehn–Nielsen–Baer Theorem in the sense that the followi ng diagram is commutative.2

Z/2Z ≈ Out(H2(Sg ; Z)) ≈ Out(H2(π1(Sg ); Z))

2For a group G, we can define Hk (G; Z) as Hk (X; Z), where X is any K(G, 1) space; see Chapter 6.

An element of Mod±(Sg ) is orientation preserving if and only if the induced element of Out(H2(Sg ; Z)) is trivial. This gives an algebraic characterization of Mod(Sg ) inside Mod±(Sg ): it is the subgroup of Out(π1(Sg )) that acts trivially on H2(π1(Sg ); Z).

The case of punctured surfaces. The Dehn–Nielsen–Baer Theorem does not hold as stated for surfaces with punctures. For example, let S0,3 be the thrice-punctured sphere. We have π1(S0,3) ≈ F2, the free group on two generators. Also, it is a theorem of Nielsen that Out(F2) ≈ GL(2, Z); see [110, Proposition 4.5] or [16, §5.3]. Thus Out(π1(S0,3)) ≈ GL(2, Z), but Mod±(S0,3) is isomorphic to the finite group Σ3 × Z/2Z (cf. Proposition 2.3).

For punctured surfaces, we will see in Theorem 4.8 below that Mod±(S) is isomorphic to the subgroup of Out(π1(S)) that preserves the collection of conjugacy classes of elements corresponding to punctures of S (the primitive parabolic elements).

4.1 QUASI-ISOMETRY PROOF

Dehn's original proof of the Dehn–Nielsen–Baer Theorem use s the notion of quasi-isometry. Again, the goal is to show that each element of Out(π1(Sg )) is induced by an element of Mod±(Sg ). The key step is to show that an element of Out(π1(Sg )), which a priori preserves only algebraic properties/objects, must in fact preserve topological ones. For example, the first step in the proof will be to prove that an element of Out(π1(Sg )) must respect the topological property of whether or not two the free homotopy classes of two simple closed curves have geometric intersection number

zero. We will prove this by studying the behavior of π1(Sg ) “at infinity” in H2.

Metrics on π1(S). Let G be a group with a fixed finite generating set S. The Cayley graph (G, S) for G with respect to S is the abstract graph with a vertex for each element g G and an edge between the vertices g and gs if s S or if s−1 S. The group G acts on (G, S) on the left by graph automorphisms.

There is a natural metric on (G, S) given by taking each edge to have length 1 and putting the path metric on (G, S), whereby the distance between two points is the length of the shortest path between them. Restricting

this metric to the vertices of (G, S) gives a G–invariant metric on G, called the word metric on G with respect to S. For g G, the distance dS(1, g) is called the word length of g. By left invariance, for any g, h G, the distance dS(g, h) is the word length of g−1h.

For a surface S with χ(S) < 0, another way to get a metric on π1(S) is to choose a covering map H2 → S that endows S with a hyperbolic metric (recall that, by “hyperbolic metric,” we mean a compl ete, finite-area Riemannian metric with constant curvature −1). If we fix a basepoint in S, its set of lifts to H2 are in bijection with elements of π1(S). We can therefore define the hyperbolic distance between two elements of π1(S) as the hyperbolic distance between the corresponding lifts.

Clearly, the word metric on π1(S) depends on the choice of generating set, and the hyperbolic metric on π1(S) depends on the choice of covering map. We would like to understand what properties of the metric do not depend on these choices. In short, the answer is that all chocies give metrics that “look the same, up to a universally bounded stretch, at large scales.” This brings us to the notion of quasi-isometry.

Quasi-isometries. A set map f : X → Y between metric spaces X and Y is a quasi-isometric embedding if there are constants K and C so that

K1 d(x, x′) − C ≤ d(f (x), f (x′)) ≤ Kd(x, x′) + C

for any choice of x and x′ in X. We say that f is a quasi-isometry if there is a constant D so that the D–neighborhood of f (X) is equal to Y . In this case we say that X and Y are quasi-isometric. It is not hard to check that quasi-isometry is an equivalence relation on metric spaces.

There is a more symmetric definition of quasi-isometry, as fo llows. Two metric spaces X and Y are quasi-isometric if and only if there are maps f : X → Y and f : Y → X, and constants K, C, and D such that

As a first exercise one can show that, given two word metrics on the same finitely generated group G, the identity map G → G is a quasi-isometry.

This fact also follows from the first statement of Theorem 4.2 ; see Corollary 4.3 below.

The fundamental observation of geometric group theory. The following

ˇ

theorem, sometimes called the Milnor– Svarc Lemma, is one of the most basic theorems in geometric group theory. It first appeared i n the work of

˘

Efremovi˘c [50], Svarc [162], and Milnor [128].

Recall that the action of a group G on a topological space X is properly discontinuous if, for each compact K in X, the set {g G : g ·K ∩K 6= } is finite. Let X be some metric space. The space X is proper if closed balls in X are compact. A geodesic in X is a distance-preserving map of a closed interval into X. Finally, X is a geodesic metric space if there exists a geodesic connecting any two points in X.

THEOREM 4.2 (Fundamental Oberservation of Geometric Group Theory)

Let X be a proper geodesic metric space, and suppose that a group G acts properly discontinuously on X via isometries. If the quotient X/G is compact, then G is finitely generated, and G is quasi-isometric to X. More precisely, there is a word metric for G so that, for any point x0 X the map

G → X

g 7→g · x0

is a quasi-isometry.

Proof. Let x0 be some fixed basepoint of X. Since the action of G on X is properly discontinuous, the metric on X induces a metric on X/G. Indeed, the distance between two points in the quotient is the infimum of the distances between any two of their preimages; the proper discontinuity implies the infimum is a minimum. As X/G is compact, it has finite diameter R. It follows that X is covered by the G–translates of B = B(x0, R), the ball of radius R about x0. Let

S = {g G : g 6= 1 and g · B ∩ B 6= }.

By the properness of X and the proper discontinuity of the action of G on X, the set S is finite.

Let d denote the metric on X. We define

S

Note that, since the action of G is properly discontinuous, and since X is proper, r is actually a minimum.

If r = 0, then G is finite, and the theorem is trivial in this case. So we may assume r > 0.

Let g G. As X is geodesic, it is in particular path connected. Given a path from x0 to g · x0, we can choose points x1, . . . , xn = g · x0 along this

path so that d(xi, xi+1) < r. Since the {g · B} cover X, we may choose g1, . . . , gn G so that xi gi · B. If we set g0 = 1 and si = gi−−11gi, we

have that s1s2 · · · sn = g. We have

d(si · B, B) = d(gi−−11gi · B, B) = d(gi · B, gi−1 · B) ≤ d(xi, xi−1) < r.

By the definition of r we see that si S {1} for all i. Thus S generates G, and G is finitely generated.

We will now show that the map g 7→g ·x0 defines a quasi-isometric embedding G → X, where G is given the word metric associated to S. In other words we will show that for g1, g2 G we have

Since G acts by isometries on itself and on X, this is equivalent to the statement that

for any g G (substitute g1−1g2 for g). In our definition of a quasi-isometric embedding, one can take K = max{λ, 1r } and C = 1. The constant C cannot be taken to be 0 because, for instance, g could be in the stabilizer of x0.

The inequality λ1 d(x0, g · x0) ≤ dS(1, g) follows immediately from the triangle inequality, the definitions of S and λ, and that fact that s S if and only if s−1 S. Thus “short” paths in G give rise to “short” paths in X.

We must now show that “short” paths in X correspond to “short” paths in G. Precisely, we will prove the inequality dS(1, g) ≤ 1r d(x0, g · x0) + 1. The argument is a souped-up version of the argument that S generates G. Let g G. Since X is geodesic we may find a geodesic of length d(x0, g · x0) connecting x0 to g · x0. Let n be the smallest integer strictly

greater than d(x0, g · x0)/r, so n ≤ d(x0, g · x0)/r + 1. We can find points x1, . . . , xn−1, xn = g · x0 in X so that d(xi, xi+1) < r for 0 ≤

1 = g0, g1 . . . , gn−1, gn = g of G so that xi gi · B. If we set si = gi−−1gi then g = s1 · · · sn. Again, by the definition of r, we have si S, and so the

word length of g is at most n. In summary, we have

1

d(1, g) ≤ n ≤ r d(x0, g · x0) + 1,

which is what we wanted to show.

By the definition of R, the R–neighborhood of the image of G is all of X,

Any Cayley graph for a finitely generated group is a proper, ge odesic metric space. Thus, by considering the action of a group G on an arbitrary Cayley graph for G, we obtain the following fact.

Corollary 4.3 For any two word metrics on a finitely generated group G, the identity map G → G is a quasi-isometry.

The following corollary of Corollary 4.3 represents the firs t step in our proof of the Dehn–Nielsen–Baer Theorem.

Corollary 4.4 Any automorphism of a finitely generated group is a quasiisometry.

By Corollary 4.3, we do not need to specify which word metric we are using in the statement of Corollary 4.4.

Proof. Let Φ : G → G be an automorphism of a finitely generated group G, and let S be a finite generating set for G. Since Φ is an automorphism, we have that Φ−1(S) = {Φ−1(s) : s S} is a finite generating set for G. What is more, we have:

dS(Φ(g), Φ(h)) = dΦ−1(S)(g, h).

In other words, the amount word-length in G is stretched under the map Φ is equivalent to the amount of stretch word-length undergoes when changing the finite generating set. The result now follows immediatel y from Corol-

Oγ = {γk · x0 : k Z}

Since γ and δ are hyperbolic isometries of H2, we may choose an N = N (R) so that each point of

OδN = {(δN )k · x0 : k Z, k 6= 0}

has distance at least R from each point of Oγ . Note that OδN is not the orbit of x0 under δN since it is missing the point x0.

Since γ and δ are not linked at infinity (and do not share an axis), we can connect the points of Oγ by an infinite piecewise-geodesic path, where each segment of the path connects two points in the orbit of x0 that lie in adjacent fundamental domains, and where each point of the path has a distance at least R from each point of OδN . We can denote such a path by its set of vertices, say {αi}. We can likewise connect the points of OδN by a piecewise-geodesic path {βi} where each βi is in the orbit of x0, so that the path {βi} stays a hyperbolic distance at least R from the path {αi}, and so that consecutive vertices βi and βi+1 lie in adjacent fundamental domains. The condition on adjacency implies that, for both paths, the length of each geodesic segment is at most 2D (any pair of points in adjacent fundamental domains have distance at most 2D). The vertices of the two paths are identified with particular elements of π1(Sg ).

Figure 4.1 Left: the polygonal paths constructed in the proof of Lemma 4.5; right: polygonal paths that are linked at infinity.

Assume, for the purposes of contradiction, that the hyperbolic isometries Φ(γ) and Φ(δ) are linked at infinity. It follows that the polygonal paths

{Φ(αi)} and {Φ(βi)} have to cross. Since Φ is a quasi-isometry with constants K and C, each geodesic segment of {Φ(αi)} and {Φ(βi)} has length at most K(2D) + C. But if these paths cross, two of the geodesic segments themselves must cross—see the right hand side of Figure 4.1. Now, each segment has at least one endpoint whose distance from the crossing point is less than or equal to (K(2D) + C)/2, and so these two endpoints lie at a distance of at most K(2D) + C.

What we have now is that there exist elements α, β π1(Sg ) with d(α, β) ≥

Rand d(Φ(α), Φ(β)) ≤ K(2D) + C. By choosing R large enough, say,

R> 2DK2 + 2CK, we obtain a contradiction with the assumption that Φ is a quasi-isometry with constants K and C. Thus it must be the case that

The induced homeomorphism at infinity. Lemma 4.5 suggests an elegant way to think about Φ, namely, through an induced action Φ on ∂H2 ≈ S1. We now define this action. If γ is an element of π1(Sg ), then the forward endpoint of the (oriented) axis of γ in H2 is identified with a point γ∞ ∂H2. We define

Φ (γ∞) = (Φ(γ))∞.

This defines Φ on a dense set of points in ∂H2. Note that Φ is well-defined on this set because the axes of two elements of π1(Sg ) can only share an endpoint at infinity if they share a common power. Indeed this argument proves that Φ is a bijection.

The following corollary uses Lemma 4.5 to prove that Φ in fact extends to a self-homeomorphism of ∂H2. While not needed for our proof of the Dehn–Nielsen–Baer Theorem, we state it because it is intere sting in its own right, and it is a fundamental idea in the study of surfaces; indeed, this idea underlies much of Nielsen's work. Also, we will apply this corollary in Sections 5.2 and 6.5.

Corollary 4.6 Let g ≥ 2. Any automorphism Φ of π1(Sg ) induces a homeomorphism of ∂H2.

In fact a much more general statement is true: any quasi-isometry of H2 induces a homeomorphism of ∂H2. Corollary 4.6 is simpler to prove because we can make full use of the algebraic structure of π1(Sg ).

Proof of Corollary 4.6. Up to postcomposing with a homeomorphism of S, we may assume that Φ fixes one point of ∂H2. We cut along this point and consider the restriction of Φ to this interval. Since, by Lemma 4.5, Φ preserves the properties of being linked and unlinked, the map Φ on this interval is strictly monotonic.

Thus the induced map on the interval is a homeomorphism on the subset of ∂H2 corresponding to the set of points

Sides. In addition to linking, we can also talk about two hyperbolic elements α, β π1(S) being on the same side of a hyperbolic element γ π1(S). That is, if α and β are unlinked with γ (and do not share an axis with γ), then their axes either lie on the same side of the axis for γ or they do not. One can also formulate this notion purely topologically at infinity, in terms of the endpoints of the axes on ∂H2.

Again, to simplify the discussion, we restrict our attention to a fixed covering space H2 → S instead of proving that this notion is independent of the covering space.

Corollary 4.7 Let g ≥ 2 and let H2 → Sg be a fixed covering map. Let Φ be an automorphism of π1(Sg ). If α, β, and γ are elements of π1(Sg ) with distinct axes, then the axes for Φ(α) and Φ(β) lie on the same side of Φ(γ) if and only if the axes for α and β lie on the same side of the axis for γ.

Proof. The axes for α and β lie on the same side of the axis for γ if and

Proof of the Dehn–Nielsen–Baer Theorem. As discussed above, we need only prove that the homomorphism Mod±(Sg ) → Out(π1(Sg )) is surjective. Let any [Φ] Out(π1(Sg )) be given, and let Φ be a representative automorphism. Also, fix once and for all a covering map H2 → Sg .